IYE1SITY
AN ELEMENTARY APPROACH TO
BOUNDED
SYMMETRIC DOMAINS
Max Koecher
HOUSTON, TEXAS
1969
Iad I * s U_^
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AN ELEMENTARY APPROACH TO
BOUNDED SYMMETRIC DOMAINS
Max Koecher
Rice University
1969
3
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11
[2]
-9 thru
-5
det 0^0 for ceZ.
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for ceZ. So the first inclusion is
proved .
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PREFACE
These notes contain the material of lectures given
at Rice University, Houston, Texas, during two months
in the spring of 1969.
In the first two chapters finite dimensional sub-
algebras O of the Lie algebra Rat V of rational functions
on a vector space V are considered. In particular the
group of automorphisms of O is investigated and a
connection with groups of birational functions is given.
The algebraic construction generalizes to arbitrary-
fields (of characteristic £ 2 and 3) the groups of
biholomorphic mappings of bounded symmetric domains,
and thereby generalizes the domains themselves . Detailed
information about these algebraic groups, their Lie
algebras, and the associated Killing forms, is obtained.
The reader should compare the results with the
examples given in chapter I, §5, and in chapter III.
In chapter IV and V the algebraic method is used
for an explicit construction of bounded symmetric domains
which covers all domains of this type.
The methods used in these notes are quite elementary
For completeness the proofs of well-known results on
linear Lie groups are included.
I wish to express my thanks to my friend H. L.
Resnikoff for his continuous interest and his valuable
discussions and suggestions. I am also grateful for
Nancy Singleton's excellent preparation of the notes.
Munchen, June 15, 1969
M. KOECHER
CONTENTS
Chapter I . LIE ALGEBRAS OF RATIONAL FUNCTIONS
§1. The Lie algebra Rat V 1
§2. Binary Lie algebras 9
§3- A description of the essential 21
homomorphisms
§4- The group of essential automorphisms 27
§5 • The case n = 1 33
Chapter II . THE CONCEPT OF SYMMETRIC LIE ALGEBRAS
§1. Symmetric Lie algebras 35
§2. The group 3(0,@) 40
§3- Construction of symmetric Lie algebras 46
§4- Killing forms 54
§5- A characterization of symmetric Lie
algebras 59
Chapter III . EXAMPLES
§1. Symmetric and skew- symmetric matrices 65
§2 . The rectangular matrices 74
§3- Jordan pairings 80
§4. The two exceptional cases 87
Chapter IV - APPLICATIONS TO BOUNDED SYMMETRIC DOMAINS
§1. Some elementary results on real linear
algebraic groups 91
§2. The group T(C) 98
§3- The group Aut(0,@ ) 104
§4. The group Aut(C,® + ) and Q 113
§5. The bounded symmetric domain Z 117
§6. The Bergman kernel of Z 126
Chapter V - AN EXPLICIT DESCRIPTION OF THE BOUNDED
SYMMETRIC DOMAINS
il. Formal real Jordan algebras 131
''2. The classification of the bounded
symmetric domains 138
INDEX OF NOTATIONS 142
REFERENCES 143
Digitized by the Internet Archive
in 2011 with funding from
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http://www.archive.org/details/lecturenotesinma19691rice
Chapter I
LIE ALGEBRAS OF RATIONAL FUNCTIONS
Let K be an infinite field of characteristic differ-
ent from 2 and 3 and let V be a vector space over K
of finite dimension n > 0. If R is an extension ring
of K, the tensor product R ® V (over K) is called the
scalar extension of V by R.
§1. The Lie algebra Rat V -
1. Let t, ,..., T be algebraically independent
elements of an extension field of K and let
K : = K(Tp...,t ) be the field of rational functions
in T i>---> r n with coefficients in K. For an arbitrary
vector space E over K, denote by E' the scalar extension
of E by K'. Choosing a basis b-, } ■ . . ,b of V we obtain
the element
X = T b, + • ' • + T b
II n n
of V . Let e i' m ' • > e m be a basis of the vector space
E over K; then the elements f of E ' have a unique repre-
sentation as
f - f , e, + • • • + f e , f . e K\
11 mm j '
i, a
and we write f = f(x). We call f a rational function
of x, moreover, f is called a polynomial or a homo -
geneous polynomial of degree r if the f . ' s have these
respective properties. Writing the f.'s as reduced
quotients of polynomials, the least common multiple 5 r
of their denominators is uniquely determined (up to a
constant factor) and it is called the denominator of f.
Let cp = cp (x) be a polynomial of K' and let
a = ou b, + • • • + a b be an element of a scalar
11 n n
extension of V- Then cp(a) is defined by replacing the
t.'s in cp (x) by the a.'s. More generally if 5^ is the
denominator of a rational function f e E' and if we
call Dom f = [a; aeV, 6^(a) f 0} the domain of f, then
f(a) is defined for a e Dom f. One says that f(a) is
obtained from f (x) by the specialization x -* a and one
writes f(a) = f(x)|
I x -> a
2.- An element f e V' is called a generic element
of V, if cp(f(x)) = 0, cp 6 K', implies cp = 0. Hence f
is generic if and only if the coefficients of f with
respect to a basis of V are algebraically independent
over K. In particular, x is a generic element of V.
Finitely many elements of V' are called generically
independent if all coefficients with respect to a
basis of V are algebraically independent.
Let g e E-, f e V' and let -^ be the denominator
of g. We say that g and f are composable if 5 (f(x))
7^ 0, i.e., if we can specialize x * f(x) in g. If g
I, §1
and f are composable then g(f(x)) is again an element
of E' which is denoted by g°f. Denote by I the
polynomial Ix = x.
Let P(V) be the set of rational functions f e V'
for which there exists a rational function f e v' such
that f and f as well as f and f are composable and such
that f°f = fof = I holds. Hence any element f o_f P(V)
is a generic element of V and therefore, we can
specialize x -> f (x) in an arbitrary rational function.
Moreover, f(V) turns out to be a group with respect
to the product (f,g) -* f°g- The elements of P(V) are
called birational functions .
For u 6 V and a rational function g e E' the
differential operator A is given by
A x s(x) : = dT §( x + Tu >
T -> O
The map u -> A g(x) of V into E' is linear, hence it
can be extended to an arbitrary scalar extension of V-
Furthermore, let f e V' and suppose that g and f
are composable. Then we have the chain rule
A^(g°f)(x) = A^ (x) g(f(x)) where w : = A^f(x)
Each f e V' induces an endomorphism — r-* — <- of V' via
5f (x) . _ A u cl s
— r- 5, — *- u : = A f(x).
dx x v '
If in addition g belongs to V', then the chain rule
becomes
1,21
a(gof)(x) = ag(f(x)) af(x)
3x 3f (x) 3x
3- For rational functions h^k e V', we define a
product h*k by
(h.k)(x) := t£& h(x) - ^Elk(x).
The map (h,k) -» h*k of v'xv' into V' is K-bilinear.
Hence V' as a vector space over K together with the
product h- k becomes a (non-associative) algebra. Using
the associator (h,k,,l) = (h*k)»l - h*(k«l), we obtain
(h,k,l)(x) - A^ (x) (h.k)(x) - A^ k,1) ( x) h(x)
. A i(x) 3hixl k( x) _ ahixi A i(x) k(x)
x 3x v ' Bx x v '
= A U A V h(x),
X X v ' '
where after the differentiation we put u = i(x) and
v = k(x). Since the last term is symmetric in u and
v, we get
(1.1) (h,k,l) = (h,l,k) for h,k,l e v'.
Algebras satisfying this condition are called right
symmetric .
Denote by Rat V the algebra over K with vector
space V' and the product [h,k] := h* k - k«h. Obviously
Rat V is anti-commutative. The identity
[[h,k],l] + [[k,l],h] + [[l,h],k]
= (h,k,l)+(k,l,h) + (l,h,k)-(k,h, !)-(!, k,h)-(h,l,k)
I,§1
shows that Rat V _is a Lie algebra over K.
4. Denote by Pol V the subset of Rat V of all
polynomials in x. Hence Pol V is a subalgebra of Rat V.
Let $ , r = 0,1/2,...., be the subspace of Pol V of all
homogeneous polynomials of degree r; then we get a
direct sum decomposition
Pol V = <P r •
r>o
Setting *p_, = we obtain
(1-2) tVV C 'r+s-l for r,s - 0,1,2,... .
From (1.2) it follows that $ = V, ?-, and T> + T5, are
v 7 o 1 o 1
subalgebras of Pol V. But
o l I
is not closed under the Lie product.
^t contains the function Ix = x. For he? the
1 r
Euler differential equation A h(x) = r h(x) shows that
[h,I](x) = ^ h(x) - h(x) = (r-l)h(x). Hence
(1-3) [h,I] = (r-l)h for h e ^.
Let h = h + h, + • • • , h e T , be an element of Pol V
o 1 r r
that commutes with I and all a € *C = V. We obtain
r o
(r-l)h = and A a h (x) = for a <e V.
Because of the linearity we can replace a by x and
obtain (r-l)h = rh =0 and consequently h =0 for
I,§1
all r. Hence only commutes with I and all a e V. In
particular, any subalgebra of Pol V that contains I and
all constant polynomials has center 0.
Denote by End V the ring of endomorphisms of the
vector space V. Since an element T of End V can be
extended to an endomorphism of any scalar extension of
V, the linear function Tx belongs to $-, . Conversely
each element of "P-, has this form. In keeping with the
notation f = f(x) for f e Rat V, we also write T
instead of Tx and I instead of Ix = x. From the context
it will be clear whether we mean the endomorphism or
the corresponding linear function. Calculating [T,S](x)
= (TS-ST)x for S,T e $, we see that the product in the
subalgebra 'P-, of Pol V corresponds with the commutator
product of the endomorphisms.
Without proof we mention that Pol V is a simple
algebra if and only if the ground field has characteris-
tic zero.
Let L be an extension field of K and suppose that
x is generic over L. Then for the scalar extension we
have
L®Rat V 5 Rat(L®V) and LSPol V = Pol(LSV).
K K K K
5. Denote by IP (V) the set of f t Rat V for which
the scalar rational function
deti^Sl
dx
is not the zero function. For h e Rat V and f e P (V)
o
I,§1
we define a function h by
(1-4) h f (x) :- (-^f^ 1 ) h(f(x)),
provided h and f are composable. Obviously h belongs
to Rat V- In the notation of 3 we have
(1.5) f.h f = h°f.
Suppose that h and f as well as k and f are com-
posable; then [hjk] and f are composable too, and we get
(1.6) [h f ,k f ] = [h,k] f .
For the proof we use (1.5) and (hof)«k = (h«k)°f in
the following calculation:
(f,h f ,k f ) = (f.h f )-k f - f- (h f -k f )
= (hof)'k f - f'(h f -k f ) = (h-k)of - f-(h f -k f )
= f- (h-k) f - f- (h f -k f ).
Formula (1-1) shows that the left side is symmetric
in h and kj hence we get f«[h,k] = f • [h ,k ] and (1.6)
is proved.
Each h e Pol V is composable with each f e P (V).
From (1.6) it follows that h • h is a ho momo r p h i s m of
the Lie algebra Pol V into Rat V.
The group P(V) of birational functions is a subset
of P (V). Hence h is defined for h e Rat V and
o
f e P(V). The chain rule implies h s = (h ) 8 for
f;g c P(V). Hence h -> h is a linear bijection of
I,§1
Rat V onto itself. Moreover h = h for all h e Rat V
implies f = 1, therefore P(V) acts effectively on Rat V.
Again from (1.6) it follows that h -* h is_ ari automor -
phism of the Lie Algebra Rat V for each f e P(V).
Setting v f (h) : = h^ we obtain an automorphism V-
of Rat V- Furthermore the map
(1.7) v : P(V) -» Aut Rat V, f -> v f ,
is a_ monomorphism of the group P(V) into the auto -
morphism group of Rat V-
6. We now construct two special types of auto-
morphisms of Rat V. Denote by GL(V) the group of
bijective endomorphisms of V. For W e GL(V) we have
the linear function Wx that is birational. Hence
GL(V) can be considered as a subgroup of P(V). We
get
(1.8) (v w h)(x) = (h W )(x) = W h(W -1 x), W e GL(V).
For b e V we define the polynomial t, by
t b (x) = x + b.
From t ot = t, , it follows that t, belongs to P(V) and
that (t b ) = t_ b holds. From the definitions it
follows that
(1-9) W°t b - t wb °W for W e GL(V), b e V.
As an abbreviation set
I, §2
t ,
(1.10) T b := 7 , hence (Y b h)(x) = (h ) (x) = h(x-b)
Again Y, is an automorphism of Rat V. Formula (1.9)
yields
(1.11) V w Y b = Y wb v w , W 6 GL(V), b e V,
and we have
(1.12) Y, Y = Y, , for b,c e V.
v ' be b+c
§2. Binary Lie algebras -
1. Suppose that the ground field K has a charac-
teristic different from 2 and 3- The elements of f = f
+ T}, + ^o are written as
q = a + T + p, a € V, T e $-, , p e %•
Here p is a homogeneous polynomial of degree 2. Hence
there exists a bilinear symmetric mapping p : VxV ■* V
and a linear map a -> S^ of V into End V such that
r a
(2.1) p(x) - p(x,x), A^ p(x) = 2p(x,a), p(x,a) = S^ x,
holds for a e V-
Let 2 be a subspace of 'P-, and V be a subspace of
?« satisfying the following conditions
(B. 1) % is a subalgebra of TS, ,
(B.2) [V,V] c I,
(B.3) [2,V] c V,
10 x > §2
(B.4) [V,V] = 0,
(B.5) I e X.
Forming the subspace a=V+I+Vof Pol V and using
(1.2) we see, that (B. 1) to (B.4) mean that O is a sub-
algebra of the Lie algebra Pol V. We call O a binary
Lie algebra if in addition (B.5) is satisfied. Using
(2.1) we get [p J ,a](x) = 2 p(x,a) = 2 S P x and hence
(B.2) is equivalent to
(B.2') S p e X for a e V and p e V.
From §1.4 we know that a_ binary Lie algebra has center
0. Let L be an extension field of K and suppose that
x is generic over L. Then L^O turns out to be a binary
Lie algebra of Pol (L®&) .
2. Let O be a binary Lie algebra and let
$ : O ^ Rat V be a homomorphism of the Lie algebras.
Hence a -> $a is a linear map of V into the vector space
Rat V. Therefore there exists a linear transformation
H $ (x) of V' that is rational in x such that
(2.2) ($a)(x) = H $ (x)a for a e V.
The homomorphism § : O -> Rat V is called essential if
the determinant of the endomorphism H*(x) is not the
zero function.
THEOREM 2.1 . Let O be a binary Lie algebra and
let I : O -> Rat V be a homomorphism of the Lie algebras .
I, §2 11
Then $ is essential if and only if there exists
f e P (V) such that $q = q for q e O.
Proof : If there is an f e P (V) such that
fq = q , then let q = a e V and in view of (2.2)
we have
-1
M*> - {^r 1
Conversely, let $ : O ^ Rat V be an essential
homomorphism of the Lie algebras and let H $ (x) be its
associated linear transformation. Set F = F(x)
= [Hx(x)] . Due to the linearity of $, we can write
(i) $q = F~ [a+b +c ], where q = a + T + p.
T -> b„ and p ^* c are linear mappings of X and V,
respectively, into Rat V.
For two elements q, and q 2 in we abbreviate
w. = $q . and write w. as in (i). In the notation of
1. 2 we obtain
w 2
($q 1 «$q 2 )(x) = (w 1 *w 2 )(x) = A x w-^x)
- -F _1 [A x 2 F(x)]w 1 (x) + F" 1 A^ 2 [b T (x) + c p (x) ] ,
by using the fact that A u [F(x)] _1 = -F _1 [A U F(x)]F _1 .
X X
It now follows that
(S[q 1 ,q 2 ])(x) = [$q 1 ,$q 2 ] (x) = [w^w^Cx)
-1 / w ? w l
= F (-[A/ F(x)] Wl + [A x L F(x)]w 2
12 I, §2
+ A™ 2 [b T (x)+c (x)] - A%b (x)+c (x)]\ .
Setting q. = a. e V, we find that [qpq 2 ] =
and we then obtain
[A c F(x)]u n = [A F(x)]u for u. = F a..
L x V/J l x v / j 2 j j
As this expression is bilinear in u, and u~, this
equation is also valid in any scalar extension of V.
The above equation simplifies to
(ii) F«[q r q 2 ] - (b^+c^).^ - (b^ c )• tq r
Now let q, = T e X and q 2 = a e V. As [q,,q 2 ] = Ta,
sfqi^qo] = F ~ Ta and $a = F ~ a ' i<: follows tnat Ta
= b * (F a).. Since both sides of this equation are
K'-linear in a, we can replace a by an arbitrary
element of Rat V and the equation will remain valid.
We thus have
(iii) b T *h - T F h for T e X, h e Rat V.
As I e X we can substitute T = I and obtain
(iv) F(x) = i|!*l for f(x) := bjCx).
As the determinant of F = F(x) is not zero, f is an
element of P (V) . Now substitute q. = T. 6 X in
equation (ii)- Since [q^q^ = [T-^.T-] e X, we have
that $[q^,q 2 ] = F br ,, and using (iii) we obtain
I, §2 13
b [T r T 2 ] = b T 1 '< F "S 2 >- b T 2 ' (rlb T 1 ) = T l b T 2 - T 2 b Tl '
Now for T, =1, L =T, ve get the relation
(v) b T = Tf for all T e X.
F
or q, = p e V, q« = a e V, we apply (2. 1) to
the calculation of [q-,,q ] = 2 S, = 2 S • Using (v)
J_ Z 3 a.
and the fact from (B.2') that S e Z we get
%$[q ] _,q 2 ] - F" 1 b g - F" 1 S a f,
so that (ii) yields
2Sf = C'$a=c. (F _1 a) .
a p p v J
Now replace a by Fh in the above equation and obtain
(vi) 2 S—f - c »h for h e Rat V.
Fh p
Finally., substitute q, =1, q« = p e V in (ii) and in
view of (1-3) and (iv), it follows that
-F$p = b T » §p - c • $1 and 2 c = c • (F f ).
^ I K p pp v/
A comparison with (v) yields
C p = S f f = P( f ' f > = P° f -
Taking this and (v) together, we get that the image of
q = a + T + p under $ is given by
(*q)(x) - [F(x)]" 1 [a+Tf(x)+p(f(x))] = (q f )(x).
14 I, §2
This completes the proof of the theorem.
3. For an essential homomorphism $ : £ ■* Rat V
there is an f e P (V) such that $q = q for q 6 O. We
define the rational function r. by
(2.3) , r.(x) := [H $ (x)] _1 (§I)(x).
Obviously r $ depends only on the images §1 and $a, a e V.
Writing $q = q for q - I and for q = a e V we obtain
(*I)(x) = (^H^)" 1 £(x),
(*a)(x) =H |( x)a = ^^) _1 a.
Hence f = r, and
I
In particular the rational function f is uniquely
determined by $ . We say that f = r, belongs to the
essential homomorphism $ .
4. Let W e GL(V) and consider the automorphism
v„ of Rat V given by (1.8). It follows that v I = I
and the image v O is again a binary Lie algebra. More-
over the restriction of v T7 to is an essential homomor-
w
phism and W belongs to it.
For b i V we consider the automorphism Y, = v
b %
of Rat V given by (1. 10) . and we show that the restric -
tion of Y to _is_ a_n automorphism of the binary Lie
I, §2 15
algebra O. Because of (1. 12) it is enough to prove
Y.O c SX Writing q = a+T+p e O = V+I+V we obtain
from (1.10)
(Y b q)(x) = q(x-b) = [a-Tb+p (b) ] +[Tx-2p (x,b) ] +p (x) .
Hence we have only to show that 2p(x,b) = [p,b](x)
belongs to X. But this is a consequence of (B-2).
Furthermore the restriction of ¥, to O is an
essential homomorphism and t, belongs to it.
5. Later we will see that the essential automor-
phisms of O form a group. As a first step we prove
LEMMA 2.2 . Let O and O 7 be binary Lie algebras
and let
$ .: 0-> O', §': D' -> Rat V,
be essential homomorphisms such that r, and r , / are
composable . Then § '$ : 0-> Rat V is essential and
we have
r, / , ~ r, o r« / •
Proof: Put f = r* and g = r, /• Since f and g are
composable, the chain rule shows that fog belongs to
P (V) too, and that h f ° 8 = (h f ) 8 holds for h e Pol V.
From Theorem 2. 1 we conclude
$'$q = (§q) g = (q f ) g = q f ° 8 for q e O.
16 l >* 2
Hence $'$ is essential and fog belongs to it.
The assumptions of the Lemma are certainly
satisfied if
r $ e Pol V, r § /e Rat V or r $ e Rat V, r $ / c -P(V).
In particular we get the
COROLLARY. If $ : O -> Rat V is essential and if
b.c e V then Y, $ Y is essential and t or.ot ,
b c -c i -b
belongs to it .
6. Let v e !pj- We define a linear transformation
Y of Rat V by
v J
(2.5) Y v =Id + ad v + %(ad v) 2 ,
where as usual the adjoint representation ad v is
given by (ad v)h = [v,h], h e Rat V. We know that
V W j W e GL(V), is an automorphism of r it V,. hence we
(2.6) V w Y v = Y u v w for W . GL(V). v 6 ^ . u := r y v.
