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On some causal and conformal groups. (English) Zbl 0873.17030

Let \(V\) be a simple Euclidean Jordan algebra, \(V^c\) its conformal compactification and Co\((V)\) the Kantor-Koecher-Tits group of conformal transformations of \(V\). Consider an involutive transformation on the set of invertible elements of the Jordan algebra of the form \(x\mapsto (\alpha(x))^{-1}\) and its induced involution on Co\((V)\), where \(\alpha\) is in the structure group of \(V\). Denote by \(G\) the corresponding fixed point subgroup of Co\((V)\) and by \(X=G\cdot 0\) its orbit in \(V^c\). \(X\) is then a symmetric space and is also called a Makarevich space. They were classified by B. O. Makarevich [Math. USSR, Sb. 20, 406-418 (1974); translation from Mat. Sb., Nov. Ser. 91(133), 390-401 (1973; Zbl 0279.53047)] without proofs. The author gives a proof by using the result of K. H. Helwig [Halbeinfache reelle Jodan-Algebren, Habilitationsschrift, München 1967] on the classification of involutive automorphisms of Jordan algebras. The author also identifies the pseudogroup of causal transformations of the space \(X\) with the group \(\text{Co}(V)\), some examples being worked out for the classical matrix Jordan algebra by elementary methods.
Reviewer: G.Zhang (Karlstad)

MSC:

17C20 Simple, semisimple Jordan algebras
53C35 Differential geometry of symmetric spaces
43A85 Harmonic analysis on homogeneous spaces
17C37 Associated geometries of Jordan algebras

Citations:

Zbl 0279.53047
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