A Quasi-Pentagon Equation for a Heisenberg Double of a Quasi-Hopf Algebra

TL;DR

This paper extends the algebraic structures of Heisenberg doubles to the quasi-Hopf setting, establishing a quasi-pentagon equation and inverse-like elements despite non-invertibility issues.

Contribution

It constructs quasi-Hopf analogues of Heisenberg doubles and proves the existence of inverse-like elements satisfying key equations.

Findings

Canonical elements satisfy a quasi-pentagon equation.

Existence of inverse-like elements despite non-invertibility.

Roles of elements are reversed in different quasi-Hopf doubles.

Abstract

For a finite-dimensional Hopf algebra $H$, the canonical elements of the Heisenberg doubles $\mathcal{H}(H^\ast)$ and $\mathcal{H}(H)$ satisfy the pentagon and Hopf equations, respectively. In this paper we construct quasi-Hopf analogues of these structures. For a finite-dimensional quasi-Hopf algebra $H$, we consider natural quasi-Hopf analogues $\mathcal{H}_1(H^\ast)$ and $\mathcal{H}_1(H)$ of $\mathcal{H}(H^\ast)$ and $\mathcal{H}(H)$. Although their canonical elements are defined just as in the Hopf algebra case, they need not be invertible. We prove that there nevertheless exist natural inverse-like elements. In $\mathcal{H}_1(H^\ast)$, the canonical element satisfies a quasi-pentagon equation and its inverse-like element satisfies a quasi-Hopf equation, while in $\mathcal{H}_1(H)$ the roles are reversed.