Symplectic groups over noncommutative algebras

We introduce the symplectic group Sp2(A,σ) over a noncommutative algebra A with an anti-involution σ. We realize several classical Lie groups as Sp2 over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups Sp2(A,σ) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of Sp2(A,σ) 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 XSp2(A,σ), and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as Sp2(A,σ)) 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.