Gauge invariants raised from commutators of unimodular Hopf algebras
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∈ H is defined by {a,b}=a1S2(b1)S(a2)S−1(b2)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 U̅q(sl2).
Related
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- Single commutators in C*-algebras
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- From gauge-covariant Noether invariants to electromagnetic velocity-dependent mechanics
BibTeX
@misc{wang2026gauge,
title = {{Gauge invariants raised from commutators of unimodular Hopf algebras}},
author = {Zhihua Wang and Libin Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24523},
eprint = {2609.24523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24523},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gauge invariants raised from commutators of unimodular Hopf algebras
AU - Zhihua Wang
AU - Libin Li
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24523
UR - https://arxiv.org/abs/2609.24523
AB - 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∈ H is defined by {a,b}=a_1S^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 U̅_q(sl_2).
ER -