Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring. - MathOverflow

Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring.

2011-06-21T02:07:05Z

Hi, Everyone:

Does anyone know anything about orthogonal and symplectic groups

associated to Gl(n,R)? I am using symplectic/orthogonal in what I think

is the standard sense; I mean, we have an R-module R_M (left- or right- ), and

symplectic /quadratic forms q_S , q_Q respectively . Then the symplectic/orthogonal group

associated with (R_M,Q) is defined to be the subgroup of Gl(n,R) that preserves q_S, resp. q_Q.

I have checked Artin, I Hungerfor(d) Algebra--even Schaum's :) .

Thanks for ideas, refs.

Answer by mathphysicist for Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring.

2011-06-21T06:39:57Z

Your goal appears to require a somewhat different approach, at least for noncommutative rings; see the paper Lie algebras and Lie groups over noncommutative rings by Berenstein and Retakh and references therein. The said paper concentrates more on Lie algebras than on Lie groups but still seems close enough to what you want.

Answer by Larry for Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring.

2011-07-07T02:41:32Z

Sorry for the delay: I am actually interested in working with the ring H_1(Sg,Z) , with Sg the orientable genus-g surface, Z the integers and q being the intersection 2-form. I have tried to extract some info from the naturality of the reduction map between H_(Sg,Z) and H_(Sg,Z/2) with little success.