Symplectic groups over noncommutative algebras - Alessandrini, Daniele; Berenstein, Arkady; Retakh, Vladimir; Rogozinnikov, Eugen; Wienhard, Anna (Selecta mathematica, 2022)
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.
Gesehen am 23.09.2022