Algebraic Structures of Symmetric Domains by Ichiro Satake

By Ichiro Satake

This e-book is a accomplished remedy of the overall (algebraic) conception of symmetric domains.

Originally released in 1981.

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Remark following § 6, Exerc. 3). [Other equalities in (8. 7) follow from this and V, § 6, Exerc. ] Conversely, let Ube a formally real Jordan algebra and define T,, by (8. 6). Put lJ= {Tx(xE U)), f= [µ, µ], and g=f++J. Then, by Proposition 7. 4 and (7. 19), g=Lie I'(U, {}) and the conditions (i)-(iii) are again satisfied (with respect to the trace fori:n ,). Hence, by Lemma 8. Q=G1e is a self-dual homogeneous cone and one has (**). Moreover, by Lemma 8. Q, q. e. d. As was noted in the above proof, in the notation of§ 7 (with V = U), one has by (7.

C(a)+n is a minimal parabolic subalgebra of g. From the theory of algebraic Lie algebras (Chevalley [2], [4]), it is easy to see that all Lie subalgebras of g containing b~ are algebraic and hence of the form br for some I'cJ. It is known that any algebraic group defined over F contains an (absolute) maximal torus j defined over F. It follows that there is an F-torus Tin C(A) such that 7=TFismaximalinC(A)F. Then Tis (absolutely) maximalinG and A coincides with the F-split part of T. Let X be the (absolute) character module of j and Xo the annihilator of A in X.

3 (and§ 1, Exerc. 16) Der(V, { }) = {TeLieI'(V, { })I T*=-T). Exercises In the following exercises, ( V, { } ) is a JTS, not necessarily satisfying the condition (JT 3). ForaeV, we define the "quadratic map" P(a)eEnd(V) by P(a)x={a,x,a). 1. YD (P(a)x), (6. 19) (P(x)P(a)x) Oa = (P(x)a) D (P(a)x) = xO (P(a)P(x)a). Hint. 18), compute {a,y, {a, x, a}}, {x, a, {x, a,y)} and {y, a, {x, a, x) }. Remark. The so-called "fundamental formula" (6. 18')). (by (JT 2)) (by (6. 17)) Conversely, if one defines the triple product { } by {x,y, z) = 21 (P(x+z)-P(,:)-P(z))y, then the condition (JT 2) follows from (6.

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