积分上Sweedler幂映射的不变量
Invariants from the Sweedler power maps on integrals
摘要：
对于具有非零左积分的有限维Hopf代数$A$，研究了$P_n(Λ)$与$P_n^J(Λ)$之间的关系，其中$P_n$和$P_n^J$分别是$A$和扭Hopf代数$A^J$的$n$次Sweedler幂映射.我们使用这种关系给出几个不变量的代表类别代表$(A)$被认为是一个张量类别。作为应用，我们区分了12维点非半单Hopf代数的表示范畴.此外，这些不变量足以区分表示类别Rep $(K_8)$、Rep $(\kk Q_8)$和Rep $(\kk D_4)$，尽管它们已经通过它们的Frobenius-Schur指示符完全区分。我们进一步揭示了$A^*$中的右积分$λ$和$(A^J)^*$中的$λ^J$之间的关系。这可以用来给出一个显著结果的统一证明，该结果说，$n$ -th指示符$ν_n(A)$是$A$的规范不变量，对于任何$n\in \mathbb{Z}$。我们还使用$λ^J$的表达式给出了已知结果的另一种证明，即Hopf代数$A$的Killing形式在扭曲下是不变的。因此，$A$的Killing根的维数是$A$的规范不变量。
英文摘要：
For a finite-dimensional Hopf algebra $A$ with a nonzero left integral $Λ$, we investigate a relationship between $P_n(Λ)$ and $P_n^J(Λ)$, where $P_n$ and $P_n^J$ are respectively the $n$-th Sweedler power maps of $A$ and the twisted Hopf algebra $A^J$. We use this relation to give several invariants of the representation category Rep$(A)$ considered as a tensor category. As applications, we distinguish the representation categories of 12-dimensional pointed nonsemisimple Hopf algebras. Also, these invariants are sufficient to distinguish the representation categories Rep$(K_8)$, Rep$(\kk Q_8)$ and Rep$(\kk D_4)$, although they have been completely distinguished by their Frobenius-Schur indicators. We further reveal a relationship between the right integrals $λ$ in $A^*$ and $λ^J$ in $(A^J)^*$. This can be used to give a uniform proof of the remarkable result which says that the $n$-th indicator $ν_n(A)$ is a gauge invariant of $A$ for any $n\in \mathbb{Z}$. We also use the expression for $λ^J$ to give an alternative proof of the known result that the Killing form of the Hopf algebra $A$ is invariant under twisting. As a result, the dimension of the Killing radical of $A$ is a gauge invariant of $A$.
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