# 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 '11 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 '11 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 '11 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 '11 at 8:09

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.

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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.

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The book by Hahn and O'Meara titled The Classical Groups and K-Theory could be useful for you.

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