Year: 2026
Type: preprint
Abstract: This paper studies commutators in finite-dimensional unimodular Hopf algebras and invariants derived from them. For a unimodular Hopf algebra $H$ with the distinguished group-like element $g$, the commutator of $a,b\in H$ is defined by $\{a,b\}=a_{1}S^{2}(b_{1})S(a_{2})S^{-1}(b_{2})g^{-1}$. We show that the span of all commutators is a left coideal and generates a normal left coideal subalgebra $N$ of $H$ for which $H/HN^{+}$ is commutative. The main result asserts that for any central element $a$ and any integral $Λ$ of $H$, the commutator $\{a,Λ\}$ and the value $λ(\{a,Λ\})$ for a right integral $λ$ of $H^*$ are invariant under twisting. This yields several gauge invariants, including ratios of characters of iterated commutators with $Λ$. In the semisimple characteristic-zero case, explicit descriptions of these invariants are given. As applications, we compute some invariants for the group algebra of a finite group, its Drinfeld double, and the restricted quantum group $\overline{U}_{q}(sl_{2})$.
Source: arXiv (Cornell University)
Authors: Wang, Zhihua, Li, Libin
Institutions: Taizhou University, Taizhou University, Yangzhou University
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3660190117
OpenAlex ID: W7214019412