By Ichiro Satake

This ebook is a finished remedy of the final (algebraic) idea of symmetric domains.

Originally released in 1981.

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**Example text**

40 Algebraic Preliminaries tension of

2) [p, q]' = n ~l (api . oqi) oxi qJ-PJ ox 1 then we have (7. 2') where the bracket on the right-hand side is the usual Poisson bracket. Thus ~ is an (infinite-dimensional) Lie algebra over F isomorphic to the Lie algebra of polynomial vector fields on V by the correspondence P<-> -p :x. Clearly one has (7. 3) In particular, 113 1 is a Lie subalgebra, identical to gf(V). Also, for Aegf(V)=ll3, and be V=ll3o, one has [A, b]=Ab. Now let ( V, { } ) be a non-degenerate JTS. For be V, we set p,(x) = {x, b, x) (=P(x)b) (7.

D}. Therefore we have p=ad(K 0 )a+, and so (5. 7) (5. 8) G 0 = K 0 P = K 0 (exp a+)K 0 • It is also known that one has G = K-A 0 N (5. 9) which is a topological direct product decomposition. The decomposition (5. 9) (which is also valid for any R-group, not necessarily reductive) is called an Iwasawa decomposition of G (cf. Helgason [l ], VI). § 6. The structure group of a (non-degenerate) JTS. Let V be a finite-dimensional vector space over a field F of characteristic zero. When there is given an F-trilinear map { } : V X V X V-.

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