The Drinfel'd double versus the Heisenberg double for Hom-Hopf algebras. (Q2797015)

For a finite-dimensional Hopf algebra \(A\), the Drinfeld double \(D(A)\) is a quasitriangular Hopf algebra whose braided monoidal module category gives rise to solutions of the quantum Yang-Baxter equation. Looking for further solutions, the authors study Hom-Hopf algebras and introduce a Drinfeld double for them whose module category is braided monoidal. They establish relations between this Drinfeld double and the Heisenberg double in this setting.NEWLINENEWLINE Hom-Hopf algebras were introduced by \textit{S. Caenepeel} and \textit{I. Goyvaerts} [Commun. Algebra 39, No. 6, 2216-2240 (2011; Zbl 1255.16032)]. A unital Hom-associative algebra is a (not necessarily associative) algebra \(A\) with a distinguished element 1 (not necessarily a unit element of \(A\)), and a multiplicative map \(f\) from \(A\) to \(A\) such that \(f(1)=1\), \(1a=f(a)=a1\) and \(f(a)bc=abf(c)\) for \(a,b,c\) in \(A\). Similar definitions are given for Hom-coalgebras, Hom-bialgebras and Hom-Hopf-algebras \((A,f)\). The Drinfeld double of \((A,f)\) is constructed by generalizing \textit{S. Majid}'s bicrossed product [J. Algebra 130, No. 1, 17-64 (1990; Zbl 0694.16008)] after obtaining a Hom-Hopf algebra structure on \(A^*\). The Heisenberg double \(H(A,f)\) involves a Hom-2-cocycle on \(D(A,f)\) and is the smash product \(A\#A^*\) with respect to the left regular action of \(A^*\) on \(A\). One relation is that \(H(A^{op})\) is the left twist of \(D(A)\) with explicitly described Hom-2-cocycle. -- Two examples are given which are not Hopf algebras. One involves a finite group and an automorphism of it. The other is a Hom-version of Sweedler's four dimensional Hopf algebra.