Quasi-quantum groups obtained from tensor braided Hopf algebras

Abstract

Let H be a quasi-Hopf algebra, $${}_H^H{{{\mathcal {M}}}}_H^H$$ the category of two-sided two-cosided Hopf modules over H and $${}_H^H{\mathcal YD}$$ the category of left Yetter–Drinfeld modules over H. We show that $${}_H^H{{{\mathcal {M}}}}_H^H$$ admits a braided monoidal structure for which the strong monoidal equivalence $${}_H^H{\mathcal M}_H^H\cong {}_H^H{\mathcal YD}$$ established by the structure theorem for quasi-Hopf bimodules becomes braided monoidal. Using this braided monoidal equivalence, we prove that Hopf algebras within $${}_H^H{{{\mathcal {M}}}}_H^H$$ can be characterized as quasi-Hopf algebras with a projection or as biproduct quasi-Hopf algebras in the sense of Bulacu and Nauwelaerts (J Pure Appl Algebra 174:1–42, 2002) . A particular class of such (braided, quasi-) Hopf algebras is obtained from a tensor product Hopf algebra type construction. Our arguments rely on general categorical facts.