Invariants from the Sweedler power maps on integrals

TL;DR

This paper explores invariants derived from Sweedler power maps on integrals in finite-dimensional Hopf algebras, providing tools to distinguish representation categories and proving gauge invariance of certain algebraic properties.

Contribution

It introduces new invariants from Sweedler power maps that distinguish representation categories and proves gauge invariance of the n-th indicator and the Killing form.

Findings

Distinguished representation categories of specific Hopf algebras.

Established gauge invariance of the n-th indicator.

Proved invariance of the Killing form under twisting.

Abstract

For a finite-dimensional Hopf algebra $A$ with a nonzero left integral $\Lambda$, we investigate a relationship between $P_n(\Lambda)$ and $P_n^J(\Lambda)$, 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 $\lambda$ in $A^*$ and $\lambda^J$ in $(A^J)^*$. This can be used to give a uniform proof of the remarkable result which says that the $n$-th indicator $\nu_n(A)$ is a gauge invariant of $A$ for any $n\in \mathbb{Z}$. We also use the expression for $\lambda^J$ 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 $A$.