Bertram, Wolfgang On some causal and conformal groups. (English) Zbl 0873.17030 J. Lie Theory 6, No. 2, 215-247 (1996). 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) Cited in 8 Documents MSC: 17C20 Simple, semisimple Jordan algebras 53C35 Differential geometry of symmetric spaces 43A85 Harmonic analysis on homogeneous spaces 17C37 Associated geometries of Jordan algebras Keywords:causal symmetric spaces; Euclidean Jordan algebras; Kantor-Koecher-Tits groups; conformal transformations; semisimple Lie groups and Lie algebras Citations:Zbl 0279.53047 PDFBibTeX XMLCite \textit{W. Bertram}, J. Lie Theory 6, No. 2, 215--247 (1996; Zbl 0873.17030) Full Text: EuDML