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

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.