The restrictions of ad v, ^ TT , Y, and Y to O
W b v
will be denoted with the same symbol if there is no
possibility of misunderstanding.
Furthermore we define a linear map B (x) of V into
Pol V by
(2.7) [B v (x)]a = (a-[v.a] + %[v, [v,a] ] ) (x)
= a - 2v(x,a) +2 v(x.v(x.a)) - v(a.v(x))
I, §2 17
for a e V. Then B (x) can be extended to any scalar
extension of V. One sees that B (x) is a polynomial
of highest degree 2 and one has B (0) = I. Hence
B (x) considered as a linear transformation of V
v
is invertible. The expression
(2.8) t v (x) := [B v (x)]" 1 [x-v(x)] 3 v e ^
is a rational function. In particular t (0) is defined and
we have
at„(x)
V°> = °- -ir
= i.
x*o
Hence t lies in P (V).
v o v
THEOREM 2.3 . Let O = V + X + V be a binary Lie
algebra and let v e V. Then
St (x) 1
(a) t v € P(V) and ^ x = [B v (x)]~\
(b) Y is an essential automorphism of O and t_
belongs to it ,
(c) t , = tot and Y , = Y Y for u £ V.
v ' u+v u v u+v u v
(d) Wot = t oW, where u = v v.
v u W
Proof : (1) Using (B. 1) to (B.4) we see that
(ad v) £ = 0. Hence from (2.5) it follows that
Y q = (exp ad v)q for q e £
18 I, §2
holds. Since ad v is a derivation of £ the restriction
of exp ad v to turns out to be an automorphism of
O. Moreover for u e V we obtain [u,[v,q]] - [v,[u,q]]
= [q,[v,u]] = from (B.4). Therefore ad u and ad v
commute on O. Hence
I f = (exp ad u) (exp ad v) = exp ad(u+v) = Y . •
(2) As an abbreviation write $ = Y . Then com-
v / v
bining (2.2) and (2.7) we get
H § (x)a = (§a)(x) = a+[v,a](x) + h[v , [v,a] ] (x)
= B_ v (x)a.
In particular Y is essential. Furthermore from (1-3)
r v
we obtain [v^I] = v and $1 = I+v. Hence r, = t So
part (b) together with (2.4) implies the second state-
ment of part (a).
(3) The determinant of B (x) is a denominator of
-v
t (x) and t (0) = 0. B (0) = I. Hence t and t
-v -u -v -v -u
are composable. Applying Lemma 2.2 we see that Y Y
is again essential and t ot belongs to it. From (1)
-v -u
and (2) we obtain Y Y = Y , and the function
u v U+V
belonging to it equals t . Hence tot = t
b b M -u-v -v -u -u-v
and part (c) is proved. In addition we see that t is
a birational function.
(4) Part (d) follows from (2.6).
In particular t_ ot - I for v e V. Using the
chain rule together with part (a) of the theorem we end
I, §2 19
up with
(2.9) B (x) B (t (x)) = I, v g V.
\ / v -v V
Finally from the definitions (2.7) and (2.8) we obtain
(2.10) B „(-x) = B (x), t (-x) = -t (x), v e V.
\ / -V v ' V -v v
Parts (a) and (d) of Theorem 2.3 yields
(2.11) W B (x) = B (Wx) W, v e V, W e GL(V), u = v v.
7. Any binary Lie algebra O = V + X + V gives rise
to a family of Jordan algebras defined on the vector
space V- We are going to prove
MEYBERG'S THEOREM. Let O = V + Z + V be a binary
Lie algebra and let veV. Then V together with the
bilinear product ab = [[a,v],b] turns out to be a
Jordan algebra .
Proof : As an abbreviation we write
[a,u,b] = [[a,u]jb], {u,a,v} = [[u,a],v] for a.beV, u^veV.
Since V and V are abelian subalgebras of O, both
triples are symmetric in the first and last entries.
In order to prove
(2.12) ia,u,{h,v,c}} - ib , v, U,u, c } }
= [ [a,u,b },v,c } - [b, [u,a,v},c },
a,b,c e v, u , v e V,
20 I, §2
one puts T = [a,u] and uses the Jacobi identity. Ana-
logously we get
(2.12') [u,a, [v,b,w}} - {v,b, [ u, a, w} }
= {{u,a,v} ,b,w} - [v,{a,u,b},w],
a,b e V, u,v,w e V.
The left side of (2. 12) is skew- symmetric in (a,u)
and (b,v) ; hence
(2.13) { (a,u,b},v,c } - [b, [u, a, v} ,c } + [ [b , v,a },u, c }
- [a, [v,b,u},c] = 0.
Choosing a = b and u = v we get
(2.14) [b ,v,b} = {b,v b ,b} where b v = {b,v,b}
and v, = [vjb.v].
b
In the same way from (2.12') it follows that
(2.14') iv b ,b,v} = iv,b v ,v}.
Using the product ab = [a, v,b} = [[a,v],b], we get
2
b - b . Now in (2. 13) we choose a = b and replaci
u by v, and v by v, • We then obtain
(2.15) 2{b, {v b ,b,v},c} - c{b,v b ,b} + {c,v b? b 2 },
2
Next we set u = v and replace a by b in (2. 12) :
2b 2 (bc)-2b(b 2 c) = 2b 3 c-2ib, iv.b 2 ,v},c}.
Using (2.14'). we put (2.15) in this equation
I, §3 21
2b 2 (bc)-2b(b 2 c) = b 3 c-[c,v b ,b 2 }.
2
Finally put u = v and replace a by b and b by b
in (2.12):
b(b 2 c)-b 2 (bc) = b 3 c-{b 2 ,v b ,c}.
2 2
This means 3[b (bc)-b(b c)] =0 and the theorem is
proved.
§3- A description of the essential homomorphisms .
1. Again let 0=V+I+Vbea binary Lie algebra
in Pol V and let $ : O^ Pol V be a linear map. Hence
we obtain a representation
(3-D $q = £ g^ where g^ e ^
q °q 'v
v>o
as a finite sum. Here q ^ g V is a linear map of O
into $ . We write g = g + g™ + g whenever
V Q a. x p
q = a + T + peO. If $ : O^ Pol V is a homomorphism of
the Lie algebras then (1.2) implies
v+1
(3 ' 2) § [q,q'] 1 [8 q'V ] ^ q ' q 6 °'
u=o
We obtain our first information about the homomorphisms
of binary Lie algebras in
LEMMA 3. 1 . Let O = V + I + V and C' = V + I '+ V '
be binary Lie algebras in Pol V- Suppose that
22 I, 3
$ : £1 -> £i' is an epimorphism of the Lie algebras with
g T = and such that a -» g is a bijection of V. Then
there exist W e GL(V) and v € V such that
$q = V W ^v q for q £ O
and O' = v &
Proof: We define the linear transformation W of
V by Wa = g . Hence W is bijective and consequently
W € GL(V). We know from §1.6 that v is a homomorphism
of Pol V that maps binary Lie algebras onto binary Lie
algebras. Setting I* : = v $, Z" := v~ o', we obtain
a homomorphism § : O -» O* satisfying g T = and
~o , ~v -1 v
g = a, where e = v TT g .
& a 6 q W & q
Hence we may assume that $ : -> O' is a homomor-
phism satisfying g T = and g = a, and we have to
prove O' = O and $ = Y for some v e V-
r v
Substituting q = T e 2, q ' = a e V in (3- 2)
we get [q.q'] = Ta and
(3-3) g° a = g T (g°) - g a (g T )-
T = I yields a = g = g (a),, hence g T =1 and
a l 1
$1 = I + v, V € V'.
For q = a + T + peO we get [I,q] = a - p from (1.3).
Consequently i (a-p) = $[I,q] = [$I.$q] implies
(3-4) la = [$I.$a]. = [§I.$T], -§p = [II, |p].
I, §3 23
Together with (3-2) the first condition leads to
2 2
£ v g V a = I [v,g^],
v=o v=o
We compare the homogeneous terms and get
1 2
g a = [v,a], g a = %[v, [v,a]] resp. [v, [v, [v,a] ] ] = 0.
That means
3
$a = a + [v,a] + ^[v,[v,a]], (ad v) a =. for a e V.
In the same way the second condition of (3-4)
yields
2 2
Y (v-1) §t = 2, [v,gjf]
Again we compare the homogeneous terms and obtain
g^ " 0. g T = [v,g T ] and [v,g T ] = 0. Formula (3-3)
1 2
leads to g T = T and hence gt = [v,T], [v,[v,T]] = 0.
This means
$T = T + [v,T], (ad v) 2 T = 0, for T e X.
Finally the third condition of (3-4) leads to
2 2
7 (v-2) g^ - y [v,g£].
v=o v=o
Hence g = g = and [vjg^] = 0. Substituting q = p e V,
q' = a e V and v = 1 in (3.2) we get
24 l '^
1 r 2 Oi _ r 2 „-i
§[p,a] = [ V § a ] " [g p' a] -
From (B.2) we obtain [p,a] e Z and therefore g„ = T
2 2
yields [p,a] = [g .a]. This means p(x,a) = g (x,a)
2
and consequently g = p as well as [v,p] = 0. Hence
$p = Y p, [v,p] = 0, for p e V.
Summing up we have
(3-5) $q = [I + ad v + %(ad v) 2 ]q, (ad v) 3 q = 0, q e £.
Because of v e V' we know from Theorem 2.3b that Y_ is
an automorphism of D'. Hence Y_ § : O -> turns out
to be a homomorphism,, but (3.5) leads to Y_ $q = q.
Therefore O' = O and i = Y q for q e O .
q v
2. As a first application we prove
THEOREM 3. 2 . Let $ : O -> O 1 be an isomorphism
of the binary Lie algebras satisfying *I = I. Then
there exists a W in GL(V) such that $q = v q for q e C
w
Proof : In the notation of (3. 1) we have g° = 0.
Moreover from $1=1 it follows that $a = §[I,a]
= [I.la] and (1.2) implies g = g =0. Hence a •* g°
a a a
is a bijection and we can apply Lemma 3.1. There is
a W <- GL(V) and v e V' such that $q = V T _ 7 q for
M W v -
q c O. Substituting q = I we get v = and the
Theorem is proved.
I, §3 25
A second application leads to the following
main result on the automorphisms:
THEOREM 3. 3 - Let O and O' be binary Lie algebras
in Pol V. Then :
a) (i) I_f i : O -> O' is an essential isomorphism ,
then § can be written as
(3-6) $ = v y Y b ? v * . where W € GL(V), b,c e V, v e V,
and £>' = v O.
(ii) The rational function r* belonging to $ is
birational and one has
(3.7) r, = (Wot, ot °t ) .
$ b v c
(iii) If there is a d e V in the domain of
definition of r z such that
(3. 8) r,(d) = _and det
Bx
i 0,
x->d
then the statements are true for c = 0.
b) Each map of the form (3.6) turns out to be an
essential isomorphism of O onto O' = v O.
Proof : 1) Suppose first that the essential isomor-
phism $ satisfies the condition of (iii). From §2.4 we
know that ¥, is an essential automorphism of &' 3 hence
■d
26 1,53
algebras which is essential because of the Corollary
of Lemma 2.2 and f = r $ ot d Delon g s to *• From (3.8) we
get
f(0) = and Q = ^^±
ax
is invertible.
xr^o
We write fq in the form (3- 1) and specialize x - in
lq = q , q = a + T + peO. Hence
< + H + §p = ( ^ )(0) = ^" la -
In particular g T = and a -> g is a bijection. We
j_ a
apply Lemma 3- 1 and obtain
?q = V y ? v q for q e O
and W e GL(V), v e. V- Hence $q = v ^ b ? v q and
r, = (W°t,°t ) ,. where b = W d according to (1.11)
So (iii) is proved.
2) Now let $ : O ■*• O 7 be an arbitrary essential
isomorphism. The Corollary of Lemma 2. 2 shows again
that $ = § y is essential and r~ = t -r, belongs to
-c i c § b
it. We choose c,d t V such that d is in the domain
of definition of r? and that (3- 8) is satisfied for r^.
Part (i) of the proof yields (3-6) and (3.7).
3) Since the functions W. t, and t are birational
b v
we obtain the statement of part b) from Lemma 2.2 and
Theorem 2.1.
Using the definition of v f in \\.5 we obtain the
I, §4 27
COROLLARY . If. $ : O -» O' is an essential isomor -
phism , then there exists a ( uniquely determined ) f e P(V)
such that $ = Vj-. Moreover f = W° t, °t °t in the
r b v c
notation of (3-6).
§4- The group of essential automorphisms .
1. Let 0=V+I+Vbea binary Lie algebra and
denote by T(O) the group of W e GL(V) satisfying v„£i = O.
Obviously, W e F(O) is equivalent to
v TT X = WXW _1 = X and v IT v" = V.
w W
In particular, a- I, ^ a e K, belongs to T(O). One
can show that T(O) is a linear algebraic group defined
over K.
Denote by Aut O the subgroup of the automorphism
group Aut O of O that is generated by the automorphsims
(4.1) V for W 6 F(O), Y b for b e V, Y for v e V
(see §1.6 and Theorem 2.3).
THEOREM 4. 1 . Let O = V + X + V be a binary Lie
algebra . Then :
a) The set of essential automorphisms of O
coincides with the group Aut O, which is Zariski-open
in Aut O.
b) Each $ in Aut >c & can be written as
28 I, §4
$ = v T7 *. ? Y , where W e r(C), b,c e V, v e V,
W D V C
and the rational function belonging to § is given by
r, = (W°t, ot °t )~ .
$ b v c
c) i ->• r -i gives a monomorphism of Aut"0 into P(V).
Proof: The generators (4. 1) are essential auto-
morphisms of O and the rational functions belonging to
them are birational. Using Lemma 2.2 we see that Aut"C
consists only of essential automorphisms. Conversely,
each essential automorphism of O belongs to Aut"£>
because of Theorem 3. 3- Since an automorphism $ is
essential if and only if det H $ (x) ^ (see §2.2), the
set of essential automorphisms turns out to be a
Zariski open subset of Aut O. So parts a) and b)
are proved.
From Lemma 2.2 we get r. '* = r *° r s ' • Hence we
need only prove that $ -> r. is an injection. Consider
$ e Aut*0 such that r^ = I. From (2.4) it follows
that Hx (x) = I and consequently $1 = I. Applying Theor
3.2 we obtain I = V T1 for some W e GL(V). Hence r, = W
and W = I.
2. Sometimes it is useful to consider the image
(O) of Auf'O under the injection
(4.2) $ • r _-. . I € Aut"C.
§ L
i.e.. the set
em
1
I, §4 29
S(Q) = [r, ; $ e Aut*0}
Part c) of Theorem 4.1 shows that 2(D) is a subgroup of
P(V) and (4. 2) turns out to be an isomorphism of Aut' v O
onto H(O). Comparing $q = q for f - r $ and the defini-
tion of v.p (see §1.5) we see
(4.3) v f = i" 1 » f = r 4 .
Hence
v : H(O) -» Aut Oj f -* v f ,
is the inverse map to (4.2).
In case $ equals v , f, , or ? i the function r $
belonging to $ equals W , ^v,) = t -h' or ^v^ = t -v J
respectively. Hence the group H(O) is generated by the
birational mappings
(4.4) w e r(O), t b for beV, t for v e V.
Moreover from part b) of Theorem 4.1, we know that
each f in ^(O) can be written as
(4.5) f = W°t,°t °t where W £ T(&) , b,c e V, v e V-
v b v c
Using the chain rule together with part a) of Theorem
2. 3 we get
(4.6) 2|&1 = w[B v (x+c)] _1 .
As a first application we prove a lemma that is trivial
30 I, §4
in the case of characteristic zero.
LEMMA 4.2 . Let f and g be in S ( O) . Then d f * » x '
= 8\ x / if and only if there exists an element a e V
g x z
suc h that f = t og.
a °
Proof: We set h = fog and we obtain —5-* — *- = I
to ox
using the chain rule. Therefore it is enough to prove the
statement in the case g = I. Writing f = Wot ot ot we
to b b v c
obtain W = B (x+c) from (4-6) . So x = -c leads to W = I
and to B (x) = I. From (2.7) we get v(x,a) = for a e V
and hence v = and f = t, , .
b+c
Clearly f - t.g implies M&1 = %^ ■
3. ox ox
Remark . From (4-5) we obtain a decomposition
H(Q) = r(0)oEoEoE
where E = [t ;aeV} and E = [t ;veV} are abelian subgroups
of S(O). From (1.9) and part d) of Theorem 2.3 we get
r(0)o£ = E°T(0) as well as E(S)c ? = EoE(O). This de-
composition induces an equivalence relation on the set
V: For u,v e V we define u~v whenever t € r(0)cEot C E.
u v
In particular u~v and WeE(O) implies v u~v v.
W W
Moreover from (4.6)
I, §4 31
turns out to be a polynomial whenever f e 2(0). Denote
by Df the set of a e V such that uu f (a) f 0. Further-
more the chain rule yields
(4.7) w gof (x) = w g (f(x)) a) f (x) for f,g e H(O).
Writing f ■ r $ e H(O) the formulas (2.3) and (2.4)
lead to
f(x) = m&L (§D(x).
$1 belongs to O and consequently $1 is a polynomial.
Therefore we get: For f e H(O) we have D f c Dom f.
3- Note that a representation (4.5) is not unique.
But setting
H° = a°(0) - Cf ; f e H(o), uu f (0) f 0},
we get the
THEOREM 4. 3 . Let O be a binary Lie algebra . Then
a) The elements of 2 (o) are exactly the functions
(4.8) f = Woyt , where W e r(0) J b e V, v e V,
and this representation is unique .
b) The image of H (O) under the map f -» v f is
Zariski-open in Aut"C
Proof : a) From B (0) = I and from (4. 6) we conclude
that any f given in (4.8) belongs to H (O). Conversely,
32 I,$4
let f e H°(0). Hence is in Dom f. Put d = f(0)
and use (4.7) for g = f~ in order to get uu (d) t 0.
Hence d is in Dom g. Choose $ e Aut*0 such that
f = (r $ ) _1 = r x , i.e., g = r § . Hence d is in Dom r $
and we get r ffi (d) = f~ (d) = as well as uu (d) ^ 0.
This is exactly the condition (3- 8), so Theorem 3-3
implies f = (r,)~ = W° t,° t , where W e GL(V), b e V
and v e V. Consequently W e F(O) and (4. 8) is proved.
Because of (1.9) and part c) of Theorem 2.3, to
prove uniqueness we need only consider the equation
Wot,°t = I. From t (0) = we get b = and then
b v v °
B (x) = W using (4.6). The definition (2.8) yields
x = Ix = x - v(x). Hence v = and W = I.
b) For f = r x , § e Aut"D, the equation (2.4)
leads to uu f (x) = det H $ (x) and the proof is complete.
COROLLARY 1 - Let a e V and v e V. If det B (a) ^
then B (a) belongs to F(O).
Proof: We put f ~ t ° t and obtain x r (x) = det B (x+a)
r v a f / v
from part a) of Theorem 2. 3- Hence uu-(0) ^ and conse-
quently f € 3 (O). Part a) of the theorem yields t °t
V 3.
= f = W°t°t for some W e T(C), b e V and u e V. The
equation (4.6) yields B (x+a) = B (x)W and x = leads
to B (a) = W e T(C). In view of (4.6) we obtain the
COROLLARY 2 . Suppose that a is in D f for some
af(x)
f e 3(0). Then
dx
belongs to r ( C)
x-^a
I, §5 33
§5- The case n = 1 .
As an illustration we consider the case that
V = K is the one dimensional vector space over K.
Hence the generic element x is an indeterminate over
K. Denote by 3DL the group of invertible two-by-two
matrices over K and set
*M<*> " ^ » h «e M - (y !)
e a» 2 .
Then M -» f M defines an epimorphism of 3JL onto 1P(K)
having the kernel (otl; f aeK} where I denotes the
unit matrix of SDL.
The Lie algebra Rat K is given by the vector space
K(x) of rational functions together with the product
[h,k] = h'k-hk'j where h' denotes the derivative of h.
The only subalgebras of Pol K that contain V = K are
K, {a+bx;a,beK} , and the binary Lie algebra
2
= *p = [q; q(x) = a+bx+cx ; a,b,c e K}.
In fact, O is (up to isomorphisms) the split three-
dimensional simple Lie algebra. We see that r(O)
coincides with the mulitplicative group of non-zero
elements of K.
A verification shows h e O whenever q e O and
f e P(K). Hence f ->• v f maps P(K) into Aut a We
observe further that each automorphism of O is essential.
According to Theorem 3-3 it follows that V : P(K) -> Aut O
is an isomorphism of the groups . In particular we
34 I, §5
get Aut*C = Aut O and H(o) = P(K) (see §4.2).
Finally let us consider the generators of the
group S(O) = P(K) according to §4.2. At first we have
Wx = wXj / w e K, and t (x) = x+a, a e K. In order
~ 2
to describe t where v(x) = bx . b e K, we observe
v
according to (2.7) and to (2.8)
B (x) = (1-bx) 2 and t (x) = x(l-bx)" 1 .
v V
Indeed we obtain the usual set of generators of P(K).
II, §1 35
Chapter II
THE CONCEPT OF SYMMETRIC LIE ALGEBRAS
§1. Symmetric Lie algebras .
1. A pair (O, ®) is called a symmetric Lie algebra if
(i) Q=V+2+V is a binary Lie algebra,
(ii) 9 is an automorphism of O of period 2 such
that ©V = V.
