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.

87 pages