Cocycle twisting of E(n)-module algebras and applications to the Brauer group. K-Theory 33

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf auto-morphism group. This is applied to detect subgroups of the Brauer group BQ(k,E(n)) of E(n) that are isomorphic. For any triangu-lar 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 Symn(k), the group of n × 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 kernel 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 Symn(k)