2
From © = Id we get ©V = V • In order to prove
(1.1) 03: = X and ©I - - I
we put ©I = a + S + p • But [I,b] = b implies [a,@b] =
for beV and hence a = 0. From (I; 1.3) we get [I, ©I] =
- p and - ©p = ©[I, ©I] = [©1,1] = p yields p = 0. Next
©v = 9[v,I] = [©v,S] for veV leads to [S,b] = - b for
beV and hence to S = - I. Since I is in the center of
I we obtain 32 c Z from (I; 1.3). The elements of a
symmetric Lie algebra (£>,©) we write as q = a + T + ©b,
where a,b e V and Tel.
For a symmetric Lie algebra we are able to express
the automorphisms HL, of O by © and the automorphisms ¥, •
LEM MA 1.1 . For beV we have ¥«. = ©Y, ©.
Proof : From (I; 1.10) we observe that
36 II, U
la = a, ¥,I = - b + I, where a,b 6 V.
b b
Consider the automorphism $ := ®¥@t® °^ ° ' Since
¥,-,, is the identity on V and since ¥ : , I = I + 3b
holds, we observe
ia = a, $1 = - b + I.
In particular I is essential. In I, J2.3 we have seen
that an essential automorphism is uniquely determined
by the images $1 and §a for a e V. Hence $ = ¥,.
2. The automorphism 8 induces involutions,
i.e., involutorial anti-automorphisms, of the groups
F(G) and 2(D). First we show
LEMMA 1.2 . Let ( O, ©) be a symmetric Lie algebra .
Then there exists an involution W • W of ^(C) such
that
a) v TT © v „ = © for all We r (£).
W W -
b ) F ( O) acts as a group of inner automorphisms
on the Lie algebra X. moreover
(WTW" L ) = W* _1 ( T)W* for T < I and W e F(C).
Proof : Consider the automorphism § = ~c T c of C
w
Using v w I = I we get $1 = I. Hence I. Theorem 3-2.
can be applied- There is a W~ GL(V) such that
•tt = $ = v _ -, • Since $ is an automorphism of C
W
we get W~ , P(C). Passing to the inverse we see that
II, §1 37
# #
V . 9 V TI s @ and consequently (W ) = W. The statement
W # W
b) follows by applying a) on the elements of X.
Nov/ we can extend the map W ■* W to the group
H(O) of birational mappings.
THEOREM 1.3 - Let (O, ©) be a symmetric Lie algebra .
#
Then there exists an involution § -> $ ' of Aut^O and
f •* f of_ H ( D) , respectively , such that
a) §©$* = 9 for $ e Aut*D,
b) v f ©v . = © and (v f )* = v „ for f e H(O).
Proof : Since V : H(O) -> Aut"£ is an isomorphism,
it is enough to prove part a). Let $ be in Aut"£, then
§ = v TT Y ?-, Y , where W e r(O), a,b,c e V, because
Wad be N ' *
of part b) in I, Theorem 4. 1. Using Lemma 1. 1 we
observe that
©$© = @V TT Y 8 Y,© Y ©
W a b c
= v"^(©Y ©)Y,(@Y ©) e Aut*a
w f a b c
Hence $ ■* (®§0) turns out to be an involution of O.
# -1
Now the statement follows by setting $ = (®§0)
Using the theorem we calculate
(1.2) (V h )* = Y_^ b and (t.) = t_ £b for b e V.
3- Now we are going to prove some basic relations
and identities. Let (£, ©) be a symmetric Lie algebra,
38 H,«l
O = V + I + V. For a,b £ V we put
B(a,b) = B-(a,b) = B @b (a)<
Hence B(a,b) is a polynomial in a and b which is
defined for all a,b in any scalar extension of V,
because it is a polynomial of degree < 2 in a and in b
and because the characteristic of K is not 2. From
(I;2.10) we get B(-a,-b) = B(a,b).
From (I;2.8) it follows then that
t @b (x) = [B(x,b)] -1 [x-(eb)(x)], b e v;
and part d) of I, Theorem 2.3, implies
W°t 0b = t Q oWj where W e T(O) and c = W b.
From (I;4.5) we see that f e H(£t) can be written as
(1.3) f = W°t a °t Qb °t , where W € r(£t) and a,b,c € V.
Moreover (I;4. 6) yields
(1-4) m& = W[B(x+c,b)] _1 .
This formula remains valid in any scalar extension for
which x is generic.
LEMMA 1.4 . Let a and b be in some scalar extension
L V of V. such that det B(a,b) £ 0. Then
a) B(a.b) 6 r(L--Q) and [B(a,b)]* - B(b.a).
b } *@b° t a = W ° t c° Vd ' where w = [ B (a . b ) ]
-1
II, §1 39
c = a- (0b) (a), and d - t @ (b)
c) B(x+a,b) - B(x,t @a (b)) B(a,b).
d) B(Wa,b) = WB(a,W*b)W for W e r(L®0).
Proof : It suffices to prove the lemma in the case
that a,b are generically independent elements of some
scalar extension Lg>V and that x is still generic over L.
We put f = t,-,, t and obtain ov(x) = det B(x+a,b)
from (1-4) and part a) of I, Theorem 2.3- Hence
^f(O) f and consequently f e H (L5JO). Part a) of
I, Theorem 4.3, yields
(1-5) L<>t = w° t »L
v ' ®b a c ®d
for some W e F(LSO) and c,d e L®V. Passing from f to
f we obtain
(1. 6) t rs Ot . = t ,0t Q OW*.
v ' -@a -b -d -0c
Using (1-4) twice we conclude
(1.7) B(x+a,b)W = B(x,d), W # B(x-b,-a) = B(W # x,-c).
Choosing x = we obtain
W = [B(a,b)] _1 and W* = [B(b,a)] _1 .
So part a) is proved. Substituting x = in (1.6)
and using (I;2.10) we get d = t -, (b). Moreover x =
in (1.5) leads to Wc = t , (a), i.e., c = a-(~b)(a)
according to (I;2.8). So part b) is proved. Now part
40
11,52
c) follows from (1.7).
In order to prove part d) we apply (I;2.11)
and get B(Wa,b)W = WB(a,c) where u = ®b, v = ©c and
-1 #
u = v TT v. Hence c = 9v I7 &b = v „b = W b.
According to (1.4) and part a) of the lemma,
E e H(O) the en<
r(KVs) and we have
for f 6 H(O) the endomorphism — r^ — *- belongs to
m)
( Sf(x) l \
I ^ u J
= f 5f(x)
x-> a v x— a
for a € Dom f.
§2. The group 5(Q, Q) .
1. Again let (&,©) be a symmetric Lie algebra,
= V + X + V, and let f -* f* be the involution of
S(C) induced by © according to Theorem 1.3- We consider
the group
3(0,0) = [f;fe3(0), f*°f=l} = [f,fe3(o), v f ©=2v f }.
Clearly f -> f ' maps this group onto itself. Further-
more, let
r(o,e) = H(o J ®)nr(o) = [w;Wer(o), w # w=i}.
Next we define the subset
D(£,©) = [a;aeV, there exists WeT(O) such that B(a . -a)=W i W}
of V. Clearly belongs to it and a e 0(0,?) implies
Wa -. D(T;, ! ) for W r(C, S) because of part d) of
II, §2 41
Lemma 1.4. For a e D(£t, ®) we choose W o e T(O) (not
3.
canonically) such that
B(a,-a) = wf W .
a a
Obviously W is uniquely determined up to a left-
factor out of r(O,0). We may choose W = I, W = W .
Moreover, a e Dom t ,-, for a £ D(&, ®). Hence the
' -8a v j /
element
a = W a t_ @a (a), a e D(a,0),
of V can be defined. Finally we put
s = t~°W B t fl , a e D(O,0),
a a a @a N ' '
and we obtain an element of the subset a (O) of H(O).
Clearly, -a e Dom s so s (-a) = and s (0) = a. More-
3. 3 3.
over, -s (x) = s (-x).
' a v ' -a v '
2. We show that f e H(0,©)nH°(o) is equivalent
to f = W°s where W e r(O,0) and a e D(0, e).
a — ■"■ ~"~-^—
According to I, Theorem 4.3, part a), the elements
of a (D) are exactly the functions
f = Wot,°L for a,b e V and W e r(O)
b e)a
Using (1-2) we observe
f*" 1 - » H .Ln .
©b a
Hence f of = I is equivalent to
42 II/§2
(2.1) W#w °V^a = ^b ot a-
Using (1.4) we get
(2.2) B(x,a) - B(x+a,b)W # W.
Conversely suppose that (2.2) is satisfied. Hence
M^M . „H [B(rtjb)] -i . w[B(x , a)] -i . 2|i*l
#-1
and I, Lemma 4.2, yields f = t of for some c e V.
> > j c
~ #-1 ~ ~
Therefore f - t m of = t fl °t °f and we get t r , = t
©c ©c c & 9c -c
from which we obtain c = 0. Hence (2.2) is equivalent
to f*of = i, f = Wotuot n •
— b @a
In particular,
s = UoW oL o "WotoL , b = W" 1 a = t - (a) ,
a a a (sa a b ca a --a
belongs to H (£l,@) whenever
B(x,a) = B(x+a,t__ (a))B(a,-a)
holds. But we get this identity by replacing a by -a,
b by a and x by x+a in part c) of Lemma 1.4. So we
proved s e 3 (C, 9)0-° (£) .
Now in the notation above let f be in ~ (£.,?) .
Substituting x = -a in (2.2) we get a t D(£. -) and
doing the same in (2.1) we get W*W(b+t r (-a)) = 0,
i.e., b = t_ @a (a) = W~ a. Using (I; 1.9) we observe
that f - W 1 °s a . W x e T(O). But we know already that
. <-. 4
s a satisfies s a o s a = I, so W-, e T(C, ?).
II, §2 43
3- Suppose that x and y are generically indepen-
dent elements of a scalar extension of V. Let G be
the group of f e H(O) satisfying the differential
equation
(2.3) B(f(x),-f(y)) = ^1 B (x,-y) (-^-j # .
We prove first , that s e G for a 6 D(0, @).
For the proof we rewrite part c) of Lemma 1.4
as follows
(2.4) B(t a (x),y) = B(x,t@ a (y))B(a,y)
Applying the involution * and interchanging x and y
we get
(2.4') B(t@ (x),y) = [B(x,a)] _1 B(x,t a (y))
Because of 2 we have
s = UoW Ota = S' x = t„^W ff °t
a a a ^a a Ud a i
and hence using (2.4)
B(s a (x),-s a (y)) = B(s a (x),-s a _1 (y))
= B(W a ot 3a (x),W a _1 t_ a (-y))W a Q(y)
where Q(y) does not depend on x. Now part d) of Lemma
1.4 and (2.4') yields
,(s a (x),-s a (y)) = W a B(t @a (x),t_ a (-y))Q(y)
44 II* §2
= W a [B(x,a)] -1 B(x,-y)Q(y)
Using f = s Q we have
B(f(x),-f(y)) = 2|Jfel B(x,-y)Q(y).
Applying # and interchanging x and y we get
B(f(x) J -f(y))= [Q(x)]* B (x,-y)(^i)' .
Specializing x ^> -a we have f (0) = 0,
(Mfel) -vf-M)]- 1 .**- 1
N ' x-> - a
# #-1
and therefore we obtain I = [Q(y)] B(a,y)W and
this means that Q(y) = PffiM * H
ence s e G.
a
4. As in 1,54.2, we denote by Df the set of
a e V such that uu f (a) ^ 0; we know D f c Dom f. We
introduce now the condition
(A) D(0,©)nD f + for all f e H(O),
which is certainly satisfied, if D(0, ®) meets all the
Zarski-open subsets of V.
THEOREM 2.1 . Let (£, 8) be a symmetric Lie algebra
Q = V + £ + V, and suppose that (A) is satisfied . Then
for f e H ( O) the following conditions are equivalent :
a) f*°f = I, i. e. , f e E(£J,©).
II, §2
45
b) B(f(x),-f(y)) « .3§Jfel B(x ,_ y) (l|IlI) # s
where x and y are generically independent .
c) f = W°s os, , where W e I*(0,8) and a,b 6 D(D,@)
3 D
In this case f = W e r(O,0) if and only if
e D f and f (0) = 0.
COROLLARY . If f e H(0,@)nH°(D) then f = Wo s
where W e r(£l,©) and a e D(0, ©) and this representation
is unique.
Proof : a) => c) : Let f of = I, then by (A) there
- 1 #
exists a e D(£t,®)fiD.e« Forming g = f°s_ we get g'°g = I
because of 2 and e D • Applying 2 we have g = Wo s, ,
where W e r(0, ®) and b e D (£>,©). So c) is proved.
b) =» c): Again by (A) there exists a e D(O,0)riD f .
Specializing x»a and y^a in (2. 3) we get
B(f(a),-f(a)) = W B(a,-a)W # , W = -^1
Hence b = -f(a) e D(£, ©). Forming g = s, of we see that
g satisfies (2.3) because of 3 and g(a) = 0. Specializ-
ing x—a and y^a in (2.3) we get a e D(0,®) and
h = go s = s,c f o s satisfies h(0) = and (2.3).
Specializing y-'O in (2.3) we see that — r^ — *- does not
depend on x. Hence h = t W by I, Lemma 4.2, and
h(0) = yields h = W- Again (2.3) leads to W « r(S,®),
46 II, §3
So c) and in addition the last statement of the
theorem are proved.
The conclusions c) =» b) and b) =» a) follow from
2 and 3-
§3- Constructions of symmetric Lie algebras .
1. A bilinear map (a,b) -» aob of VxV into End V is
called a pairing of V. Let X = X be the subspace of End V
spanned by a^b for all a,b e V. The symmetric bilinear
form a = a Q of V is given by
• a(a,b) = trace (anb+bna).
We call a the trace form of the pairing n. Suppose
that
(P. 1) a is non degenerate.
Then by T* we denote the adjoint endomorphism of
T e End V with respect to j. Denote by [T,S] = TS - ST
the commutator product in End V. We assume that in
addition the following conditions hold:
(P. 2) (anb)c = (cab) a for a,b, c £ V.
(P. 3) [T.aab] = Tanb - aQT*b for a,b 6 V and T 6 X
(P. 4) (acb)* = bna for a,b t V.
II, §3 47
From (P. 3) we observe that X turns out to be a Lie
algebra of endomorphisms of V, for which T -> -T*
is an automorphism of period 2. According to (P-4)
we get a(a,b) = 2 trace adt>.
Using (P- 2) and (P-4) we observe that a((anb)c,,d)
= a((cnb)a,d) = a(a, (bnc)d) = o((cQd)a J b). Hence by
linear extension of
a z (T,S) - a(Tc,d) = a(Sa,b) for T=anb, S=cnd,
we may define the symmetric bilinear form a~ of X, which
is also non degenerate. A verification shows that a~
is an associative bilinear form of X satisfying
(3.1) a z (T*,S) = c x (T,S*) for T,S e X.
LEMMA 3.1 . The identity I belongs to X and
trace T = %a^(TM) for T e X.
Proof : Since ct~ is a non degenerate bilinear form
of the vector space X, to the linear form trace T
there corresponds J e X such that trace T = a~.(T,J).
From trace T* = trace T together with (3- 1) we observe
J* = J. For T = ad) it follows that
a(a,b) = trace (aQb+baa) = a^CJ^aob+bna) = 2a(Ja,b).
Hence 2 J = I e X.
2. Let Xjy,z be generically independent elements
of some scalar extension of V. We define endomorphisms
48 II,§3
P(x,y) and P(x) of a suitable scalar extension by
(3.2) P(x,y)z = %(xDz)y and P(x) = P(x,x), respectively.
Because of (P. 2) the endomorphism P(x,y) is symmetric
and linear in x and y. Moreover
P(x+y) = P(x) + 2P(x,y) + P(y).
Let a e V be such that P(x)a = 0, hence (xoa)x =
and by linearizing we get (xDa)y = and hence
boa = for all b e V. By (P. 4) we observe aob =
and consequently a(a,b) = for all b e V. So we
proved that
(3-3) P(x)a = 0, a £ V.. implies a = 0.
Note that nevertheless the determinant of P(x) can be
the zero function.
Replacing a by x and b by y in (P. 3) and applying
the result on x we get T(xDy)x- (xny )Tx = (Txny)x- (xnT*y)x
and consequently
(3.4) 2P(x,Tx) = TP(x)+P(x)T* for T e X.
By using (P. 2) and (P. 4) in scalar extensions of
V we observe that
-(P(x)a,b) = r((xn a )x,b) = j(x, (aPx)b) = j(x, (bax)a)
= j((xnb)x,a) = -(a.P(x)b)
and consequently P(x) is self-adjoint with respect to -.
II, §3 49
3- Next we consider for a given pairing
□ : VXV -*■ End V the direct sum of vector spaces
q = 2 = £ © V © V
D
and we write the elements of S as u = T©a©b where T e 2
and a,b e V. By
(3.5) 3 U ( X ) = -a + Tx + P(x)b
we obtain a linear injection u -> q of Q into the sub-
space y = \ + \ + ^2 of Po1 V- We a S ain identify 2 with
the space [Tx;t e I] of linear functions and put
V = V n = {P(x)b;beV}.
Hence the image of Q under the map u •» q is given by
0=0 - V + 2 + V.
It follows from (3- 3) that b -> P(x)b is a linear
bisection of V onto V.
THEOREM 3-2 . Suppose that the pairing a : VxV -> End V
satisfies the conditions (P. 1) to (P. 4). Then O = O q
turns out to be a binary Lie algebra , for which in
addition Z = [V,V] holds .
Proof : We have to prove that the conditions
(B. 1) to (B.5) of I, §2,1, are fulfilled. (B. 1) is
clear because 2 is a Lie algebra of endomorphisms of V.
(B.5) follows from Lemma 3.1. Let p(x) = P(x)b and
50 II, §3
q(x) ~ P(x)c be elements of V. For a e V we observe
that
[p,a](x) = 2p(x,a) = 2P(x,a)b - 2(anb)x,
hence [p,a] = aDb e Z and (B.2) is proved.
For T € X we obtain
[p,T](x) = 2p(x,Tx)-Tp(x) = [2P(x,Tx)-TP(x)]b = P(x)T*b
by using (3.4). Hence [p,T] e V and (B. 3) is proved.
Finally we have
fc[p,q](x) = p(x,q(x))-q(x,p(x)) = P(x, P(x)c )b-P(x,, P(x)b) c
We apply (P. 3) for T = and on a and obtain
2P(a,P(a)b)d-P(a,P(a)d)b = P(a)P(b J d)a.
Since the right side is symmetric in b^d we observe
3P(a,P(a)b)d = 3P(a,P(a)d)b. Hence [p,q] = and the
theorem is proved.
We apply Meyberg's Theorem (I, §2.7) to this case.
For v(x) = -P(x)c we obtain [[a,v],b] = 2[P(x,a)c,b]
= 2P(a,b)c = (adc)b and hence we have
THEOREM 3. 3 - Suppose that the pairing
d : VxV -* End V satisfies the conditions (P. 1) to (P. 4).
Then for any given c e V the product (a,b) -> P(a,b)c
defines a Jordan algebra in V.
Using the bisection u * q of 2 onto Q we lift
the product of Q to Q. Hence 2 turns out to be a Lie
II, §3 51
algebra with respect to product u = [u-,,u~] = T©a©b,
u. = T.©a.@b., that is given by
1111 & y
(3.6)
T = [T,,T 2 ] + a 1 ab 2 " a 2 Db i>
a = T- L a 2 - T 2 ap b = Tp^l - T'/b 2
Algebras of this type are considered in [ 8 ] in a more
general set-up.
4. For the given pairing □ : VxV -> End V we write
r if and only if
r = T D = r(0 Q ). Remember that W e GL(V) belongs to
WXW _I = X and v TT V = V
W
(see I, §4-1).
LEMMA 3-4 . Let W be in GL(V). Then the following
conditions are equivalent :
a) We ?,
b) P(Wx) = WP(x)W*.
c) W(a^b)W _1 = Wa-W"" L b.
Proof : First of all, W -. 7 is equivalent to
W(a=b)W _1 e X and WP(W _1 x)b = P(x)Wb
for a,b a V, when W is some endomorphism of V. The
second condition means WP(x) = P(Wx)W and this is
equivalent to
52 II, §3
(3.7) W(aDb)W~ - WaOUTb for a,b e V.
Hence (3.7) is equivalent to W e T. Going over to the
trace in (3-7) we get a(a,b) = cr(Wa,Wb) and consequently
W = W* .
5. We define a bisection 8 of O = 3 by setting
(®q)(x) = -b-T*x-P(x)a where q(x) = a+Tx+P(x)b.
2 ~
Hence ® = Id and ©V = V, ®2 = 2. A verification shows
that ® is an automorphism of the Lie algebra and
(Q,®) turns out to be a symmetric Lie algebra . In
particular one has
©T = -T*, [a,@b] = aob.
Again we write q = a + T + ©b for the elements in C
The symmetric Lie algebra (£>,©) induces an
involution W -*■ W of F according to Lemma 1.2. In
order to prove
W # = W'
we apply V ®V „ = © to a e V and observe that
W
- 1 #
WP(W x)W a = P(x)a. Hence the statement follows
from Lemma 3.4.
Using the abbreviation given in §1.3 we are goinj
to prove
(3.8) B(a.b) = B,,(a) = I + acb + P(a)P(b).
