Symplectic groups over noncommutative algebras

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.