2025/10/13 by Rogozinnikov, Eugen
#16W10 #32M15 #53C35 #Differential Geometry (math.DG) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2510.11267
We introduce the symplectic group Sp2(G, σ) associated to a Lie subgroup G of a (possibly noncommutative) associative algebra A equipped with an anti-involution σ. Our construction recovers several classical Lie groups as special cases, and in particular provides new realizations of spin groups as instances of Sp2(G, σ) for suitable subgroups G of the Clifford algebra. This case is not covered by the framework, which focuses on the specific situation G = A^×, and is thus of particular interest. We construct and study geometric spaces on which Sp2(G, σ) 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 Sp2(G,σ) and provide several models for it.