II, §3 53
We put T = [8b,a] = -anb and get
[®b,[®b,a]] = [3b, T] = ®[b,®T] = ®[T*,b] = ®(T*b)
where T*b = -(baa)b = -2P(b)a. Using the definition
(I;2.7) we observe that [B @b (x)]a = a+(aDb)x+P(x)P(b)a
= [I-bcnb+P(x)P(b)]a, hence (3-8) is proved.
From Lemma 1.4 we obtain
B(a,b) e T if det B(a,b) ±
and hence Lemma 3.4 yields
(3.9) P(B(a,b)x) = B(a,b)P(x)B(b,a) , a,b e V-
In part c) of Lemma 1.4 we compare the terms that
are of degree two in x and observe that P(x)P(b)
= P(x)P(t = (b))B(a,b) whenever det B(a,b) + 0. Now
(3.3) leads to
P(y) = P(t 0a (y))B(a,y).
But from the definition (I;2.8) and (3-9) it follows
that
P(y+P(y)a) = B(y,a)P(y)
and again comparing the terms of highest degree in y
we end up with
(3.10) P(P(a)y) = P(a)P(y)P(a).
Hence our method is powerful enough to prove non-trivial
identities about the pairings.
54 n ^ 4
6. We generalize 8 by setting
® = 0v = v Q, whenever JeT and J* = J.
Using the abbreviation
T J = J T*J, T £ End V,
we have more explicitely
(3.11) @jq = J _1 b - T J + ®(Ja), where q = a+T+@b e O
Again 2 is an automorphism of period two satisfying
V = V. Hence for any J e T, J* = J, the pair (O, £ )
is a symmetric Lie algebra . Using I, Theorem 3-2,,
one can easily show that these are the only automor-
phisms of O which lead to a symmetric Lie algebra.
Note that T ■* -T is an automorphism of the Lie
algebra X. By the same argument that we used in 5,
one shows that the involution of T induced by 0,
# J
is given by W = W .
'4. Killing forms ,
1. Let © be a Lie algebra over K. Denote by
ad u the adjoint representation. For a linear trans-
formation A of © mapping a subspace b of © into itself
we denote by A b the restriction of A to b. Let
(u,vL = trace (ad u) (ad v)
denote the Killing; form of © •
II, §4 55
Suppose that there is a direct sum decomposition
© = q + b, [ct,a] c a, [o,b] c b, [b,b] c Q .
Then we prove first
LEMMA 4. 1 . a) The subspaces a and b are ortho -
gonal with respect to the Killing form of ®.
b) For g = a + b, aea, beb,, one has
2 2
(g,g) @ = <a,a> Q + trace (ad a) b + 2 trace (ad b) b -
Proof : a) From (ad a) (ad b)a c b and (ad a) (ad b)b
c a it follows that <a,b) = trace (ad a) (ad b) = 0.
b) For ge@ put (ad_jjg) (a+b) = [g,a] and (ad_g)(a+b)
= [g,b]. Hence ad g = ad,g + ad_g. A verification
yields
(4.1) (ad a) 2 = (ad + a) 2 +(ad_a) 2 ,
(ad b) 2 = (ad + b)(ad_b)+(ad_b)(ad + b).
2
Since fed.a) is zero on b and equals the square of
2
the adjoint representation on o we get trace (ad.a)
2
= (a, a) . Moreover, (ad_a) is zero on a and equals
2 2 2
(ad a), on b, hence trace (ad a) = trace (ad a), .
So we obtain the statement for g = a using (4.1).
2
Again from (4. 1) we obtain trace (ad b)
= 2 trace (ad ,b) (ad_b) . But (ad, b) (ad_b) is zero on
2
a and equals (ad b). on b. Hence the lemma is proved.
56 II, §4
2. Now let Q : VxV • End V be again a pairing
satisfying the conditions (P-l) to (P. 4) and let a
be its trace form. We consider the Lie algebra
O = O = V + X + V together with the involution ®.
Let cw be the associative bilinear form of X given
in §3.1- For q = a v + T v + ®b v e O we put
a (q 1 ,q 2 ) = (^(TpTg) + a(a 1 ,b 2 ) + aCa^b^.
Clearly a is a symmetric non degenerate bilinear
form of O and a verification shows that a n is an
associative bilinear form for O.
LEMMA 4. 2 . The Killing form of the Lie algebra
£ is non degenerate and coincides with j_. In addition
we have
(T,T)j + 2 trace T 2 - ^(T.T) where TeX .
Proof : We apply Lemma 4. 1 to the case © = C,
q = X, b = V+V. Applying
[T,®b] = ®[©T,b] = ®[-T*,b] - -®(T*b)
we calculate for c,d e V
(ad T) (c+@d) = [T,Tc-8(T*b)] = T 2 c + 9(T* 2 b).
Hence
trace (ad T) y+ ^ - trace (T 2 +T* 2 ) = 2 trace T 2 .
Furthermore we have for T = and-c^b
II, §4 57
(ad(a-f^b)) 2 (c-HBd) = [a+€b, [a,@d]+[©b,c ] ] = [a+6b,T]
- -Ta+@T*b = +(aDb)c-(aad)a-H5(-(bnc)b+(bDa)d)
and consequently
2
trace [ad(a+t3b)] ,~ = trace (aab+bna) = a(a,b).
Summing up we get
<q,q> = (T,!)^ + 2 trace T 2 + 2a(a,b) where q = a+T+©b.
Since the Killing form of a Lie algebra is associative^
we obtain an associative bilinear form X by setting
Uqpq 2 ) = ^p^o ~ a o^i jq 2^' But x Cq x ^ q 2 ) = x(t 1 ,t 2 )
implies X(T,ao D ) = X(T, [a,8b]) = X(Ta,3b) = 0. So
X = and the lemma is proved.
3. In order to give a sufficient condition for
£ = Cl to be simple we prove first
LEMMA 4-3. A subset ^ of O is an ideal of O
if and only if
v + z + ev, ,
o o 1
where V and V. are subspaces of V, X an ideal of X
o 1 c o
such that
XV c V, I V c V , X*V cv p [V,0V ] ex
v v o o o 1 v J o
holds for v = 1,2.
Proof : Let 3 be an ideal of C and let
q = a + T + r ~-b c . 3- We observe [I,q] - a-Qb e 3
and [I,[I,q]] = a+Vb f 3- Hence T, a and b belong
58 II, §4
to 3 and we have 3 = V Q + X Q + ®V r A verification
leads now to the conditions listed in the lemma.
THEOREM 4-4 - If 2 acts irreducibly on V then O
is a simple Lie algebra .
Proof : Let 3 be an ideal of . Then 3 = ^ +z + ®^i
according to Lemma 4-3 and we have £V c V . Hence the
V 's are invariant under X. By assumption the only
invariant subspaces of V are and V itself.
The case V = or V-, =0 yields X =0 and hence
o 1 J o
[V,®V ] = 0. For a € V and b e V we obtain anb =
and hence a(a,b) = 0. That means that V = or
V, = implies V = V, = and hence 3=0.
In the case V = V, = V we get [V,®V] c Z and
o 1 ° o
hence 3 = a -
4- According to the criterion of Killing-Cartan
in case of characteristic zero, any pairing gives
rise to a semi -simple Lie algebra.
As a further application of the lemma we prove
LEMMA 4.5 . An endomorphism T _of V belongs to X
if and only if
2P(Tx,x) = TP(x) + P(x)T*.
Compairing this result with Lemma 3-4 we see that 2
coincides with the Lie algebra belonging to the linear
II, §5 59
algebraic group T = T .
Proof : Because of (3-4) it is enough to consider
a T satisfying the condition above. By linearization
we get [T,aOb] = TaQb - aoT*b and hence [T,X] c Z. On
the other hand for v(x) = P(x)a we get [v,T](x)
= 2P(Tx,x)a - TP(x)a = P(x)T*a e V. Therefore
[T.O] c o and hence q -» [T,q] turns out to be a
derivation of O. But a Lie algebra with non degenerate
Killing form has only inner derivations (see N.Jacobson,
Lie algebras,, page 74). Hence [T,q] = [q ,q] for
some q e 0. Since a binary Lie algebra has center
we end up with T = q and hence T e X.
§5- A characterization of symmetric Lie algebras.
Essential parts of the following results are due
to K. Meyberg and U. Hirzebruch.
1. Let ® be a finite dimensional Lie algebra
over a field K of characteristic different from 2 and
3- Suppose there exists a direct sum decomposition
= i) + q + b
as vector spaces having the composition rules
(1) [t),t,] - t, = [»,&], [$,b] c a, [^b] c b,
[a, a] = [b..b] = 0.
60 II, §5
The elements of ® are in an obvious notation written
as u = h+a+b. Suppose further
(2) The Killing form (u,v) of © is non degenerate.
(3) There exists an automorphism t of © of period 2
satisfying
Tt) = t) , TQ = b, Tb = Q.
If a^ pairing □ : VxV -> End V satisfies the
conditions (P. 1) jto (P. 4) of §3.1, then O = O q = V+X+V
together with the automorphism t = defined in §3-5
satisfies the conditions (1) to (3) for fc> = X, a = V
and b = V (see Theorem 3-2 and Lemma 4.2).
2- Suppose now that ® satisfies the conditions
(1) to (3). We are going to prove some propositions:
PROPOSITION 1 . There exists h in the center of b,
- o
such that
[h ,a] = a and [h ,b] = -b
o o
for aea, beb
Proof : Using (1), a verification shows that the
map h+a+b -> a-b is a derivation of @. Because of (2)
any derivation of © is inner (see N. Jacobsen [ 6 ],
page 74), hence there exists u e © such that [u ,u]
o L o J
= a-b for u = h+a+b e @. Hence
II, §5 61
[h ,h] =0, a = [h ,a]+[a ,h], -b = [h ,b]+[b ,h].
L o o o o o
We observe a = b =0 and the proposition is proved.
PROPOSITION 2 . Let u. = h.+a.+b. e 0. Then
= x ill
one has
(UpiO = (h 1 ,h 2 )+<a 1 ,b 2 )+<a 2 ,b 1 >
and the following implications
<b,a> = =» b = <a,b> = =* a = 0,
[h,a] = =* h - 0, [h,b] = =» h = 0,
where heb,, aea, beb.
Proof : Using Proposition 1, we observe that
(h,a) = (h, [h ,a]> = ([h,h ],a) = and similarly
(h,b) = 0. Furthermore <a-,,a ? ) = < [h ,aJ,aJ
= -(a,,[h ,a„]> = -(a,,a^), hence (a^aJ = and
similarly we obtain (b,jb«) = 0. So the Killing form
of @ has the form indicated in the statement. From (2)
we obtain the first two implications. Finally suppose
[h,o] = 0. Then = <[h,o],b> = <a,[h,b]> and
[h,b] = 0. Similarly, [h,b] = implies [h,o] = 0.
Next [h,t>] = [h,[a,b]] = [[h,o],b] + [a,[h,b]] =
and h is in the center of ©. But © is centerless
because of (2).
3- Next we put V = a and we write now the elements
of ©as h+a+"b, where he*) and a,b i V. Setting
62 IMS
a(a,b) = <a, Tb>, a,b e V
we obtain a symmetric bilinear form of V because of
<0u,,0u ? ) = (u-.,u„). By Proposition 2, a is non
degenerate -
For helj we define an endomorphism TV of V by
T, a = [h,a] , a e V.
n
By Proposition 2, h -» T, is a linear injection of ^
into End V. Moreover, a pairing □ : VxV -» End V is
given by
aDb = Tr , I where a,b e V.
PROPOSITION 3 - The pairing □ : VxV ■> End V
satisfies the conditions (P.l) to (P. 4) of §3-1
and a is its trace form - Furthermore , h -* T, defines
an isomorphism of the Lie algebra t) onto the Lie
algebra Z associated with the pairing -
Proof : Since V = q is an abelian subalgebra of @
we observe (P. 2). Then using Lemma 4-1 we have for
a,b e V
<a,Tb> = 2 trace [ad(a+-b) ]?,« .
By a verification, the right side equals trace (anb+bca)
Hence 3 is the Lrace form of the pairing and (P. 1)
is proved. For (P-4) we have
II, §5
63
j(T h a,Tb) = <[h,a],Tb> = -<a,[h,Tb]>
= -<a,T[Th,b]> = -a(a,T Th ,b)
and consequently
h Th
Hence (P-4) is satisfied, too. From the definition
of T, we observe
h
T [h,k] = tW' h ' k e *■
Finally, using the Jacobi identity we get
[T h ,adb] = [\,T [aJh] ] = T [hAa}jh]]
= [h,a]nb + an[Th,b]
= T, anb - anijb.
h h
Hence (P. 3) is valid, too.
4. We construct the binary Lie algebra
C n = V + 2 + V associated with the pairing □ : VxV -
End V, where V = ©V and where ® is the automorphism
of £X_ defined in §3. 5- Hence
h + a + Tb -» a + T u + ©b
h
turns out to be a linear bijection of © onto O .
Now a verification shows that this map is a homomor-
phism of the Lie algebras. Summing up we have
64 ii, §5
THEOREM 5-1 - If the Lie algebra @ satisfies the
conditions (1) to (3), then <g is isomorphic to a
symmetric Lie algebra O , where the pairing □ satisfies
the conditions (P. 1) to,(P-4), and vice versa .
5. For example let Z be a bounded symmetric
domain in a complex vector space. (See S. Helgason
[ 3 ], Chapter VIII, §7. )
Denote by G the group of biholomorphic mappings
of Z onto itself and denote by ® the complexif ication
of the Lie algebra of G- Then ® considered as a
Lie algebra over R satisfies the conditions (1) to (3).
Hence we get
THEOREM ,5.2 . If Z is a bounded symmetric domain
then @ considered as a real Lie algebra is isomorphic
to a symmetric Lie algebra £ = £l.
Ill, §1 65
Chapter III
EXAMPLES
§1. Symmetric and skew- symmetric matrices .
1. Let 2K be the vector space over K that consists
of all r by r matrices with entries in K. For e = ±1
denote by V = V the subspace of ae^ such that a = ea,
where a stands for the transpose of a. Hence the
dimension of V equals %r(r+e). Furthermore let GL(r.,K)
be the group of invertible matrices of 2(1 and let e
be the unit matrix of ^ •
r
For ueDOi we define an endomorphism T of V by
r r u J
T x = u x+xu. Hence u -> T is a linear injection of
u u J
2H into End V. Note that this is not true in the case
r
e = -1 and r = 2. A verification shows
(1.1) [T ,T ] = T r ■> for u,v e fl» .
u v LujVj r
PROPOSITION 1. For ueSQl one has trace T
■- r u
= (r+e)trace u.
Proof : Define a linear form X of 301 by X(u)
= trace T • Hence X.(uv) = X(vu) because of (1.1).
Since the bilinear form of IR that is given by
(u,v) -* trace(uv) is non degenerate, there exists an
element ae!Dl such that X(u) = trace (au) and we get
trace(auv) = trace(avu) = trace(uav). Hence au = ua
for ue3J! and consequently a = ae where acK. So we get
66 "Ml
trace T = a trace u. For u = e we find trace T u
= 2 -dim V and trace e - r, hence a = r+e .
PROPOSITION 2 . The set {T ;ue2K } of linear transforma -
tions of V acts irreducibly on V.
Proof : We have to show that and V are the only
subspaces of V that are mapped into itself under the
maps x -* u x+xu for ue3JI . Let u be the matrix with 1
at the first entry of the diagonal and zero elsewhere.
Then x- (u x+xu) is obtained from x by replacing the
first row and first column (except the first diagonal
element) by zeros. Now an induction argument completes
the proof.
PROPOSITION -3 - The vector space SCR is spanned by
elements of the form ab where a^b e V.
Proof : Let e = 1. Since 3JI is spanned by the
matrices that have non-zero entries only at the inter-
sections of two rows and the corresponding two columns
it suffices to prove the proposition for 2 by 2 matrices.
But in this case the statement follows from
u 3\ foe y\ [1 0[ , j3
Y 5/ W \0
1
In the case e = -1 a similar argument shows that
it is enough to prove the statement for three by three
matrices, for which again one uses a verification.
There is another type of endomorphisms of V given
Ill, §1 67
by elements of 3D? . For ue2S define W by
J r r u J
W x = u xu, xeV.
A verification yields
(1.2) W W = W for u,v e 332 .
v u v uv r
1 — hp
PROPOSITION 4. For ueDJi one has det W = ±(det u) .
= r u
Proof : Since the field K is infinite it suffices
to prove the statement for u e GL(r^K). But both sides
are (up to a sign) multiplicative, so it is enough to
prove it for a set of generators of the group GL(r,K).
Using the standard generators of GL(r,K) the proof can
be completed.
2. Suppose now that the characteristic of K does
not divide r+e . We define a pairing □ : VxV -* End V by
(anb)c = ab c + cb a, a,b,c e V-
Let X be the subspace of End V spanned by aob for
a,b 6 V. Clearly the elements of 2 are the endomor-
phism T V7here u is in the vector space spanned by
ab = -ab for a,b e V. Hence by Proposition 3 we get
x = It ;uei« }.
u r
By Proposition 1 the trace form of the pairing o
is given by
68 III,§1
^
(a,b) = (r+e) trace (ab ) for a,b e V.
Hence a is non degenerate and T = T r ■
u xi
Because of the associativity of the matrix product
and the commutativity of the trace of a matrix, one
verifies then that the pairing □ : VxV -* End V satisfies
our conditions of II, §3.1- We obtain
(1.3) P(a)b = ab t a for a,b e V.
and the associated binary Lie algebra O = O consists
of the elements
(1.4) q(x) = a+u x+xu+xb x where a,b e V and ueiOt .
Here the generic element x of V can be chosen as a
matrix x = (t;.), t.. = eT. ., where the T..'s (i<i
if e = 1 and i<j if e = -1) are algebraically indepen-
dent over K. According to II, Theorem 4.4, and to
Proposition 2 the Lie algebra is simple. Clearly
the dimension of O over K equals r(2r+e).
P(a) is an endomorphism of V provided aeV. Com-
paring (1-3) and the definition of W we observe P(a)
= eW , aeV. Hence using Proposition 4 we obtain
3.
(1.5) det P(a) = ±(det a) r+€ .
Let the automorphism 8 of be defined as in II, • 3-5-
Hence we get
H @ (x) = -P(x)
Ill, §1 69
(see I, §2. 2) and from (1.5) it follows that ® is
essential if and only if e = 1 (r>l) or e = -1 (r>3 even)
For u e GL(r,K) we consider the endomorphism W
of V (see 1). It follows from Proposition 4 that W
= u
belongs to GL(V). A verification shows that the
adjoint of W with respect to the trace form a equals
W t - Hence we obtain P(W x) = W P(x)W and according
u
to II, Lemma 3-4, we get
W u e r(O) for u 6 GL(r,K).
3- We consider now the group H(JO) of birational
functions. From (II;3.8) we know B(a,b) = I+anb+P(a)P(b)
and hence we have
(1.6) B(a,b)c = (e+ab t )c(e+b t a) where a,b,c e V.
and B(a,b) equals W for u = e+b a. Furthermore
using II, §1.3, we observe
(1.7) £ eb (x) = X ( e+Gbx ) -1 = (e+exb) -1 x.
In order to describe the group 3(0) we define a 2r
by 2r matrix Q by
-ee
and we denote by Q = Q . the group of 2r by 2r matrixes
M satisfying the condition
(1. 8) M t QM = Q.
70
III,§1
Note that M t eQ whenever MeQ. Writing
/. b\
M - , where a,b,c,d are rxr matrices,,
\c d)
a verification shows that MeQ is equivalent to
(1.9) a C c = ec^, b t d = €d C b, a t d-ec t b = e.
From (1.8) it follows that the inverse of M is given
by M = -eQM 1 ^., hence
»-'■(*', -f)
\-ec a /
Next let Q* denote the set of MeQ such that the
rminants of cx+d and of -ec x+a ai
polynomials in x. Hence we can define
determinants of cx+d and of -ec x+a are not the zero
-1 a b
f M (x) = (ax+b)(cx+d) L where M = e Q*.
From (1.9) we observe
(1.10) [f M (x)] C = e.f M (x) and — ~~ = W u where u=(cx+d) \
Moreover, for N = M the function f^ is also defined
and one verifies f , ,o f ,, = f ,,o f ., = I. Hence f,, belongs to
M N N M M &
the group P(V) of birational functions . A verification
yields now f^p f M = f„ M where M, and M~ are in Q*.
According to (1.4) we write the elements of O as
q(x) = q-. + q2X + xq2 + xq„x where q-,,q^ e V and q ?t -0i .
III,§1 71
In the notation of (I; 1.4) we obtain
(1. 11) q = u q-,11 + u q 2 v + ev q~u + ev q^v, f = f ,
where u = cx+d and v = ax+b. From (1. 10) one concludes
that q is in D whenever qeO. Since the same is true
for f instead of f, the map q -> q is a bijection of
D According to I, §1.5, we obtain an automorphism
v r of fl whenever f = f„, MeQ*. One concludes from
f — — M *
(1. 11) or from I, Theorem 2. 1, that v~ is essential .
Then from I,§4. 2, it follows that f belongs to the group
3(D). So we proved f M € 3(D) whenever MeQ*.
