Extensions of orthogonal groups of torsion quadratic forms. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:22:55Z http://mathoverflow.net/feeds/question/106201 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106201/extensions-of-orthogonal-groups-of-torsion-quadratic-forms Extensions of orthogonal groups of torsion quadratic forms. Lukasz Fidkowski 2012-09-02T21:53:57Z 2012-09-03T03:50:09Z <p>Hi. This is related to a question I asked earlier. The setup is:</p> <p>Let $L$ be an $n$-dimensional lattice with an integer valued quadratic form $q$. Fix a basis $e_i$ for $L$ and let $K_{ij} = \langle e_i, e_j \rangle$, where $\langle x,y \rangle = q(x+y) - q(x) - q(y)$ is the associated bilinear form. Let $L^*$ be the dual lattice to $L$, and $G_L = L/L^*$, and assume $|G_L|$ is odd. $q$ then descends to a $\mathbb{Q}/\mathbb{Z}$-valued quadratic form $q_G$ on $G_L$. Let $O(G_L)$ be the group of automorphisms of $G_L$ that preserve $q_G$.</p> <p>Since $O(G_L)$ acts on the finite abelian group $G_L$, it is natural to consider extensions of $O(G_L)$ by $G_L$. These correspond to $H^2(O(G_L), G_L)$. Have such extensions been considered before? Is there an algorithm or method to calculate them (given $K_{ij}$)? Any references would be appreciated.</p> <p>Thanks!</p> <p>(edit: corrected mistake above, thanks to B. Conrad for pointing it out)</p>