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IYE1SITY AN ELEMENTARY APPROACH TO BOUNDED SYMMETRIC DOMAINS Max Koecher HOUSTON, TEXAS 1969 Iad I * s U_^ ~fUjt w-a-TT AN ELEMENTARY APPROACH TO BOUNDED SYMMETRIC DOMAINS Max Koecher Rice University 1969 3 ■ ERRATA Line - 9 h -11 ? 15 ob ta in - 2 r 2, [v,gp] - 3 B(O) - 1 w # e r(o) - 2 ©v ® w - 1 a - 2 ©b e 3 1 (T )* = T . „. 3 u ,v 5 maximal compact - 1 3(D) - 2 a € V - 1 G - 1 a(c,c) , 11 [2] -9 thru -5 det 0^0 for ceZ. But Q is hermitian and Q =I>0 . Since ^c ^o Z is connected we end up with Q >0 for ceZ. So the first inclusion is proved . /en 3- tions ie liled led fiams itary Le 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 LYRASIS Members and Sloan Foundation 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. 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