4. Let MeQ* and suppose that f M equals an element
W in r(O). We get ax+b = (Wx) (cx+d) and this is equival-
ent to b = 0, (Wx)d = ax, (Wx)(cx) = 0. Then from
(1.9) it follows a d = e and hence Wx = axa as well
as c = 0. So for MeQ* we see that f M e F(D) is
equivalent to
/«* o \
M = I J for some u e GL(r,K).
\0 u" 1 /
i.e., to f M = W ■ Denote by T (D) the subgroup of
r(D) consisting of the elements W where u e GL(r,K).
Hence f„ e r(D) implies f„ € T (D). Finally denote by
3*(D) the subgroup of 3(D) consisting of the functions
f e 3(D) such that
3f(x)
ax
€ r (D)
x>d
72
III,§1
whenever d is in the domain of definition of f. From
(1. 10) we obtain that f M e -*(£) whenever Me^*. Writing
f = Wo t o t n , o t where W £ r(O), a,b,c e V
a "ib c
(see I, §4. 2) we obtain
^I^^WtBCx+cb)]" 1
from (I;4-6). In view of (1.6) we see that f belongs
to H*(o) if and only if W e F (O). Using (1.7) we
observe
K ° \
W if M = J.
u" 1 •
fw = S
t. if M =
t™ if M
CD
\0 e
/e 0^
\ e:b e ,
and in each case M belongs to Q*. Hence for f s 3*(0)
there exists a M G* such that f = f lr .
M
Summing up we proved:
(i) The elements of * (Z) are exactly the functions
(ii)
f N , where M/F".
M
Q * is a subgroup of Q and M - f defines an
epimorphism of the groups having the kernel
la(g °); j& a.K}.
(iii) ']" can be generated by the matrici
Ill, §1 73
u fc
u" 1
where u e GL(r,K) and a,b e V-
(iv) Each element in S(O) can be written as Wo f M
where W e T(O) and MeQ*. Here W can be
chosen in a given set of representatives of
r(O) modulo r (O).
5- We consider now the case e = 1. Then the
group Q coincides with the symplectic group Sp(r,K).
One can show that in this case Q - * equals Q. One has
only to prove that det(cx+d) is not the zero polynomial
a b
whenever M = ( ,) e Q. Since c can be replaced by
ucv and d by udv where u,v e GL(r^K) one can choose
c in a normal form and then det(cx+d) ^ follows from
(1.9). For more details see C. L . Siegel [15] .
PROPOSITION 5 - For W e T(O) there exists aeK and
u e GL(r,K) such that W = a-W •
v u
Proof : Assume first that K is algebraically
closed. Since any invertible symmetric matrix a can
be written as a = u u, u e GL(r.,K), it suffices to
prove the statement for W e r(Q) such that We = e.
Let e., i = \,2,... } x, be the diagonal matrices having
non zero elements only at the i row. Then given a
2
symmetric matrix a satisfying a = a f there exists
u e GL(r,K) such that u au = e, + • • • + e for some s
74 111,52
and such that u u = e. Hence we may assume that
We. = e. for i = \,2,...,r. Now a verification yields
W = I and we proved that W e T(O) implies W = W for
some u e GL(r,,K).
If K is an arbitrary field and if W e r(£t) we
apply the previous result to the algebraic closure K
of K and obtain an r by r matrix u with entries in K
such that Wa = u au for all symmetric matrices a
with entries in K. An observation yields u = 3v
where (3eK and v has entries in K. But W maps V onto
2
itself hence a = |3 eK.
From Proposition 5 and from (iv) it follows now
that S(O) consists of the elements a- f M where f aeK
and M e Sp(rjK). Furthermore from (iii) we obtain the
usual set of generators of Sp(r,K).
In particular we see that the Lie algebra C is
of type C r -
For e = -1 one can show that is a Lie algebra
of type D •
§2. The rectangular matrices .
1. Let V be the vector space of r by s matrices
with entries in K and suppose r>s. Hence the dimension
of V equals rs. We assume that the characteristic of
K does not divide r+s. Let HI be the vector space
of pairs (u,v) such that ue3J! , veKI and trace u
= trace v.
Ill, §2 75
For (u,v) e 2R we define an endomorphism T
of V by T x = ux-hcv. A verification shows that
J U, V
T x = for all x implies u = ae and v = -ae for
u, v
some aeK. Hence a- r = -a- s and we obtain a = 0.
The map x -* T x of 331 into End V consequently is a
r u,v rs n J
linear injection.
PROPOSITION 1 . For (u,v) e 5K one has
- x ' rs
trace T = (r+s) trace u = (r+s) trace v-
u,v
Proof : We consider the linear transformation
x -*■ ux, ueJR , of V. Writing x = (x-,j-..,x ) where the
x.'s are vectors we get ux = (ux,,... ,ux ) and hence
s- trace u is the trace of this linear transformation.
A similar argument shows that r- trace v is the trace
of the transformation x -> xv, veffl .
PROPOSITION 2 . The set [T ;(u,v) e EDI 3 of
- u,v rs —
linear transformations of V acts irreducibly on V.
Proof : Similar to the proof of Proposition 1
in §1.
PROPOSITION 3 - The vector space SK is spanned by
elements of the form (ab , b a) where a.b e V.
Proof: First of all, trace ab = trace b a. In
the case s = 1 we get 30i ^ = f(u, trace u) ;ue9Jt } and the
proposition follows from the fact that 2U is spanned by-
matrices of the form ab where a,b are r by 1 matrices.
76 III, §2
In the case s>2 again it is enough to show the
statement for r = s = 2. But one gets this by a
verification.
V by
For ue9JJ , ve3K we define an endomorphism W of
r s u,v
W x = uxv, xeV.
u, v
A verification yields
(2. 1) W W . = W . for a,u e S» and b,v e 5R .
v/ u,v a,b ua,bv r s
Similarly to the proof of Proposition 4 in § 1 we obtain
PROPOSITION 4 . For ue3B and veUl one has
det W = ±(det u) s (det v) r .
u,v
2. We define a pairing □ : VXV -> End V by
(anb)c = ab c + cb a, a^b^c e V-
Let £ be the subspace of End V spanned by anb for
a,b e V. The elements of X are the endomorphisms
T where (u,v) is spanned by (ab ,b a) for a,b e V.
Hence by Proposition 3 we get
X = [T ;(u,v) e DOR }.
u, v rs
By Proposition 1 the trace form of the pairing □ is
given by
c(a,b) = (r+s)trace ab for a,b e V
Ill, §2 77
and consequently a is non degenerate. Moreover (T ) + T
' u , v
A verification shows that the pairing □ satisfies
the conditions of II, §3.1- We obtain again
(2.2) P(a)b = ab fc a for a,b e V
and the associated binary Lie algebra O consists of the
elements
(2.3) q(x) = a + ux + xv + xbSc where a,beV, (u,v)e^ rs .
The generic element x can be chosen as an r by s matrix
having algebraically independent entries. According
to II, Theorem 4.4, and to Proposition 2 the Lie
algebra O is simple. The dimension of equals
(r+s) 2 -l.
Let & be the automorphism of O defined as in
II, §3. 5- We get H Q (x) = -P(x) and from (2-2) it
follows that 9 is essential if and only if r = s.
For u e GL(r,K) and v € GL(s,K) we consider the
endomorphism W (see 1) of V. From Proposition 4
it follows that W belongs to GL(V). The adioint
u,v & J
of W with respect to the trace form a equals
u, v K M
W t t and we obtain
u , v
W e r(O) for u e GL(r,K) and v e GL(s,K)
u, v v
according to II, Lemma 3.4. Now (2.1) shows that
(u,v) -» W is a homomorphism of the group
GL(r,K)xGL(s,K) into T(Q) having the kernel
Ill, §2
[(ae.^e); f aeK}.
PROPOSITION 5 - For W £ T(O) there exists
u 6 GL(r,K),, v e GL(s,K) such that W = W u .
Proof: Let a. be the i column vector of aeV-
l
Writing'
we obtain
[Wa] . = £ w. .b. , w. . e S ,
[W*b] • = E w^.b.
1 j Ji J
and a verification shows that W e T(O) is equivalent
to
(2,4) k ^ p w jk a k a J W U = k ^ W i^ a k a l w jk
where a-.,---,a are arbitrary columns. In particular
we get
„ „t t _ t
w.i a a w., - w., a a w.,
jk lk lk jk
for arbitrary column a. For a given i there is a k
such that w M ^ 0. Hence w.. =a..u.. a., e K, u. e UK
ik ij ij i- ij - i r
Replacing W by WW for suitable v e GL(s,,K) we may
assume that a.. = 6.. holds. Now (2-4) yields u. = u
for all i and the proposition is proved.
3- In order to describe the group H(£}) we obtain
from (II;3.8)
Ill, §2
79
(2.5) B(a,b)c - (e+ab t )c (e+t^a) where a,b,c e V
and B(a,b) equals W for u = e+ab , v = e+b a.
v M u,v
Using II, §1.3, we observe
(2.6) £ @b (x) = x(e+b t x)" 1 = (e-hcb t )" 1 x.
We write the elements of GL(r+s,K) as
/a b\
M =
where a e 3K , b,c e V, del
,c t d r s
Since the s by r+s matrix (c ,d) has maximal rank, the
determinant of c x+d is not the zero polynomial. Hence
for M e GL(r+s,K) we have the rational function
f M (x) = (ax+b)(c t x+d)" 1 .
A verification yields f M ° f N = fimu for M, N s GL(r+s,K)
and we obtain a homomorphism M ■* f M of GL(r+s,K) into
the group P(V) of birational functions and its kernel
consists of the diagonal matrices. In particular we
get
f M = <
u \
W if M = , , ueGL(r,K), veGL(s,K)
v" 1
u, v
if M =
, aeV,
if M =[ , beV.
ib e
But the group GL(r+s,K) is generated by the matrices we
80 III, §3
listed above and we obtain
H(O) = [f M ; M € GL(r+s,K)}.
Hence O is a Lie algebra of type A . .
§3- Jordan pairings -
1. Let V be a vector space over the field K of
characteristic different from 2 and 3 and let 31 be a
Jordan algebra defined in V with unit element e.
Denote the left multiplication by L, i. e. , ab = L(a)b,
and suppose that its trace form given by
(ab) -» trace L(ab)
is non degenerate - Hence 2J is separable and in
particular semi-simple (for details about Jordan
algebras see [ 2 ])-
We define a pairing □ : VxV -* End V by setting
(3-1) anb = 2L(ab) + 2[L(a),L(b)] where a,b-.V.
Then the trace form of this pairing is given by
(3-2) j(a,b) = 4 trace L(ab)
and hence it is non degenerate. Moreover j turns out
to be an associative bilinear form of the algebra 3'.
The adjoint of T c End V with respect to r is denoted
by T" • In particular we have L"(a) = L(a).
Ill, §3 81
It is known and easy to prove (see [ 8 ] and [ 10] )
that the pairing (3- 1) satisfies besides (P. 1) also
the conditions (P. 2) to (P. 4) of II, §3.1. We call such
a pairing a Jordan pairing of the first kind - The
examples given in § 1 are Jordan pairings of the first
kind provided e=lore=-l and r>3 is even.
2
From (anb)a = 4a(ab)-2a b we conclude that the
endomorphism P(a) defined by (II; 3- 2) coincides with
the quadratic representation of the Jordan algebra 91.
Hence T=r coincides with the structure group F(2J)
of 91 because of II, Lemma 3-4.
The results of II, §3 show that any Jordan pairing
of the first kind leads to a binary Lie algebra £l. = O
such that (D , 3) is a symmetric Lie algebra. From
the definition of ® in §3.5 we observe
H@(x) = -P(x)
(see I, §2. 2). Thus det H @ (x) ± because of P(e) = I
and hence ® is essential.
2. Since % is essential, there exists a birational
2
function i in 3 (£L.) such that 3 = V .. Here 3 = Id
9J -J
implies joj = I and because of I, §2. 3, the function j
is given by
j(x) = -[H3(x)] _1 (8l)(x) = -[P(x)] _1 x = -x" 1 ,
where x stands for the inverse of x in some scalar
extension of the Jordan algebra "I.
82 III, §3
We are able to express the birational functions
tat, by J an d the translations t, where beV.
LEMMA 3-1 - Suppose that □ : VxV -* End V is a
Jordan pairing of the first kind . Then for beV one has
t @b = J°t_ b oj and B(x,b) = P(x)P(x +b).
Proof : We know from II, Lemma 1.1, that ¥ @ , = ®Y,®
holds. Moreover the birational functions belonging to
¥, or L, are t_, or t_au , respectively. Hence we get
t_Q, = (~j )° t_, ° (-j ) and this proves the first formula.
The second formula now follows from part a) of II,
Theorem 2.3, together with — r — = -[P(x)]
As a conseuqence we see that the group H(£\ ) is
generated by the functions W, t and j where W e F(D )
3. JJ
= r(<U) and aeV. In particular, H(D ) coincides with
the group S(?I) considered in [11]. For more results
see also H. Braun [ 1 ] and [12].
3- Let □ : VXV ■* End V be an arbitrary pairing
satisfying the conditions (P. 1) to (P. 4) of II, §3.1.
Denote by (O n ,3) the induced symmetric Lie algebra.
Let dcV and denote by 21, the algebra defined on the
vector space V by the product (a,b) -> P(a,b)d. We
know from II, Theorem 3-3, that 21 is a Jordan algebra .
Denote by L d and P d the left multiplication and the
quadratic representation of >v , respectively. Thus
Ill, §3 83
L d (a) = ^and.
We are going to prove
(3.3) P d (a) = P(a)P(d).
Indeed, we apply (P. 3) for T = cdb on c and obtain
(anb)(cQb)c - (cOb)(aOb)c = [(cnb)anb]c - [cn(bna)b]c.
Hence it follows that
I^O^CcOc) - I^(c) = l£(c) - P(c)P(b).
Since the square of c in % equals L, (c)c we get
P b (c) = P(c)P(b) and (3-3) is proved.
4. Two pairings □ and □' of V are said to be
isomorphic if the associated binary Lie algebras
and O are isomorphic under an isomorphism $ : Z -* z'
that satisfies $1 = I. According to I, Theorem 3-2,
the two pairings are isomorphic if and only if there
exists a W e GL(V) such that Z' = v TT 0.
W
THEOREM 3-2 . Let a be a pairing of V satisfying
the conditions (P. 1) to (P. 4) of §3-1 and let (Z, 3)
be the associated symmetric Lie algebra . Then the
following statements are equivalent :
a) I. is essential .
b) det P(x) + 0.
c) There exists deV such that 31 ■, has a unit
element.
84 III, §3
d) The pairing Q is isomorphic to a Jordan pairing
of the first kind .
Proof : From the definition of ® it follows that
H @ (x) = -P(x)
holds. Hence a) and b) are equivalent (see I, §2. 2).
But b) implies the existence of deV such that det P(d)
^ 0. Hence det P d (d) + from (3-3). Hence the equi-
valence of b) and c) follows from [ 2 ], chapter IV,
Theorem 2.7.
It suffices to show that c) implies d) . Choose
deV such that 91, has a unit element c. Consider the
d
binary Lie algebra £}' = v„0 where W = P(c). The
algebra O can be considered as a binary Lie algebra
defined by a pairing □' of V such that the endomorphism
P' is given by
P'(x) = WP(W _1 x)
and © ' = v turns out to be the corresponding
automorphism of O'. We obtain I = P,(c) = P(c)P(d)
and using (II;3-10) we observe
P'(c) = WP(W _1 c) = P(c)[P(c)] _1 = I.
Hence we may assume that there is ceV such that P(c) = I.
Using (3.3) we end up with P(a) = P (a). Using [ 2 ],
chapter IV, Theorem 2.5, we know that the square e of
c in 91 is the unit element of V Q and
Ill, §3 85
P(e) = P (c 2 ) = [P (c)] 2 = I. We write 9J = 91 and
obtain a Jordan algebra 91 with unit element such that
P is the quadratic representation of 91. Thus the pair-
ing a is given by (3. 1) and the trace form of 9J is
non degenerate because of (3-2).
5. Next we are going to define the Jordan
pairings of the second kind. We start again with a
Jordan algebra 21 in V with unit element and an auto-
morphism a ^ a' of II of period 2. Suppose again that
the trace form of 21 is non degenerate.
The automorphism a -* a ' of 91 induces a direct
sum decomposition
91 = 21, + 9J_ where 21 = [a ;ae9J, a ' = ±a }.
Here 91, is a subalgebra of 91, 21 _ f and one has
9I + ?I_ c 9J_, 9I_9J_ C a
Moreover we conclude a (a', b') = c(a,b) for a,b e 91.
Hence %, turns out to be orthogonal to 9J_ with respect
to a and the restrictions of a to 21, and 2l_ are
non degenerate.
The bilinear form of 21, given by u ■> a(e,u) is
normal . Hence 21, is non degenerate and consequently
semi-simple (see [ 2 ], chapter I, §8 and §9).
We define a pairing □' of 91_ by
(3.4) (an'b)c = 2 (ab)c + 2a(bc) - 2b(ac) where a,b,ce2I
86 III, §3
Clearly ao'b e End 9I_ and ao'b is the restriction of
aDb given by (3.1) to 9J_. Let Z' be the vector space
spanned by an'b where a,b e 3I_.
If A is a linear transformation of 9J then we
denote by A its restriction to 91,. Hence the trace
form a' of the pairing □' is given by
(3.5) rr'(a,b) = trace(aa'b +bo'a) = 4 trace L_(ab)
where a^b e 91_ .
PROPOSITION . There exists an element d in the
center of 91 , such that a ' (a ., b ) = a(da,b) where a,b e 91 _ .
Proof : Let u,v e 91, and set \(u,v) = 4 trace L_(uv)
Then X is a symmetric bilinear form of 91,. Using the
basic identities about Jordan algebras one observes
that X is associative. But the restriction of a to
91, is associative and non degenerate. Hence there
exists d in the center of 91, such that X(u,v) = j(du,v)
because of Theorem 6.4 in [ 2 ], chapter I. Now we
obtain from (3.5)
o'(a,b) = X(e^ab) = a(d,ab) = a(da,b)
where a^b e fl J.
LEMMA 3-3 - Suppose that the trace form of 9! and
the bilinear form a ' are non degenerate . Then the
pairing □' given by (3-4) satisfies the conditions (P. 1)
to (P. 4) of II, §3.1.
Ill, §4 87
Proof : Clearly, we have to prove only (P. 3)
and (P. 4). We write the condition (P. 3) for the pair-
ing (3- 1) where T = uov, T ■ vQu. Choosing all
elements in 21 _ we obtain
[ud'vj an'b] = T'ao'b - aa 'Sb where T'=un'v, S=vn' U}
and in a similar way
[ba'a, vd'u] = bn'T'a - Sbn'a.
Taking the trace we get ?'(T'a,b) = a'(a,Sb). Hence
S equals the adjoint of T' with respect to a' and
consequently (P. 3) and (P. 4) hold.
We call D a Jordan pairing of the second kind .
The examples given in § 1 for e = -1 and in §2 for
r>s are Jordan pairings of the second kind.
§4. The two exceptional cases .
I. Let E be a Cayley algebra over K and suppose
that the characteristic of K is different from 2 and
3- Thus E is alternative and there exists a non
degenerate bilinear form u and a linear form X of S
such that
2
a = 2\(a)a - u(a,a)e for a e E,
where e is the unit element of S and X(a) = u(a,e),
X(e) = lj holds (for details see [2 ], chapter VII, §4)
The map a -' a' = 2\(a)e-a defines an involution of E
88 III, §4
and one has
a(b'c) + b(a'c) = 2u(a,b)e.
The dimension of E over K equals 8.
Denote by S~(E) the vector space of 3 by 3 matrices
1 a 3 a 2
a a a, , a. e K, a. e E.
3 2 1 i l
a 2 a l a 3
If ab means the usual matrix product, &a(E) becomes a
central simple Jordan algebra over K with respect to
the product ao b = i;(ab+ba), a so-called exceptional
algebra (for d.etails see [2 ], chapter VII, §6).
Associated with the Jordan algebra &o(E) we
obtain a Jordan pairing of the first kind
□ : »„(E)xSo(E) -• End §o(E). The binary Lie algebra
£ = £ is an exceptional Lie algebra of type E (see
[8 ])•
2. Let e, be the (absolute primitive) idempotent
that is given by a, =1 and zero elsewhere. Using
at = [a;aa^(S), e,:a = va }
for v = 0.%,1 we obtain the Peirce decomposition
3 K o k 1
Ill, §4
89
which is a direct sum of vector spaces. The map a -> a*
which changes the sign of the component in 2Ii is an
automorphism of the Jordan algebra 31 and the eigenspaces
are
31 , = 31 + 31, , 31 =31!
+ o 1 - %
Set V = £_©£_ and write the elements of V as a = a,©a ? .
We define a pairing of V by
[(anb)c] 1 = [2\s(a l ,b l )+\J.(a 2 ,b 2 )]c l + [2\i(c l ,b 1 )+\s(c 2 ,b 2 )]a 1
- 2u(a 1 ,c 1 )b 1 +%[c2(b 2 a 1 )+a2(b 2 c 1 )-b2(a 2 c 1 +c 2 a 1 )] ,
[(aDb)c] 2 = [u(a 1J b 1 )+2n(a 2 ,b 2 )]c 2 +[u(c 1 ,b- L )+2n(c 2 ,b 2 )]a 2
- 2u(a 2 ,c 2 )b 2 +%[ (a 2 b 1 )c^+(c 2 b 1 )a ; [-(a 2 c 1 +c 2 a 1 )b-[].
