Center construction and duality of category of Hom-Yetter-Drinfeld modules over monoidal Hom-Hopf algebras

Abstract

*Abstract:Let (H,α) be a monoidal Hom-Hopf algebra. In this paper, we will study the category of Hom-Yetter-Drinfeld modules. First, we show that the category of left-left Hom-Yetter-Drinfeld modules \begin{array}{l}H\\ H\end{array}HYD is isomorphic to the center of the category of left (H,α)-Hom-modules. Also, by the center construction, we get that the categories of left-left, left-right, right-left, and right-right Hom-Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Second, we prove that the category of finitely generated projective left-left Hom-Yetter-Drinfeld modules has left and right duality.

Keywords

Monoidal Hom-Hopf algebra / Hom-Yetter-Drinfeld module / center construction / duality

References

[1] Bulacu D, Caenepeel S, Panaite F. Yetter-Drinfeld categories for quasi-Hopf algebras. Comm Algebra, 2006, 34: 1–35

[2] Caenepeel S, Goyvaerts I. Monoidal Hom-Hopf algebras. Comm Algebra, 2011, 39: 2216–2240 (2011)

[3] Caenepeel S, Wang D G, Yin Y M. Yetter-Drinfeld modules over weak bialgebras. Ann Univ Ferrara Sez VII Sci Mat, 2005, 51: 69–98

[4] Chen Y Y, Wang Z W, Zhang L Y. Integrals for monoidal Hom-Hopf algebras and their applications. J Math Phys, 2013, 54: 073515

[5] Gohr A. On Hom-algebras with surjective twisting. J Algebra, 2010, 324: 1483–1491

[6] Joyal A, Street R. Tortile Yang-Baxter operators in tensor categories. J Pure Appl Algebra, 1991, 71: 43–51

[7] Joyal A, Street R. Braided tensor categories. Adv Math, 1993, 102: 20–78

[8] Kassel C. Quantum Groups. Grad Texts in Math, Vol 155. Berlin: Springer, 1995

[9] Liu L, Shen B L. Radford’s biproducts and Yetter-Drinfeld modules for monoidal Hom-Hopf algebras. J Math Phys, 2014, 55: 031701

[10] Majid S. Representations, duals and quantum doubles of monoidal categories. Rend Circ Mat Palermo (2) Suppl, 1991, 26: 197–206

[11] Makhlouf A, Panaite F. Yetter-Drinfeld modules for Hom-bialgebras. J Math Phys, 2014, 55: 013501

[12] Makhlouf A, Silvestrov S D. Hom-algebras structures. J Gen Lie Theory Appl, 2008, 2: 52–64

[13] Makhlouf A, Silvestrov S D. Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras. In: Silvestrov S, Paal E, Abramov V, Stolin A, eds. Generalized Lie Theory in Mathematics, Physics and Beyond. Berlin: Springer, 2009, 189–206

[14] Makhlouf A, Silvestrov S D. Hom-algebras and Hom-coalgebras. J Algebra Appl, 2010, 9(4): 553–589

[15] Montgomery S. Hopf Algebras and Their Actions on Rings. CBMS Reg Conf Ser Math, No 82. Providence: Amer Math Soc, 1993

[16] Yau D. Hom-algebras and homology. J Lie Theory, 2009, 19: 409–421

[17] Yau D. Hom-bialgebras and comodule algebras. Int Electron J Algebra, 2010, 8: 45–64

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Higher Education Press and Springer-Verlag Berlin Heidelberg