Cocycle twisting of E(n)-module algebras and applications to the Brauer group

Abstract

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group $BQ(k,E(n))$ of E(n) that are isomorphic. For a triangular structure $R$ on E(n) we prove that the subgroup $BM(k,E(n),R)$ of $BQ(k,E(n))$ arising from $R$ is isomorphic to a direct product of $BW(k)$, the Brauer-Wall group of the ground field $k$, and $Sym_n(k)$, the group of $n \times n$ symmetric matrices under addition. For a general quasi-triangular structure $R'$ on E(n) we construct a split short exact sequence having $BM(k,E(n), R')$ as a middle term and as a left term a central extension of the group of symmetric matrices of order $r<n$ ($r$ depending on $R'$). We finally describe how the image of the Hopf automorphism group inside $BQ(k,E(n))$ acts on $Sym_n(k)$.