Using the injection
a
cp : V -> © 3 (S), cp(a 1 ©a 2 )
1 a 2
we see that cp (V) equals 31 = 9J,. Furthermore a verifi-
cation shows that
%p((aob)c) = [cp(a)cp(b)]cp(c) + cp (a) [cp(b)cp(c) ]
- cp(b)[cp(a)cp(c)]
90 III, §4
holds. Hence the image of the pairing of V under cp
coincides with the pairing of 2J_ given in (3-5).
An observation shows that the trace form of n
is given by
a(a,b) - 48[n(a 1 ,b 1 )-Hi(a 2 ,b 2 )].
Hence a is non degenerate. According to Lemma 3. 3
we obtain a pairing □ : VxV -> End V of the second
kind. Denote by O = O the associated binary Lie
algebra. One can show (see K. Meyberg [13]) that O is
a Lie algebra of type E fi .
IV, §1 91
Chapter IV
APPLICATIONS TO BOUNDED SYMMETRIC DOMAINS
§ 1. Some elementary results on real
linear algebraic groups -
1. For an arbitrary finite dimensional vector
space V over P we denote by V = C 5? V its complexifi-
° F F °
cation and by V the space V considered as vector
ID
space over R. Note that V and V are the same sets.
R
The elements of V (and of V ) are written as u = a + ib
R
where a.b £ V • The vector spaces V , V and V are
o r o
equipped with the natural topologies. Let D ^ be an
open subset of V and let f : D -*■ V ' (V ' being a vector
space over C) be a map. Then f is called holomorphic
if in the representation
m
f (z,b..+ • • • + z b ) = 7 f, (z, , • • • ,7. )b/
x 1 1 n n L k x 1 n 7 k
k=l
(bi , • • • , b and b-,' , . . . ,b ' being a basis of V and V ' )
K 1 n 1 m &
the functions f, are holomorphic in the complex
variables z-, , . . . ,z ■
1 n
Note that the multiplication by i and the conjuga-
"D
tion u -» u = a - ib belong to End V . For A.B £ End V o
the endomorphism A + iB of V is given by
(A+iB)(a+ib) := (Aa-Bb) + i(Ba+Ab). Conversely for any
92 IV, §1
W e End V there exists A,B e End V Q such that
W = A + iB. For W e End V denote by W* the induced
1R
endomorphism of V . Moreover, if 8 is a subset of
End V we denote S R = [W ; WeS}. Clearly W - W defines
p
a monomorphism of the ring End V into End V". For
W e End V we define W by Wu = Wu where u e V . Clearly
?- w R .
P
An arbitrary endomorphism T of V can be written
as T(a+ib) = (Aa+Bb) + i(Ca-H)b) where A,B,C,D e End V .
We obtain
(1.1) trace T = trace A + trace D.
IP
The endomorphism T of V is C-linear if and only if
D = A and C = -B, i.e., if T = W" for some W e End V.
Furthermore T- commutes with the conjugation u -> u
if and only if B = C = 0. Note that in both cases
the conditions are linear equations over P. From (1.1)
we observe
(1.2) trace (A+iB) F = 2 trace A, where A,B € End V .
ID
For a subset S of End V denote by Sj_ the set of
elements of S that are C-linear. Hence there is a
subset U of End V such that S^, = Jt R .
p
For an endomorphism T of V , V or V we define
o
the exponential by
CO
exp T := Y \ T m .
m.
IV, §1 93
Hence we obtain a map exp of the endomorphism space
into the corresponding general linear group. For
T e End V we have exp(T ) = (exp T) '.. Note that the
exponential map is bijective in a neighborhood of zero.
2. Let 3 be a hermitian positive definite form
of V. The adjoint of T e End V with respect to 3 is
denoted by T . An element T of End V is called unitary
(with respect to 0) if T T = I and hermitian (with
respect to 0) if T = T. Moreover we call T positive
definite (with respect to p) and we write T > if
T = T and P(Tu,u) > for =f= ueV. For a subset S of
End V we write S P = IT 3 ; TeS}.
It is well known that the exponential map maps
the hermitian endomorphisms of V bijectively onto the
positive definite endomorphisms of V.
ID
3- A subgroup Q of GL(V ) is called a real linear
algebraic group if there exists a non-empty set p of
polynomials in an endomorphism variable of V" having
real coefficients such that W e GL(V" ) belongs to Q.
if and only if rr(W) = for all rr e p. Note that any
P.
real linear algebraic group is closed in GL(V ), hence
it is a real Lie group. The subgroup Q v is again a
real linear algebraic group and there is a subgroup H
of GL(V) such that Q Vc = U . A subgroup H of GL(V) is
called a real linear algebraic group if 9 has the
corresponding property.
94 IV, §1
Let Q be a closed subgroup of GL(V"). Because
of the following lemma we call
(1.3) Lie Q := [T; T e End V P \ exp ?T e Q for § e R]
the Lie algebra of Q.
LEMMA 1.1 . If Q is a closed subgroup of GL(V')
P
then Lie Q is a Lie algebra of endomorphisms of V J .
Proof : We use the formulas
(exp ?T)(exp ?S) = expU(T+S) + 0{% 2 )},
(exp §T) _1 (exp cS) -1 (exp IT) (exp 5 S) = exp{§ 2 [T,S] +0(: 3 ) }
For a given % e F and positive integer m we replace
5 by ?m and raise the first formula to the power m
2
and the second to the power m . Then the limit m -> =°
yields T + S e Lie Q and [T,S] e Lie Q.
Remark . Suppose that H is a closed subgroup of
F
GL(V) and let T e Lie U . Hence exp ?T and consequently
T itself is C-linear. We obtain Lie W P = (Lie H) R
where
(1.4) Lie M := [S ; S £ End V, exp SS e tt for | e r).
Note that Lie M can be considered as a Lie algebra
over C.
4. Let P be a hermitian positive definite form
of V. For a closed subgroup H of GL(V) the condition
IV, §1 95
8 R
U = H implies (Lie W) = Lie H. The group of unitary-
elements of GL(V) clearly is a real linear algebraic
group ; its Lie algebra consists of the T in End V
o
such that T = -T. Moreover, this group is compact.
LEMMA 1.2 . Let 3 be a hermitian positive definite
form of V and let & be a real linear algebraic subgroup
of GL(V) satisfying ti® = U. Then
a) the unitary elements of it form a maximal compact
subgroup K o_f H that is again a real linear
algebraic group .
b) Each element of W can be uniquely written as
UP where U e K, P = exp T > 0, T 3 = T e Lie U.
c) If_ W € Wj W > 0, then there exists a uniquely
\, \, i- 2
determined W 2 e U such that W 2 > and (W 2 ) = W
Proof : Each W e GL(V) can be uniquely written as
W = UP where U is unitary and P > 0. Hence P = exp T
where T^ = T and we obtain WW = P = exp 2T- From
3 2
H - H it follows that P belongs to U.
Consider the curve W(?) := exp 2§T, § e R, in
GL(V). For any polynomial rr we obtain a finite sum,
■<W(S)) = Y a r
e
m
by using the "minimal decomposition" of the (semi-simple)
endomorphism T. Now let p be a set of polynomials that
defines the real linear algebraic group U. For integer
96 iv, a
k we have W(k) = P 2k c U and hence -(W(k)) = for
rr € to and all k. Hence a = " f or all ra and we obtain
r m
W(§) e U. This means T e Lie U and U e ». So part
b) is proved.
2
In order to prove part c) we write W = P where
P > and W = exp T, T e Lie ti, according to part b).
Since the positive definite square root is unique we
get P = exp \T e U.
For part a) let K ' be a compact subgroup of U
such that K c k'. Let W 6 k'; hence part b) implies
W = UP where U e K and P > 0, P e Jt. The proof of
part a) will be complete if we show that P e U 1 , P > 0,
implies P = I.
■K ' being a compact subgroup of H means that K'
is compact irt GL(V). By a known result there exists
a hermitian positive definite form y of V such that
Y(Wu,Wu) = y(u,u) for u e V and W e K ' . Writing
Y(u,v) = 3(Bu,v) we obtain B > and W^BW = B. Choose
2
C > such that B = C and put W = P, D = CPC Hence
2 2
D = B and therefore D = B which means P = I. So the
lemma is proved.
5. Again let M be a real linear algebraic subgroup
of GL(V) and suppose that 3 is a hermitian positive
definite form of V such that A p = &. Hence (Lie H) D
= Lie H and T i: -T is an automorphism of the real
Lie algebra Lie M. We set
IV; §1 97
a = [A; A e Lie U, A = -A},
b = (B; B e Lie Jt, B = B}
and we obtain a direct sum decomposition
Lie M = a + b, [a, a] c q, [o,b] c b, [b,b] c a.
Again let K be the maximal ; o t -.nct subgroup of M
consisting of the unitary elements of tt. Hence we
obtain
W = X-exp b , Lie K = o.
Finally we prove
LEMMA 1.3 - The restriction of the Killing form
of Lie M _to a o_r b is negative semi-definite or positive
semi-definite, respectively .
Proof : For S,T e Lie H we put y(S,T) := trace ST •
From trace T = trace T it follows that y is hermitian
positive definite on the vector space Lie H (considered
as a vector space over C) . A verification yields
v([ad T]S L .S ) = Y(S r [ad T ]S 2 )- Hence for the
v R
adjoint with respect to y we observe (ad T) = ad T
and
vT,T 3 ', Lie u = trace (ad T) (ad T) Y > 0,
which completes the proof.
6. Let W be in End V . Then the extension of W
- o
98 IV, §2
to V is an endomorphism of V and we will identify End V
with the sub-ring [W; W e End V, W = W} of End V. Hence
for subgroups Q of GL(V ) we have the notion of a real
linear algebraic group. Suppose that a is a symmetric
positive definite bilinear form of V ■ Denote the
r o
extension of it to V also by a. Then (u,v) -> a.(u,,v)
defines a hermitian positive definite form of V-
Clearly our results are valid for a real linear
algebraic subgroup Q of GL(V ) and the endomorphisms
of Lie Q as well as the decompositions in Lemma 1.2
can be chosen as endomorphisms of V •
r o
§2. The group T(Q) .
1. Let V be a finite dimensional vector space
over R. We suppose that □ : V XV -* End V is a
" o o o
pairing satisfying the conditions (P. 1) to (P-4) of
IIj§3.1j such that its trace form a is positive
= Q C
definite .
Denote by X the Lie algebra spanned by aob where
a,b t V . The adjoint of T e End V with respect to
o J o r
is denoted by T*. Furthermore let P (a) be the
o J o
endomorphism induced by a according to (II;3.2).
Using the identification of "generic elements" of
V with "vector variables" we consider the Lie algebra
Rat V q of rational functions in the real variable x of
V • Denote the binary Lie algebra associated with the
IV, §2 99
pairing by O , i.e..
O = V +1 + V where V = IP (x)b;beV }.
o o o o o o o
2. Now let V be the complexif ication of V • By
linearity the pairing □ of V extends to a pairing
D : vxv ■* End V having the trace form c- A verification
shows
(2.1) -(a+ib,c+id) = 7 o (a,c)-a o (b,d)+i[c o (a,d)+a o (b,c)],
hence a coincides with the extension of 3 to V. In
o
particular, a is a non degenerate bilinear form of V,
and (u,v) -»■ a(u,v) defines a hermitian positive definite
form of V.
Clearly the vector space (over C) spanned by uDv,
where u,v 6 V, coincides with the complexif ication X of
I . Again the adjoint of T e End V with respect to a
is denoted by T* • Hence we have (A+iB)" = A* + iB*
where A,B e End V • By linearity the pairing
□ : VxV ■> End V satisfies the conditions (P. 1) to_
(P. 4), too. Moreover, let P(u), ueV, be the endomor-
phism of V induced by the pairing. Then for a*-. V
the restriction of P(a) to V equals P (a) and we have
o M o
P(a+ib) = P(a) - P(b) + 2iP(a,b), a,b e V ■
In particular P(u) = P(u) for u f V.
We choose a complex variable z of V and consider
the Lie algebra Rat V of rational functions in z. By
100 IV /§ : 2
construction all functions in Rat V are holomorphic in
z. Denote the binary Lie algebra associated with the
pairing □ of V by £L, i.e.;
a. = V + X + V where V = [P(z)u;ueV}.
Note that £L is not the complexif ication of s but £^
can be considered as the holomorphization of O in the
r o
following, sense: Let 2 be any subspace of Rat V •
& o o
Then the holomorphization 2.,. of 2 is obtained by the
c « o
complexif ication 2 of 2 and by replacing the real
variable x by z.
3. There is a third pairing induced by the
1R
original one. Let V be the space V considered as
vector space over P. Then the pairing o of V can be
IP i
considered as a pairing of V having the trace form r
Using (1-2) we obtain
(2.2) a (a+ib..c+id) - 2[o (a.c) - o (b..d)]
= 2 Re -(a+ib,c+id).
R ID
Again o is non degenerate and (u,v) -* a' (UjV) defines
a symmetric positive definite bilinear form of V . The
n
vector space (over R) spanned by uov where u.v e V
F F
coincides with X . Denote the adjoint of T e End V"
R 1T
with respect to a by T . For T = A+iB e End V where
A,B ' End V we have the induced endomorphism T of V'
(see §1.1) and it follows that (T" ) = (T*) . In the
IV, §2 101
F
notation we will not distinguish between T and T" if
there is no possibility of misunderstanding.
1R 1R 1R
Hence by linearity the pairing □ : V xV -* End V
satisfies the conditions (P. 1) to (P. 4). The induced
endomorphism P (u) coincides with P(u) considered as an
R 1R
endomorphism of V . Note that the endomorphisms of Z
are C-linear.
Let x,y be real variables of V • We consider the
J o
1R
Lie algebra Rat V of rational functions in x and y.
Setting z = x + iy we get a complex variable of V and
IP IP
(Rat V) is a subspace of Rat V . The Cauchy-Riemann
differential equations show that it is in fact a sub-
algebra.
Denote the binary Lie algebra associated with the
IP
pairing o of V by Q, i.e.,
0= V R + £ R + V F ' = (q v ) R .
Note that O is a real Lie algebra, but the product is
C-linear, so O can be considered as a complex Lie
algebra.
4. Next we consider the groups T associated with
the pairings. According to II, Lemma 3-4, we have first
r(0 ) = [W; W e GL(V ), P(Wx) = W P(x) W* }
and second
:(C.J = [W; W t GL(V), P(Wz) = W P(z) w*}
102 IV, §2
We observe that T(0 ) is the subgroup of T(%) consist-
ing of the W's such that W = W. Finally we get
r(D) = [W; W £ GL(V F ), P(Wz) = W P(z) W f }.
R IR
Using the injection W ■* W of GL(V) into GL(V' ) we
obtain as the image of r(£^) the subgroup
r*(0) = [W R ; W 6 GL(V), P(Wz) = W P(z) W*}
of T(O) of the C-linear elements (see §1.1). Hence
IP
[r(Qj,)] = r >If (D). Using the identification mentioned
above we also write r(Q.,,) = T. (O). Furthermore from
P(u) = P(u) it follows that the conjugation J given by
Jz := z
belongs to r(Q) and one has J = J. For a C-linear
endomorphism W the endomorphism W = JWJ is C-linear,
too. Hence W and W* belong to r^(£i) whenever W e r,_(0).
LEMMA 2.1 . The Lie algebra of 1^(0) coincides with
X R .
Proof: An element S e End V belongs to 2 if
and only if there is a T e X such that S = T J . Accord-
ing to IIj Lemma 4-5, it suffices to prove that
(2.3) 2 P(Tz,z) = T P(z) + P(z) T*
for T e End V is equivalent to exp ? T t IXQ^) for
5eF, i.e., to T 6 Lie r(£k). We put
IV, §2 103
W - W(§) = exp :T,
Q = Q(§) = P(Wz) - W P(z) W*
and denote the derivative with respect to I by '.
One gets W' = WT = TW and
Q' = 2P(w'z,Wz) - W' P(z) W* - W P(z) W'-.
If T e Lie "(O*) then Q(§) = and hence Q'(0) = 0,
so (2.3) holds. Conversely suppose (2.3)- We have Q*
= Q and hence Q '* = Q'. But
Q' = 2P(TWz,Wz) - TWP(z)W* - WP(z)W*T* = TQ - Q*T*
yields Q '* = -Q ' and hence Q' = 0. One observes Q(?)
= Q(0) = and consequently W e Y (%) > so T e Lie P^).
TD ID
5. Clearly T* e X ' whenever T e 2 . Hence T - v -T*
jR
becomes an automorphism of the Lie algebra X and we
have the induced direct sum decomposition as vector
spaces over P.:
(2.4) x r = b + i . d = [D; D 2~\ D* = -D],
I = [L; LpX F , L* = L).
Using II, Lemma 4.1, we see that b and I are orthogonal
with respect to the Killing form ', , ) of I . We put
K = i.W; W f r *(£) . W*W = 1}
and we see that K is a real linear algebraic subgroup
104 . IV, §3
of GL(V R ).
LEMMA 2.2 . The group K is maximal compact in r^(£)
having b as a Lie algebra . Any element of ^(O) has a
unique representation as U- exp L where U e K and Lei.
Moreover one has
(2.5) <D,D> < 0, <L,L) > for Deb and Lei .
Proof : Defining g by p(Ujv) = o(u,v) we obtain a
hermitian positive definite form of V. For W e GL(V)
the adjoint W with respect to P equals W*. Let
U = {W; W e r(Q fe ) J W*W = I). Then K = » R and Ji 3 = U.
So we are able to apply Lemma 1.2 and Lemma 1.3 from
which the statements follow.
§3. The group Aut(0, ® ).
1. We apply now our results of chapter I and II
to the binary Lie algebra
O = V R + £ R + V P c (Rat V) R
(see §2.3). According to II, §3. 5, there is an auto-
morphism 6 of O of period two given by
(3.1) (Sq)(z) :- - b - T*z - P(z)a
where q(z) = a + Tz + P(z)b e O,
and (O, ®) is a symmetric Lie algebra in the sense of
II, §1.1. Again we write the elements of Das
IV, §3 105
q = a + T + ®b where a,b e V and Tel. In particular
we have
@ T = _ t* anc i [ a ,Hb] = aab.
Note that ® is C- linear. From II, §3- 5, we know that
the involution W -*• W of T(O) induced by 9 (see II,
f
Lemma 1.2) is given by W . Hence for W 6 F., (O) we
obtain W* = W* e r *(°)- As in II^U-3, we set
B(a,b) = B^(a,b) = Bg b (a)
and we obtain from (II;3-8)
(3.2) B(a,b) = I + aab + P(a)P(b).
Hence from II, §1.3., we observe
(3-3) t @b (z) = [B(z,b)] _1 (z+P(z)b) where b e V.
From II, §1.3, it follows that
Wot., = t °W where c - W* _1 b, b e V, W e F,,(0).
9 b ;c *
Using (3.2) and [P(a)]* = P(a) we get [B(a,b)]* = B(b,a)
and II, Lemma 1.4, yields
(3.4) B(a.b) e r *(£) whenever det B(a,b) £ 0.
the function t , is holomorphic in its domain of defini-
tion because it is rational in z. Clearly the same is
true for t , a V. From (I;4.5) we see that the ele-
ments of ( C) are exactly the functions
106 IV, §3
(3-5) W°t °t s ,ot where W e F(O) and a } b,c e V.
R
Let § be a subset of Rat V • Then in comformity with
the previous notations let &^ be the set of functions
in S which are holomorphic in its domain of definition.
Henc e the subgroup S . ( O) of_ 3 ( O) consists of the
elements (3-5) where now W e T^CO). Applying (I;4.5)
to £L instead of O we observe
3*(0) = H(^).
LEMMA 3.1 . Suppose § = v f for some f e H(£).
Then f belongs to -^(O) if and only if § i_s C- linear.
Proof : If f e a^(O) then V f e Aut £L and $ is
C-linear. Conversely, let $ be C-linear. We write
f = g , g = Wo t ot.,ot where We r(£f) and a,b,c e V.
b & ■ a @b c v
Hence §q = q g for q e O and from (I; 1.4) it follows
that the inverse of the Jacobian of g, B(z+c,b) W ,
is C-linear. Therefore W is C-linear and g e Hj,(£>).
2. We apply now II, Lemma 4.2, to O. Since the
ID
trace form is given by a , the Killing form of O is
ID ro
given by (o ) „. But a' is non degenerate and hence
(a )» is non degenerate, too. Hence by the criterion
of Killing-Cartan, O is a semi-simple Lie algebra .
Furthermore, II, Lemma 4.2, yields
(q r q 2 > - (T V T 2 ) R + 2 trace T^
+ r F ( ai ,b 2 ) + cv R (a 2 ,b 1 )
IV, §3 107
where q, = a, + T, + @b, and where the trace is taken
over Z .
