E（n）-模代数的余圈扭及其在Brauer群上的应用
Cocycle twisting of E(n)-module algebras and applications to the Brauer group
摘要：
本文对Hopf代数E（n）的余拟三角结构在懒惰上圈和Hopf自同构群作用下的轨道进行了分类。这被应用于检测E（n）的Brauer群$BQ(k,E(n))$的同构子群。对于E（n）上的三角结构$R$，我们证明了由$R$产生的$BQ(k,E(n))$的子群$BM(k,E(n),R)$同构于$BW(k)$和$Sym_n(k)$的直积，$BW(k)$是基域$k$的Brauer-Wall群，$Sym_n(k)$是$n \times n$对称矩阵的加法群.对于E（n）上的一般拟三角结构$R'$，我们构造了一个分裂的短正合序列，该序列以$BM(k,E(n), R')$为中间项，以$r<n$阶对称矩阵群的中心扩张为左项（$r$依赖于$R'$）.我们最后描述了如何在$BQ(k,E(n))$内的Hopf自同构群的图像作用于$Sym_n(k)$。
英文摘要：
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 $Sym_n(k)$.
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