论文标题
Sweedler Power图的不变性在积分上
Invariants from the Sweedler power maps on integrals
论文作者
论文摘要
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 $A^J$.我们使用此关系来提供表示为张量类别的表示类别的几个不变性。作为应用，我们区分了12维指向非hopf代数的表示类别。同样，这些不变性足以区分表示类别的代表$（k_8）$，rep $（\ kk q_8）$和rep $（\ kk d_4）$，尽管它们已经完全通过其Frobenius-Schur指标区分了。我们进一步揭示了正确的积分之间的关系$ a^*$中的$λ$和$（a^j）^*$的$λ^j $。这可以用来给出一个显着结果的统一证明，该证明说$ n $ th指示器$ν_n（a）$是任何$ n \ in \ mathbb {z} $的$ a $的规格不变性。我们还使用$λ^j $的表达式来提供一个已知结果的替代证明，即Hopf代数$ a $的杀戮形式在扭曲下是不变的。结果，$ a $的杀戮激进分子的尺寸是$ a $的规范。
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 $A$.