R
Next we use the decomposition X — b + 1 introduc-
ed in §2.5. For Deb, Lei we have, according to (1.2),
clearly trace DL = 2 trace Re DL where trace means
J o o
the trace over V . But Re DL = Re DL = - Re D* L*
o
together with trace Re D* L* = trace Re DL imply
trace DL = 0. Hence we obtain
(3.6) <qi>q 2 ^a = (D l' D 2 N t.+I +2 trace D i D 2
R
+ <L 1 ,L 2 > D+I + 2 trace L^L 2 + a (a^tO
+ n J (a^b^),
where T, = D. + L, .
k k k
R —
3- Using the conjugation J of V given by Jz = z
we introduce two more automorphisms 3, and ®_ of O by
(see II, §3.. 6). According to (II;3-10) we have the
explicit definition
(3.7) S q = ± b - T* ± 9a where q = a + T + 0b e a
Again (C 3.) and (C, 3 ) are symmetric Lie algebras and
one verifies that B, 8, and s commute by pairs. Further-
more one sets
108 IV, §3
Note that ® is C-linear but 6, is not •
o ±
The involutions of I*(0) induced by © + or 8_
coincide and are given by W ■* W* (see II, §3- 6).
LEMMA 3-2 . The symmetric bilinear form g of O
given by
is positive definite on O.
Proof : From the definition of the Killing form
follows <§q, , $q~> o = (q-i , qo > o for each automorphism i
of Q. Hence 3 is symmetric. Using (3.6) we obtain
8_(D+L) = D-L and from (3-7) it follows
<q; _q> o = < D ^ D > b +i " 2 trace °d*
B/ — \ B,, r>
<L,L> b+I - 2 trace LL* - a (a, a) - a (b,h),
where q = a + (D+L) + 0b e O. Here we have trace DD* >
and trace LL* > 0. Moreover, the definition (2.2)
yields a (a,a) > for aeV. Hence (2.5) implies
<q,®_q) < for qeC Since the Killing form of £k is
non degenerate, the same is true for the bilinear form
P and we end up with (3(q,q) > for £ qeO.
4. For Y e Aut O we set
Aut(D, Y) := U; $ e Aut O, $Y = Y§}.
Clearly Aut O and Aut(0,Y) are real linear algebraic
IV, §3 109
groups in GL(O). The Lie algebra of Aut O coincides
with the Lie algebra of all derivations of O . But O is
semi -simple and hence
Lie Aut O = [ad p; peO}.
Moreover, ad p e Lie Aut(0, Y) is equivalent to exp 5 ad p
= Y-exp z ad p-Y = exp ?[Y-ad p-Y ] and since exp
is bijective in a neighborhood of zero, we obtain the
equivalent condition Y-ad p = ad p-Y, i.e., Y[p,q]
= [Pj^q] for qeO. This means [Yp,q] = [p,q] for qeD
and since O is centerless (see II, §2.1) we obtain Yp = p.
Hence
(3.8) Lie Aut(0,Y) = tad p; peO, Yp = p}
5= [p; peC, Yp = p}.
The adjoint of $ £ Aut O with respect to g is
given by
(3.9) $ P = ®_ $~ 1 9_ •
Hence $ ■* i maps Aut O as well as Aut(0, 3.) and
Aut(0, 3 ) onto themselves (because 8, and 3_ commute).
Since the Killing form of a Lie algebra is associa-
tive we observe
(3.10) (ad p) 3 = - ad 3_p = - ©_-ad p-3_ •
We write $ > if $ = $ and if $ is positive definite
with respect to 3 (see §1-4). As an abbreviation, set
110 IV, § 3
hJ(O) := a°(o) n 3^(0)
where 3°(0) is defined as in "L, §4. 3- Now we are able
to prove
THEOREM 3 ; 3 • a) Aut(0,©_) is a maximal compact
subgroup of Aut O.
b) Each element in Aut O can be uniquely written
as Y$ where Y € Aut(£t, ® ) and where
$ = $ = exp ad p > 0, peO, 3_p = - p.
c) I_f § e Aut O and $ > then $ is essential and
C- linear . Furthermore there exists f e »j.(0) such that
$ = v f .
Proof : We apply Lemma 1.2 to it = Aut O. In
view of (3.9) the unitary elements of Aut O are exactly
the elements of Aut(0, @_), so we already proved part
a).
Moreover, we have a unique representation Y$ of
the elements of Aut C where Y e Aut (O, <3 ) and $
= exp ad p, peO, such that (ad p) = ad p. Hence
9_p = - p and part b) is proved.
Finally let $ e Aut D, $ > 0. Hence p($q,q) >
for ^ qeO. Choose q = aeV and put $a = b + T + ;c
where b,c e V and T e X R ; Clearly b = (§a) (0) = Aa
where A = H*(0) in the notation (I;2.2). We observe
@_a = - ©a and (3.6) yields
IV, §3 111
($a,a) = -(lajQ^a)^ = <$a,0a) o
R (b,a) = , P (Aa,a)
From Lemma 3.2 we conclude that det FL (0) f 0. In
particular, $ is essential. According to I, Theorem 2.1,
there exists an feH(O) such that §q=q . But § = exp ad p
is C-linear and hence fe3^(d) because of Lemma 3-1. From
(^ i )" 1 =H j(2 )
we obtain it f (0) = det H § (0) $ (see I,'}4.3) and
f e 3 (O). Using the result for $
end up with $ = V- where f € H.,. (O)
LEMMA 3-4 . Suppose $ = v f for some f e 3*(0)
p
Then $ = $ if and only if
f = t °W°t, 3 -, where ceV and W* = W e T... (O).
Proof : In view of I, Theorem 4.3, we write
f = t °W°t -j where c , d e V and W e T. (D). Hence
c id "
§ = Y V TT 3 Y , ©
c W d
because of II, Lemma 1.1, and I, Theorem 2.3b). Usinj;
(3-9) we see that i = $ is equivalent to $ i_$ = Q_,
hence to K = S §- '3 where §-,=¥, v T Y v TT
1 1 1 d -J c W
= ! . - V „,. As an equivalent condition we get
d-c - Jw
-1 ..,_ s „ r -1 j "I
y -JW r c-d " ' f d-c 7 -(JW) T " 's(d-c) -(JW) f
112 IV, §3
because of II, Lemma 1.1 and Lemma 1.2. In terms of
the rational functions belonging to it we have
(JW) -1 ot d _ r (jw) f = c @(d . F) .
But this is equivalent to d = c and JW = (JW) = W^J,
i.e. , W* = W.
THEOREM 3.5. Let I be in Aut C. Then the follow-
ing two conditions are equivalent :
a) § P = $ > 0.
b) There exists W e r^(£i) and ceV such that
$ = v f and f = t c °W°t - and W* = W > 0.
Proof : In view of Lemma 3-4 we know that § = $
is equivalent to $ = v f where f = t °W°t fl - and
W* = W e i;, f (0). We obtain
V.c = Y y TT Y -
f c W ©c
and
(Y )° = ® Y ® = ® v T Y v T ® = ® Y- © = Y~-
c' -c -J -c -J c 8c
because of (3-9). Hence
$ = V. = Y V TT (Y ) P
f c W v c y
and $ > means v IT > 0. From v TT > it follows
w w
p(v y a,a) = -{Wa,e_a> = - F (Wa,a) >
for ^ aeV. Hence W > 0. Conversely let W > 0. Then
IV, §4 113
W - U 2 where U* - U > and v y = v^v^ > 0.
(4. The groups Aut(Q, ©,) and (J.
1. Next we consider the group Aut(0, 6 1 ,) and its
subgroup
rn := Aut(Q, 3 + ) n Aut(D^_).
As in §2.5 we denote by K the group of unitary elements
of ^(O), i.e., K = (U; U e I*(D), U*U - I}- Moreover
let tn and X be the identity component of ft and K.
Hence Lie K = b.
o
THEOREM 4.1 . a) tn is a maximal compact subgroup
of Aut(D, 9.) and its Lie algebra is given by ad b.
Moreover , for De b we have
(exp ad D)q = v q where W = exp DeK and qeO,
The map v : K -> ft is an isomorphism of the groups .
b) Each element in Aut(C, 3.) can be uniquely
written as ¥$ where Y elTi and # e Aut(D, ©,) such that
(4.1) 5 = § = exp ad p > 0, p = a + 3a, aeV.
c ) Each element in the identity component of
Au t ( Z, 8 1 ) is essential , C - linear and it can be uniquely
written as v..$ where IKK and $ e Aut(D, »),) satisfying
(4.1).
114 IV, §4
Proof : We apply Lemma 1.2 to ti = Aut(0, 3.) and
we use the bilinear form 3 of Lemma 3-2. The unique
representation together with Theorem 3-3 shows that tU
is a maximal compact subgroup of Aut(0, ®.) and the
elements of Aut(0, ©,) have a representation Y$ where
YetTi and § = exp ad p, ©_p = -p. But ad p e Lie Aut(0, ® + )
yields © ,p = p because of (3-8). Hence p = a + ©a
where aeV. So part b) is proved.
The Lie algebra of fu consists of the elements of
the form ad p where © ,p = p. Hence p = Del). For
Deb and q = a+T + ®beOwe observe
CO
(exp ad D)q = Y ~r [D m a + (ad D) m T + ©(D^b)}
m=o
because of (ad D)(@b) = [D,®b] = ©[©D,b] = ©(Db). From
(ad D) m T - V (-l) k (£) D m_k T D k
k=o
it follows that
(exp ad D)q = Wa + WTW~ + ®(Wb) where W - exp DeK .
o
Using W*W = I we obtain W = W* and consequently
exp ad D = v . Hence part a) is proved because
exp ad D, Deb, generates the identity component
m of m.
o
According to part b) the identity component of
IV, §4 115
Aut(0, £,) consists of the elements v IT $ where UeK and
+ U o
$ satisfies (4.1). In particular v is essential and
C-linear. According to part c) of Theorem 3-3 the
same is true for §.
2. Let Q = Q a be the set of f e H(D) such that
v f is in the identity component of Aut(0, 0,). Accord-
ing to the parts c) of Theorem 3-3 and Theorem 4.1,
Q is a group of birational functions contained in
-jl(C) and Q is isomorphic to the identity component
of Aut(£t, 0,). Using Theorem 3-5 we see that the ele -
ments of Q are exactly the functions Uog, where UeK
and where
(4.2) g = t oWo t„- where W e T.(£), ceV
\ / & c 8c *
such that W* = W >
and v , = , v .
g + + g
Furthermore , the representation of the elements of Q
as U= g is unique .
We prove that in (4.2) the condition v 3, = 0, v
f — g + + g
can be replaced by
(4.3) g°(-D°g - -I.
Indeed, it suffices to show that for I = v , $ = $
g
(see Theorem 3.5) the condition v 0, = ©. v is
g + + g
equivalent to (4-3). But this follows from (3.9) and
116 IV, §4
©_ = & + v_ ] ..
3- We define the subalgebras + and C_ of by
D ± = [p; peQ, 9^ = p}.
We know from (3.8) that
(4.4) Lie Aut(O,0 ± ) = ± .
Using the isomorphism f -*■ V_ of Q onto the
identity component of Aut(0,3,) we may consider rj
as a Lie group. Then its Lie algebra will be isomorphic
to the Lie algebra of Aut(0, ®, ) and (4.4) yields
Lie Q - 0_
+•
Next we prove
THEOREM 4.2 . The complexif ications of + and D_
are isomorphic to Q, considered as complex Lie algebras
Proof: We write the complexif ication of O , e = ±.
2
as O + iD where the sum is direct and where i = -1.
e J e J
Define a map
cp : O — s- O + j O
by
cp(q) ■ 2-(q+@ € q) - J[J(q-3_q)] where qea
From iq = i® q we obtain cp(q) e + JO and :p(iq)
= jcp(q). Furthermore, cp is injective and P-linear.
IV. §5 117
For arbitrary q-i^q^ 6 Owe set q = q. + iq~ and we get
cp(q) = q, + jq^- Hence cp becomes a bijection. A
verification shows that cp is a homomorphism of the Lie
algebras.
Since O is semi-simple we obtain
COROLLARY 1 . Q is a (connected) semi-simple Lie
£roup_.
From part a) of Theorem 3-3 together with (4.4)
we get the
COROLLARY 2 . Aut(D, ®_) is a semi-simple compact
Lie group .
§5. The bounded symmetric domain Z .
1. We use now the results of II., hi, about the
symmetric Lie algebra (O,®.). In terms of B(a,b) the
endomorphism corresponding to S, is given by
B + (a,b) = B- b (a) = B(a,b) where a,b e V
because of £,b = £b.
The involution of T(O) induced by % is given by
the adjoint W of W with respect to j (see §2.4 and
13.1). Hence the involution r(O) induced by 0, is
— f , f
given by W . For W e f*(0) we have W = W*j where \
stands for the adjoint of W with respect to - .
118 IV, §5
Rewriting the definitions of II, §2, for ® + instead
of 3 we obtain
H0M + ) = [f; feH(o), v f + = 3 + v f },
r(o,«) + ) = [w; wer(o), w f w = i},
D(D, © + ) = [c; ceV, there exists W e T(0)
such that B(c,-c) = W W}.
Note that B has to be replaced by B,. Clearly
v f e Aut(a, ®.) for f e H(D, ® + ) and the subgroup of
r(0 J ©,) of the C-linear elements equals K (see §2.5).
We know from II, §2.1, that K maps D(Q,®,) onto itself.
From (3-2) we see that B(a,-a) is hermitian with
respect to the hermitian positive definite form of V
that is given by (u,v) -> ?(u,v). Again we write A >
if the endomorphism A of V is hermitian positive
definite.
PROPOSITION 1 . D(B,® + ) equals [c; ceV, B(c,-c) > 0},
being an open subset of V, and the condition (A) o_f II,
§2. 5 j is satisfied . Moreover , to c e D(0, ©,) there
_ 2
exists a unique B > such that B(c,-c) = (B ) , B e 1^(0)
In particular, II, Theorem 2.1, can be applied.
Proof : Let c e D(0,® + ). Hence B(c,-c) = WW for
some W e r(O). But B(c,-c) is C-linear and therefore
(2.2) yields
IV, §5 119
2a(B(c,-c)u,u) = cj F (W f Wu,u) = ^ F (Wu,Wu) >
of + ueV. Hence B(c,-c) > 0.
Conversely let B(c,-c) > for some ceV. Hence
B(c,-c) e r^(O) because of (3-4) and part c) of Lemma
x.
1.2 shows that B := [B(c,-c)] 2 belongs to r*(D). In
_ 2
particular B(c,-c) = (B ) and c € D(D,@,). Hence
D(0, ©.) is open in the natural topology of V and the
condition (A) is fulfilled.
2- Denote by Z = Z the connected component of
D(0, ©,) that contains zero. Hence Z equals the
connected component of [z; det 3(z,-z) ^ 0} that
contains zero. In particular,, Z is open in the natural
topology of V- Clearly, z ■* z as well as z -* Uz,
U e K , maps Z onto itself,
o r
We define
(5.1) g c := t c °B c °t g - for c e D(0,© + ).
Clearly g belongs to 3^(D).
Let Q be the group of birational functions as
defined in §4.2. Let D be a non empty open subset of
V. A mapping f : D -> D is called biholomorphic if f
is bijective and if f as well as the inverse mapping
f is holomorphic in D. The domain D is called
symmetric if
(i) the group of biholomorphic mappings of D
onto itself acts transitively on D,
120 IV, §5
(ii) there exists deD and a biholomorphic map f
of D such that d is an isolated fixed point
of f and f ° f = I.
THEOREM A .
a) Z is a bounded symmetric domain in V.
b) The elements of Q are exactly the birational
functions f = U°g where U e X and c e Z. Moreover,
a c
this representation of f is unique .
c) Each feQ is holomorphic in Z and Q acts on
Z via QxZ ■* Z, (f,z) ->■ f(z), as a transitive group of
biholomorphic mappings .
d) The isotropic subgroup of Q with respect to
zero equals X , i.e. , f (0) = for feQ is equivalent
to f = U e X •
— o
3- The proof is divided into several propositions.
If X is a topological space then we write cp~f for cp, \|i e X
provided there is a continuous curve in X connecting
cp and \Ju
PROPOSITION 2. Let c be in Z. Then
a) g c e Q and g^ = g_ c ,
b) B c = c - P(c)c.
c v
Proof : As in II, §2, we define (now in a canonical
way)
IV, §5 121
c = B ot -r(c) , s = t~°B °t -. c e D(D, ©,).
c -3c c c c Wc +
Note s (0) = c. Let c be in Z. Then V-, f=s r commutes
with ®, because of II., Theorem 2.1 (notice, that
B has to be replaced by B , ) . But c ~0 implies B ~I,
c~0 and s ~I. Hence s eG.
c c *
In part b) of II, Theorem 2.1, we choose f = s ,
x = y = and obtain
B(cVc) = (B c ) 2 = B(c,-c~).
Hence ceZ and B~ = B ,
c c
We define f = s ~os and we obtain an element of G.
-c c ^
From s (0) = c and s,(-b) = follows f(0) = 0. Using
the chain rule and (I;4-6) we see that the Jacobian of
f at the point equals I. Hence uu f (0) f and the
last statement in II, Theorem 2.1, yields f e T(0, ®, )
and consequently f — I. So we proved s = s_~.
Since s c*} we may apply (4.2). Hence there exists
deV, UeK and W e T, (O) such that
o
s = Uot,oWot - where W* = W > 0.
c d 6d
The uniqueness result of I, Theorem 4.3, yields d = c
and thus we have t~oB = U°t,°W = t TTJ UW- It follows
c c d Ud
c = Ud and B = UW. Here U is hermitian and B as well
c c
as W is positive definite. Hence the uniqueness of
Lemma 1.2 yields U = I and W = B . So c = c and
J c
s = g •
c c
122 IV, §5
PROPOSITION 3 - a) Each element f in Q can be
uniquely written as f = Uo g where UeK and ceZ. More -
over f (0) = is equivalent to c = 0, i. e. , to f = UeK .
b) For feQ we have Z c D f ( see I, §4. 2) and z -» f(z)
maps Z biholomorphically onto itself .
Proof: The corollary of II, Theorem 2.1, shows
that feQ can be uniquely written as f = Uo s where
U e r(O,0.) and c e D(O,0 + ). But f~I yields f _1 (0)~0
and hence c~0. So c belongs to Z and s equals g in
view of the proof of Proposition 2. From f(0) =
follows Uc = and hence c = 0. So part a) is proved.
Let feQ and beZ. Then f°g h belongs to CJ and part
a) yields h = U°g for some UeK and ceZ. It follows
' J °c o
that
\(z) = «J f (g b (z)) id (z)
^c
according to (I;4.7). Since Q is contained in =■*(&)
we have ull(0) f 0. Hence b = g b (0) e D f . Thus f is
holomorphic in Z and f(b) = h(0) = Uc belongs to Z.
So z ^ f(z) maps Z into itself. Since f is birational
it is biholomorphic .
PROPOSITION 4 . Z is a bounded symmetric domain
and
Z £ [z; zeV, I-P(z)P(z) > 0}
c [z; zeV, 21 - zal > 0}.
IV, §5 123
Proof: Let ceZ and set g = g . From Theorem 3.5
follows v > (with respect to the bilinear form 3)
and hence 3(q^,q) > for £ qeO. Choose q = a + (tb
where a,b e V and set q 8 = a, + T-, + %-.. Thus
p(q g ,q) - <a x + T L + ®bj_,b + ®a)
= a (a-^a) + a (b,^)
according to §3.2. A verification leads to
a ± = (q g )(0) = B^[a - P(c)b],
and from + q g = 9 + v q = v © + q = (® + q) g it follows
that
b\ = (3 + q S )(0) = (® + q) S (0) = B^(b - P(c)a).
Hence choosing a = P(c)b we get a, = and
< g(q g ,q) = ^(b^B'^b) if b t
where Q = I - P(c)P(c"). In particular, det Q, t ^-
ceZ. But Q is hermitian and ..
x c
is connected we - with Q for c-Z. So the
x c
first \ r ^usion is proved.
Next for ceZ we have
< B(c,-c) = I - cnc + P(c)P(c) < 21 - cac
and the second inclusion holds. Taking the trace in
21 - c-c > we obtain 2- dim V > -(c,c). Thus Z is
bounded.
124 IV, §5
Since f(0) runs through all of Z if feQ (see
Proposition 3a), Q induces a transitive group of biholo-
morphic mappings of Z and the symmetry z ■* -z is
contained in Q. Hence Z is a symmetric domain.
Putting the propositions together we complete
the proof of Theorem A.
4- As a generalization of the representation of a
complex number in polar coordinates we give a theorem,
for which the proof is based on an idea of U. Hirzebruch
[4]. Introducing the condition
(*) If x,y e V such that xoy + yOx = and
a(Lx,Lx) > a(Ly,Ly) for all Lei, L* = L,
then y = .
we have
HIRZEBRUCH 's Theorem . Suppose that the pairing, a of
V satisfies in addition the condition (*) . Then to
o v
each weV there exists U in the identity component
K of K such that Uw belongs to V •
o — a o
It is not known whether or not the condition (*) is
a consequence of our assumptions on the pairing of V .
We will see later, that (*) holds whenever Q is a
Jordan pairing of the first kind.
Proof : Since K is a compact group there exists
z = x + iy in the orbit K w such that
o
IV, §5 125
a(y,y) < o(Im Uw, Im Uw) for all U e K •
According to Lemma 2.2 the Lie algebra of X equals b.
Hence U = exp D, Deb, belongs to K . We obtain
< 2a(y,Im Dz) + a(y,Im D 2 z) + o(Im Dz,Im Dz) + ••
Replacing D by otD, < aeP, we get a(y,Im Dz) > and
hence
2
a(y,Im Dz) = and a(y,Im D z) + a(Im Dz,Im Dz) >
for all Deb. Choosing D = iL where L = L e X we
obtain
2
a(y,Lx) = and ct(Lx,Lx) > cr(y,L y) = a(Ly„Ly).
