Constructing new braided T-categories over monoidal Hom-Hopf algebras

Abstract

Let \documentclass[12pt]{minimal}\begin{document}${\sl Aut}_{mHH}(H)$\end{document}AutmHH(H) denote the set of all automorphisms of a monoidal Hopf algebra H with bijective antipode in the sense of Caenepeel and Goyvaerts [“Monoidal Hom-Hopf algebras,” Commun. Algebra 39, 2216–2240 (2011)] and let G be a crossed product group \documentclass[12pt]{minimal}\begin{document}${\sl Aut}_{mHH}(H)\times {\sl Aut}_{mHH}(H)$\end{document}AutmHH(H)×AutmHH(H). The main aim of this paper is to provide new examples of braided T-category in the sense of Turaev [“Crossed group-categories,” Arabian J. Sci. Eng., Sect. C 33(2C), 483–503 (2008)]. For this purpose, we first introduce a class of new categories \documentclass[12pt]{minimal}\begin{document}$_{H}\mathcal {MHYD}^{H}(A, B)$\end{document}MHYDHH(A,B) of (A, B)-Yetter-Drinfeld Hom-modules with \documentclass[12pt]{minimal}\begin{document}$A , B \in {\sl Aut}_{mHH}(H)$\end{document}A,B∈AutmHH(H). Then we construct a category \documentclass[12pt]{minimal}\begin{document}${\cal MHYD}(H) =\lbrace {}_{H}\mathcal {MHYD}^{H}(A, B)\rbrace _{(A , B )\in G}$\end{document}MHYD(H)={MHYDHH(A,B)}(A,B)∈G and show that such category forms a new braided T-category, generalizing the main constructions by Panaite and Staic [“Generalized (anti) Yetter-Drinfel'd modules as components of a braided T-category,” Isr. J. Math. 158, 349–366 (2007)]. Finally, we compute an explicit new example of such braided T-categories.

Drinfel’d double for monoidal Hom-Hopf algebras

“…In what follows, we will recall from [1,4,5,7] the definitions of monoidal Hom-associative algebras, monoidal Hom-coassociative coalgebras, monoidal Hom-modules and monoidal Hom-comodules. The reader could also refer to [14,16,17].…”

Drinfel’d construction for Hom–Hopf T-coalgebras

“…For this, Panaite and Staic [10] gave some interesting constructions in the case of Hopf algebra. Motivated by this, Yang [19], [20], Liu [4] and You [21] generalized their work to multiplier Hopf algebra, weak Hopf algebra and monoidal Hom-Hopf algebra cases, also the authors obtain some new classes of braided crossed categories which provided some solutions to the quantum Yang-Baxter equation over π.…”

Hom-Yang-Baxter equations and Hom-Yang-Baxter systems

“…In [29,31] Yau proposed the definition of quasitriangular Hom-Hopf algebras and showed that each quasitriangular Hom-Hopf algebra yields a solution of the Hom-Yang-Baxter equation. Meanwhile, several classes of solutions of the Hom-Yang-Baxter equation were constructed from different respects, including those associated to Hom-Lie algebras [5,25,29,30], Drinfeld (co)doubles [2,34,35] and Hom-Yetter-Drinfeld modules [3,10,13,14,18,26,33].…”