Symplectic groups over noncommutative algebras

TL;DR

This paper generalizes classical Lie groups as symplectic groups over noncommutative algebras, introduces geometric actions and invariants, and constructs models of symmetric spaces, extending hyperbolic geometry to noncommutative settings.

Contribution

It introduces the symplectic group over noncommutative algebras, generalizes classical Lie groups, and constructs geometric models and invariants in this new framework.

Findings

Realization of classical Lie groups as symplectic groups over noncommutative algebras

Construction of geometric spaces and invariants such as the Kashiwara-Maslov index and cross ratio

Models of symmetric spaces as noncommutative hyperbolic geometries

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.