We choose L = anb + boa where a.b e V and the first
o
conditions imply xt=iy + ydx ■ 0. Hence y = follows
from (*) •
5. Let D be an arbitrary bounded symmetric domain
in a complex vector space V and denote by Q the group
of biholomorphic mappings of D onto itself. The complex-
ification of the real Lie algebra of Q is denoted by @.
We have seen in II, Theorem 5.2, that there exists a
pairing of the vector space V satisfying the conditions
(P.l) to (P. 4) such that @ is isomorphic to the binary
Lie algebra associated with the pairing.
THEOREM B - If D is a bounded symmetric domain in
a complex vector V space then there exists a real form
126 IV, 56
V of V and a pairing □ of_ V satisfying the conditions
of §2.1 such that D is linearly equivalent to the
domain Z •
We give a sketch of the proof • From S- Helgason [ 3 ],
chapter VIII, §7, it follows that there is a real form
V of V such that the restriction of the pairing □ to
V satisfies the conditions in §2.1. Furthermore, let
o =
O be the binary Lie algebra associated with the pairing
of V then the coniugation T coincides with 3 and the
bounded domain Z is linearly equivalent to D.
§6. The Bergman kernel of Z .
1. Let D be a domain in V and put D = {z;zeD}.
Denote by Bih D the group of all biholomorphic mappings
of D onto itself. A function p : DxD -*• C is called a
Bergman kernel of D if
(i) p(f(z),f(w))-det 2f|2l -det ^2i = P (z,w)
holds for z,w e D and f e Bih D,
(ii) p(z,z) > for z e D and d(z,w) = p(w,z").
We need the following theorem due to St. Bergman.
THEOREM 6.1 . If D is a bounded domain in V then
there exists a Bergman kernel of D .
IV, §6 127
For a proof see S. Helgason [ 3 ], Chapter VIII, §3-
COROLLARY • Suppose that the function § : DxD -> C
satisfies the condition (i) for all f in a transitive
subgroup of Bih D as well as (ii) • Then each Bergman
kernel of D equals y§ where y is a positive constant -
Proof: Let p be a Bergman kernel of D and put
T) = l/p. Then r\(£ (z) , f (w) ) = rj(z,w) for z,weD and all f in
the given transitive subgroup of Bih D. Hence r\
does not depend on z. But r|(z,w) = ri(w,z) shows that
Ti is constant.
2- Now let Z = Z n be the bounded symmetric domain
given by the pairing D of V ■ Since the subgroup Q of
Bih D is contained in H(0, 6.) (see §5.1) we conclude
(6.1) B(f(z),^f(wT) =^B( Z) -i) (rW 1 )
for feQ from II, Theorem 2.1. Notice that B has to be
replaced by B, (see §5.1). We define the holomorphic
function C : DXD -> C by
(6.2) C(zjw) = det B(z,-w) for ZjW e Z-
From B(z,w) = B(w,z") we conclude G(z.,w) = C(w,z).
Furthermore, since B(z,-z) J( zeZ, is hermitian positive
definite we obtain £(z,z) > 0. Hence the function £
satisfies (ii) and (i) for feG- Hence the Corollary of
128 IV, §6
Theorem 6.1 yields
THEOREM 6.2 . Each Bergman kernel of Z equals yC"
where y is positive constant .
Since Z is bounded the function C(z,w) is bounded
for z,w e Z. Hence we obtain the
COROLLARY . Each Bergman kernel of Z is bounded
away from zero .
For bounded symmetric domains this result is due to
H. L. Resnikoff [14]. We are going to prove
LEMMA 6-3. Let z,w e Z and a,b e V- Then
(6.3) 4 A-.log C(z,w) = - oCfBCz^-w)]" 1 a,b),
and (6.1) holds for all f in Bih Z
Proof: Note first that
A z cp(z) = A- cp(z)
holds whenever cp is holomorphic in z. Hence the left
side of (6-3) defines a hermitian form X of V. Since
z j w
(a,b) -> o(a,b) defines a hermitian positive definite
form of V there exists an endomorphism Q(z,w) of V that
is hermitian and rational in z,w such that
X z,w (a ' b) = " ^(Q(z,w)a,b)
IV, §6 129
Since the condition (i) holds for p = £ we obtain
^Q(f(z),f^)) ^ - Q(z,w)
for f e Bih Z because of the chain rule. Hence the
function
R(z,w) := B(z,-w) Q(z,w)
satisfies
(6.4) R(f(z),f(w")) = l||5i r( Zj w) p||2l for feQ
because of (6.1).
From the definition of Q we observe -X ~(a.,b) =
trace aob = a(a,b) . Hence Q(z.,0) = I and (6-4) yields
R(f(z) J fT0T) = I for feQ.
Since Q acts transitively on Z we conclude R(z,w) = I
and the lemma is proved .
3- Denote by k' the subgroup of X consisting of
the transformations W which map Z onto itself (see §2.5)
Clearly, the connected component K of K is a normal
subgroup of K ' of finite index.
THEOREM 6.4 - The group Bih Z of biholomorphic
mappings of Z onto itself consists exactly of the
functions Uo g where UeK ' and ceZ and this representa -
tion is unique - The index [Bih Z : Q ] = [X ' : K ] is
finite .
130 IV, §6
Proof : Let f be a holomorphic map of Z onto itself.
We choose g , ceZ, such that the function h = f o g satis-
fies h(0) =0. By Lemma 6-3 the condition (6.1) holds
for h. Substituting w = we see that ^ z < is constant.
Hence h(z) - Uz where U e GL(V) . But again (6.1) yields
U U = I and hence UeX ' .
As a consequence we see that the Lie algebras of
Bih Z and of Q coincide. Using (4.5) we obtain the
COROLLARY • The real Lie algebra of Bih Z is iso -
morphic to the subalgebra O, = [p; peD, ®_jp = p} of £).
In a similar way we observe
THEOREM 6 ■ 5 • Let □ and □ ' be two pairings of V
that satisfy the , conditions of §2-1 and let Z and Z '
be the corresponding bounded symmetric domains • Then
the following statements are equivalent :
a) There exists a biholomorphic map f : Z -» Z ' •
b) There exists a W e GL(V) such that Z' = WZ .
c ) The pairings □ and n ' are isomorphic ( in the
sense of III, §3-4).
V,§1 131
Chapter V
AN EXPLICIT DESCRIPTION OF THE BOUNDED SYMMETRIC DOMAINS
§1. Formal real Jordan algebras .
1. Let 31 be a finite dimensional semi-simple
Jordan algebra over R. Hence 21 contains a unit ele-
ment e and its trace form (a,b) -> trace L(ab) is non-
degenerate (see III, §3, and [2], chapter XI). We
obtain a pairing o of the vector space 21 by
aDb := 2(L(ab) + [L(a),L(b)])
that is a Jordan pairing of the first kind (see III, §3) •
Using [ ], chapter XI, Satz 3-4, we see that the pairing
has a positive definite trace form
a (a,b) = 4 trace L(ab)
2 2
if and only if 21 is formal real, i.e., if a + b =0
implies a = b = 0.
Suppose now that 21 is formal real . We know from
III, §3-1, that the endomorphism P(a) associated with
the pairing coincides with the quadratic representation
of 21 , i.e.,
o '
P(a) := 2L 2 (a) - L(a 2 ) .
For aeSI the exponential exp a is given by
132 V,§1
exp a
L m.
and one has
P(exp a) = exp 2 L(a)
(see [2], chapter XI, Satz 2.2). Since z q is an
associative bilinear form of 31 , the endomorphism L(a)
is self adjoint with respect to a • Furthermore the
group r equals the structure group r(3i ) of 31 .
2. Let
= 21 +1 +31, = [P(x)b; beSi },
o o o o o L v/'
be the binary Lie algebra associated with the pairing O
of the vector space V =34 (see IV, §2.1). We know
K o o =
from III, §3-1, that the automorphism © of O is
essential and III, Lemma 3-1, shows that 3 (£ ) is
generated by the birational functions
W, t and j where W € T(3J ), ae3J ,
' a J v o o
and where j is given by j (x) = -x . As mentioned in
III, §3, the group "(O ) coincides with the group S(SJ )
considered in [ ]_]_] .
In the notation of I, §4.2, we have the
THEOREM 1.1. Each automorphism of £ is essential
___________ e
and
V, §1 133
v : H(O q ) ■* Aut O q , f ■* 7 fJ
defines an isomorphism of the groups ■
For a proof see [12] •
3- Let Y = Y(2I ) be the domain of positivity
given by the formal real Jordan algebra 21 . According
to [ 2 ]) chapter XI, Satz 3-6 and Satz 3-7, we have the
descriptions
Y = exp 81 = [a; ae2I , L(a) > 0}
2
and the closure of Y equals [a ; ae2I J. Furthermore, Y
is an open convex cone and equals the connected com-
ponent of the set [z, ze^l, det P(z) ^ 0} containing e.
Denote by M = U (51 ) the group of W € r(2I ) = r(0 ) such
that a -> Wa maps Y onto itself. Then M acts transitively
on Y and the index of it in r(2I ) is finite.
v o
4. Denote by SI the complexif ication of the formal
real Jordan algebra 21 . Hence 21 is a semi-simple complex
Jordan algebra. Let
H = H(9J ) = 91 + iY = [z; ze2I, Im z e Y],
o o
then H is a domain in the complex vector space 21. It
is known (see U. Hirzebruch [4 ], [ 7 ]) that the subgroup
of S(0 ) generated by W, t„ and i where WeW, ae2l . acts
— v o u u a J o
as a transitive group of biholomorphic mappings on H .
In particular, for f in this subgroup one has H <= Dom f
134 V,§1
and z ■* f(z) maps H onto itself.
p
The real pairing □ induces a pairing of 21 (see
IV, §2-3) and we obtain the binary Lie algebra
D = 2J F + £ F + JJ R , 9J F = (P(z)b; be2J P }.
Again the group 3(d) is generated by the birational
functions
W, t and j where W e r(O) , ae'i,
a
and 5(0 ) becomes a subgroup of S(O). The pairing
induces a bounded symmetric domain Z = Z in 91 accord-
ing to IV, §5-2, and to Theorem A. Using the element
p of Hj^O) given by
p(z) = (z-ie)(z+ie) = e - 2i(z+ie) ,
i.e., p = t o2iIojot. , we are going to prove
THEOREM 1.2 . The function p maps H biholomorphi -
cally onto the bounded symmetric domain Z .
Proof: Let z be in H. Hence z+ie e H and p is
holomorphic in H because j is holomorphic in H. A
verification yields
p (w) = i(e+w)(e-w) = - ie + 2i(e-w)~
provided e-w is invertible in 21 . Thus the imaginary
part is given by
Im p (w) = - e + (e-w)~ + (e-w)~ .
V,§1 135
We use the well-known formulas
L(a _1 ) = L(a)[P(a)] _1 = [P(a) ] _1 L(a) ,
P(a _1 +b _1 ) = [P(a)]" 1 P(a+b)[P(b)]~ 1
where a,b e 3J are invertible. Writing a = e-w, b = e-w
we obtain
P(e-a' 1 -b" 1 ) = I - 2L(a" 1 +b" 1 ) + PCa^+b" 1 )
= [P(a)]' 1 [P(a)P(b) - 2L(a)P(b) - 2P(a)L(b)
+ P(a+b)][P(b)] _1
provided a and b are invertible. A verification yields
now
P(Im p _1 (w)) = [P(e-w)] _1 [I - wcw + P(w)P(w) ] [P(e-w) ] ~ 1
provided e-w is invertible.
Denote the image of H under p by Z . Clearly e-w
is invertible whenever weZ- Thus weZ if and only if
Im p (w) lies in Y, i.e., lies in the connected
component of the set [y; ye?I , det P(y) f 0} containing
e. Hence w is in Z if and only if w is in the
connected component of the set [w; weM, det B(w,-w) ^ 0}
containing zero which equals Z.
5- We are going to give some more descriptions
of the bounded symmetric domain Z that is associated
with a Jordan pairing of the first kind induced by a
formal real Jordan algebra. First we have
136 V,§1
THEOREM 1.3 To each we 21 there exists U in the
identity component K of K such that Uw belongs to
the closure Y of Y.
Proof : We apply Hirzebruch's Theorem and we have
to show that the condition (*) in IV, § 5.4, holds.
From xDy + yOx = 4L(xy) we get xy = . Choosing
2 2
L = L(y) the second condition in (*) yields o (y ,y )
= and hence y = because 21 is formal real. Hence
J o
there exists UeK such that Uw belongs to 21 • Let
o & o
= 1\ V X v e F '
Uw =
V
be the minimal decomposition of Uw (see [ 2 ] , chapter
XI, §3) where the c 's form a complete orthogonal
system of idempotents of 21 . We choose cp e F such
±cp v
that e X > and set
v
-I
lcp
e v c
v
Thus q is invertible and q = q . Clearly P(q) e K
o
and P(q)Uw has a minimal decomposition with non-negative
eigenvalues. Hence P(q)Uw belongs to Y.
In view of Theorem 1.2 we may apply Theorem 12
in [ 7 ], chapter VII. We use the orderings ">" and
">" of 21 which are given by
a > b » a-beY, a> b » a-beY.
V,§1 137
THEOREM 1.4 - For ze^I the following conditions
are equivalent :
a) zeZ,
b) z = Ur where UeK and re21 such that e > r > 0,
' o o
c) I - P(z)P(z) > 0,
d) 21 - zDz > 0.
Note that part c) and d) state a sharper result than that
given in Proposition 4 in IV, §5.
Here A > means that the endomorphism A is
positive definite with respect to the hermitian form
(u,v) -> a Q (u,v) •
Proof : As an abbreviation set
Q 1 (z) = B(z,-z), Q 2 (z) = I - P(z)P(z), Q 3 (z) = 21 - zDz.
Hence
U Q k (z) U* = Q k (Uz) where Ueh and k = 1,2,3-
In view of Theorem 1.3 it suffices to prove the equival-
ence of the conditions a) to d) for z = rtil such that
' ' o
r>0 . We obtain
Q 1 (r) = P(e-r) 2 , Q 2 (r) = I - P(r 2 ), Q 3 (r) = 2L(e-r 2 ) .
Let
r=Y)v c , < \ e R,
V V V
V
138
V,§2
be the minimal decomposition of r. From the definition
2
of Z it follows that reZ is equivalent to e-r >
(see 3) and hence to e > r . Using [ 2 ], chapter VIII,
Satz 1.3, we see that Qo( r ) > is equivalent to 1 > ^
for all v and hence to e > r . But Qo(r) > means
2
e-r > 0, too.
§2. The classification of the bounded
symmetric domains .
1. Let 3fl be the space of rxs complex matrices
— r , s
and denote by e the rxr unit matrix. Cartan's classifi-
cation shows that each irreducible bounded symmetric
domain is linearly equivalent either to a domain in
the following list
nota:
Cartan
;ion
Helgason
domain
dim c
I
r, s
A III
{z; zeW
-t
z z
<e s'
rs
II
r
D III
[z; zelDl r r ,
z^z
< e r , z =-z}
r(r-l)
2
III
r
C I
[z; zeffl!
' r,r'
-t
z z
< e r? z =z}
r(r+l)
2
IV r
BD I
(q=2)
[z ; zee , z
c z <
^(l+lz^l 2 ) < 1}
r
or to an exceptional domain of type E^- or E 7 of dimension
16 or 27 respectively.
V,§2 139
Each of these domains can be obtained as a domain
Z_ (see IV, §5.2) where n is some pairing of a real
vector space satisfying the conditions of IV, §2.1.
For a real vector space V let V be its complexif ication
Type I : Let V be the real vector space of rxs
jl r,s o F
matrices with real entries. As pointed out in III,
§2.1, we obtain a pairing D of V by
(anb)c = ab c + cb a
having
a (a,b) = (r+s) trace ab
as trace form. Clearly, a is positive definite and
therefore the pairing satisfies our conditions.
According to (III; 2. 5) the endomorphism B(a,b)
is given by
t t
B(a,b)c = (e+ab )c(e+b a) where a,b,c e V,
and Proposition 4 in III, §2, shows that det B(a,-a) ^
is equivalent to
det(e-ai" t ) £ and det(e-a t a) f 0.
Using
le \ [e - a fc a o\ e 6\
M _r M = M c M where M =
\0 e - aa c \ e la ej
we see that the last two conditions are equivalent. Hence
140 V,§2
the domain Z n associated with our pairing coincides
with the set of z ' s such that det(e-z z) J* and hence
with the domain listed under I .
Type II and III : For e = ± 1 denote by V" the
;r r r o
vector space of rxr real matrices a satisfying a = ea.
According to III, §lj we obtain a pairing o of V^ by
(aDb)c = ab c + cb a
having
a (a.b) = (r+e) trace ab
o
as trace form. Again a is positive definite and the
o o
pairing satisfies our conditions. From (III; 1.6) we
conclude that the domain Z Q associated with the pairing
coincides with -the domain listed under II provided
r r
e = -1 or listed under III provided e = 1.
r r
2. We use now our results of §1. Let 2J be a
= o
formal real Jordan algebra of dimension n and let
be the induced Jordan pairing of the first kind, i.e.,
aab = 2(L(ab) + [L(a) ,L(b) ] ) .
We know from §1.1 that its trace form is positive
definite. In the following list we write all simple
formal real Jordan algebras (in the notation of [ 2 ] ,
chapter XI, §5) and the type of the domain Z Q associat-
ed with the pairing:
V,§2
141
21
o
[X,u,e]
& (R)
r
6 r (C)
6 («/ )
r v 4
S3 3 (S g )
type
IV n
III
r
X r,r
II
2r
E 7
Hence all irreducible bounded symmetric domains except
the domain of type E,- are constructed by a pairing.
3- Finally we show that the domain of type
E, is also covered by our construction. According
to III, §4.2., let S = S ft be the Cayley division algebra
over F and put V = S©&. Then there is a pairing □
of V having the trace form
o &
a (a,b) = 48[u(a,,b,) + u(a 2 ,b 2 ), a = a,®a 2 , b = b,©b> 2 eV
Since the bilinear form u of S is positive definite
we see that a is positive definite, too. Hence the
o r
pairing satisfies our conditions . According to a
recent result of K. Meyberg [ 13] the Lie algebra O = O
is of type Er • The Lie algebra of the group of biholo-
morphic mappings of the associated domain Z is isomorphic
to O, (see the Corollary of IV, Theorem 6.5) and hence
of type Er (see IV, Theorem 4.2).
Summing up we see that all bounded symmetric
domains are linearly equivalent to a domain Z where
the pairing □ is a Jordan pairing of first or second
kind satisfying the conditions of I V , 2 2.1.
FINIS
142
INDEX OF
NOTATIONS
Sets : Mappings etc . :
Aut(0, ®)
page 108
B (x)
V
9
Aut"0
27
B(a,b)
52
Bih D
126
H $ (x)
10
Dom f
2
h f
7
D f
31
v, v fJ v w
8
D(Q,0)
40
fc b
8
* - %
115
t
V
17
Lie Q
94
r § (x)
14
Pol V
5
^x
3
tP(V)
3
Y b
9
P (v)
o v
6
V
16
T = <P o + ? 1 + T 2
5
w f (x)
30
Rat V
4
W #
36
z = z D
119
w 11
100
r(o)
27
s*,f*
37
r(o J@ )
40
w R
92
r *(£)
102
Sf(x)
3
H(Q)
28
Sx
H-(O,0)
40
3°(D)
31
3 *(^)
106
143
REFERENCES
[I] H. BRAUN, Doppelverhaltnisse in Jordan-Algebren,
Hamb . Abh . 32 (1968), 25-51.
[2] H. BRAUN and M. KOECHER, Jordan-Algebren , Springer
1966.
[3] S. HELGASON, Differential Geometry and Symmetric
Spaces , Academic Press 1962.
[4] U. HIRZEBRUCH, Halbraume und ihre holomorphen Auto-
morphismen, Math . Ann . 153 (1964), 395-417.
[5] , liber Jordan-Algebren und beschrankte
symmetrische Gebiete, Math . Z. 94 (1966), 387-390.
[6] N. JACOBSON, Lie Algebras , Interscience 1962.
[7] M. KOECHER, Jordan Algebras and their Applications ,
Lecture notes, Minneapolis, Univ. of Minnesota 1962.
[8] , Imbedding of Jordan algebras into Lie
algebras I, Amer ■ J. Math . 89 (1967), 787-816.
[9] , Imbedding of Jordan algebras into Lie
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[10] , On Lie Algebras Defined by Jordan
Algebras , Aarhus Universitet, Matematisk Institut,
(1967), dupl.
[II] j Uber eine Gruppe von rationalen
Abbildungen, Inv. Math . 3 (1967), 136-171.
[12] , Gruppen und Lie-Algebren von rationalen
Funktionen, Math - Z_. 109. (1969), 349-392.
[13] K. MEYBERG, Jordan-Tripel-Sys teme und die Koecher-
Konstruktion von Lie-Algebren, to appear.
[14] H. L. RESNIKOFF, Supplement to "Some remarks on
Poincare series," Compositio Mathematica 21 (1969),
No .2 , to appear .
[15] C. L. SIEGEL, Symplectic geometry, Amer . J. Math ■
65 (1943), 1-86.
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