Quasi-quantum groups obtained from tensor braided Hopf algebras

Let H be a quasi-Hopf algebra, HHMHH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_H^H{{{\mathcal {M}}}}_H^H$$\end{document} the category of two-sided two-cosided Hopf modules over H and HHYD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_H^H{\mathcal YD}$$\end{document} the category of left Yetter–Drinfeld modules over H. We show that HHMHH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_H^H{{{\mathcal {M}}}}_H^H$$\end{document} admits a braided monoidal structure for which the strong monoidal equivalence HHMHH≅HHYD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_H^H{\mathcal M}_H^H\cong {}_H^H{\mathcal YD}$$\end{document} established by the structure theorem for quasi-Hopf bimodules becomes braided monoidal. Using this braided monoidal equivalence, we prove that Hopf algebras within HHMHH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_H^H{{{\mathcal {M}}}}_H^H$$\end{document} 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.