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

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.

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Is there something specific you want to know? It is a rather big subject, and it is hard to say anything nontrivial about totally general rings... – Andy Putman Jun 21 2011 at 2:33
Are these linear algebraic groups defined over Z? If so you can consider them as group schemes over any ring. – Andy B Jun 21 2011 at 4:59
To give you an idea of how hard it is to do linear algebra over rings that aren't fields, let's forget symplectic/orthogonal groups for a second and consider $G_n = SL_n(\mathbb{Z}[t,t^{-1}])$. In an absolute tour de force, Suslin proved that $G_n$ is generated by elementary matrices for $n$ at least $3$ (and, in particular, is finitely generated). However, for $n=2$ it is not known whether or not $G_n$ is generated by elementary matrices or finitely generated, though a deep theorem of Krstic-McCool shows that it is not finitely presentable. – Andy Putman Jun 21 2011 at 5:47
In addition to Andy Putnam's comment, you might want to have a look at the related question mathoverflow.net/questions/59884/… that I asked some time ago. – Tom De Medts Jun 21 2011 at 8:09