Symplectic groups over Lie subgroups of involutive algebras

Abstract

We introduce the symplectic group $\mathrm{Sp}_2(G, \sigma)$ associated to a Lie subgroup $G$ of a (possibly noncommutative) associative algebra $A$ equipped with an anti-involution $\sigma$. Our construction recovers several classical Lie groups as special cases, and in particular provides new realizations of spin groups as instances of $\mathrm{Sp}_2(G, \sigma)$ for suitable subgroups $G$ of the Clifford algebra. This case is not covered by the framework, which focuses on the specific situation $G = A^\times$, and is thus of particular interest. We construct and study geometric spaces on which $\mathrm{Sp}_2(G, \sigma)$ acts. In particular, we define the space of $G$-isotropic elements and the corresponding space of $G$-isotropic lines, which generalize the classical projective line. We analyze the group action on these spaces and introduce natural invariants, such as the notion of positive triples and quadruples of $G$-isotropic lines and a generalized cross-ratio of positive quadruples of $G$-isotropic lines. Finally, when the Lie algebra of $G$ is Hermitian, we define the associated Riemannian symmetric space of $\mathrm{Sp}_2(G,\sigma)$ and provide several models for it.