Symplectic groups over noncommutative algebras

Abstract

We introduce the symplectic group \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\) over a noncommutative algebra A with an anti-involution \(\sigma \). We realize several classical Lie groups as \({{\,\mathrm{Sp}\,}}_2\) over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\) on isotropic A-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic A-lines as invariants of this action. When the algebra A is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space \(X_{{{\,\mathrm{Sp}\,}}_2(A,\sigma )}\), and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\)) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A.