Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:26:20Z http://mathoverflow.net/feeds/question/68331 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68331/info-on-symplectic-orthogonal-groups-of-gln-r-r-a-ring-not-necessarily-divisi Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring. Larry 2011-06-21T02:07:05Z 2011-07-07T02:41:32Z <p>Hi, Everyone:</p> <p>Does anyone know anything about orthogonal and symplectic groups</p> <p>associated to Gl(n,R)? I am using symplectic/orthogonal in what I think</p> <p>is the standard sense; I mean, we have an R-module R_M (left- or right- ), and </p> <p>symplectic /quadratic forms q_S , q_Q respectively . Then the symplectic/orthogonal group</p> <p>associated with (R_M,Q) is defined to be the subgroup of Gl(n,R) that preserves q_S, resp. q_Q.</p> <p>I have checked Artin, I Hungerfor(d) Algebra--even Schaum's :) .</p> <p>Thanks for ideas, refs.</p> http://mathoverflow.net/questions/68331/info-on-symplectic-orthogonal-groups-of-gln-r-r-a-ring-not-necessarily-divisi/68345#68345 Answer by mathphysicist for Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring. mathphysicist 2011-06-21T06:39:57Z 2011-06-21T06:39:57Z <p>Your goal appears to require a somewhat different approach, at least for noncommutative rings; see the paper <a href="http://pages.uoregon.edu/arkadiy/noncomloopalg.pdf" rel="nofollow">Lie algebras and Lie groups over noncommutative rings</a> 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.</p> http://mathoverflow.net/questions/68331/info-on-symplectic-orthogonal-groups-of-gln-r-r-a-ring-not-necessarily-divisi/69689#69689 Answer by Larry for Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring. Larry 2011-07-07T02:41:32Z 2011-07-07T02:41:32Z <p>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_<em>(Sg,Z) and H_</em>(Sg,Z/2) with little success. </p>