Invariants from the Sweedler power maps on integrals

Abstract

For a finite-dimensional Hopf algebra A with a nonzero left integral Λ, we investigate a relationship between Pn(Λ) and PnJ(Λ), where Pn and PnJ are respectively the n-th Sweedler power maps of A and the twisted Hopf algebra AJ. 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(K8), Rep(kQ8) and Rep(kD4), although they have been completely distinguished by their Frobenius-Schur indicators. We further reveal a relationship between the right integrals λ in A⁎ and λJ in (AJ)⁎. This can be used to give a uniform proof of the remarkable result which says that the n-th indicator νn(A) is a gauge invariant of A for any n∈Z. We also use the expression for λJ 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.