Symplectic groups over noncommutative algebras

We introduce the symplectic group Sp _2(A,σ ) over a noncommutative algebra A with an anti-involution σ . We realize several classical Lie groups as Sp _2 over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups Sp _2(A,σ ) act. We introduce the space of isotropic A -lines, which generalizes the projective line. We describe the action of Sp _2(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 X_ Sp _2(A,σ ) , and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as Sp _2(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.