Get Algebraic Structures of Symmetric Domains PDF


By Ichiro Satake

This publication is a finished remedy of the overall (algebraic) conception of symmetric domains.

Originally released in 1981.

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Formally real Jordan algebras and self-dual homogeneous cones. In the remaining of this chapter (§§ 8, 9), we assume that F=R. A Jordan algebra A over R is calledformalry real (or compact) if the trace form r is positive definite. It is known that this condition is equivalent to saying that x 2 + f=O (x, ye A) implies x= y=O. (It is clear that the first condition implies the second. For the converse, see V, § 6, Exerc. ) To explain the connection between formally real Jordan algebras and self-dual homogeneous cones, we start with giving some basic definitions and results on open convex cones.

Is a derivation of the Jordan algebra A. ] (a, beA) is called an "inner" derivation. Now we define a trilinear product { } on V(=A) by {x,y, (6. 4) z} = (xy)z+x(yz)- y(xz), or equivalently, by (6. , T"]. Then (J 1) implies (JT 1). ] is a derivation of the Jordan algebra A, D is also a derivation for the triple product, i. , D{x,y, z) = {Dx,y, z) + {x, Dy, z) + {x,y, Dz). z), which follows immediately from (6. 2) and (6. 3). Thus the Jordan algebra A becomes aJTS with the trilinear product (6.

One can define a JTS structure on any Jordan algebra. First let us recall some definitions and basic identities. A finite dimensional (non-associative) algebra A over Fis called a Jordan algebra if the following two conditions are satisfied : Chapter I. 22 Algebraic Preliminaries (J 1) xy =yx, for all x,yeA. Y) = x(x~) For aeA, we define T.. eEnd(V) by Tax=ax, where Vis the underlying vector space of A. Polarizing the identity (J 2), one obtains (ab) (cd) + (be) (ad)+ (ca) (bd) = a( (bc)d) +b( (ca)d) +c((ab)d) (a, b, c, de A).

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Algebraic Structures of Symmetric Domains by Ichiro Satake

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