The cosemisimplicity and cobraided structures of monoidal comonads

Abstract

In this paper, we study the category of corepresentations of a monoidal comonad. We show that it is a semisimple category if and only if the monoidal comonad is a cosemisipmle (coseparable) comonad, and it is a braided category if and only if the monoidal comonad admit a cobraided structure. At last, as an application, the braided structure and the semisimplicity of the Hom-comodule category of a monoidal Hom-bialgebra are discussed.

Key words： Comonads; braided cateogries; monoidal Hom-Hopf algebras

References

1 MacLane S. Homologie des anneaux et des modules. Colloque de topologie algebrique, Louvain, 1956

2 Beck J. Distributive laws. in: Seminar on Triples and Categorical Homology Theory, B. Eckmann (ed.), Springer LNM, 1969 80, 119–140

3 Street R. The formal theory of monads. J. Pure Appl. Algebra, 1972, 2(2): 149–168

4 Blackwell R, Kelly M, Power J. Two-dimensional monad theory. J. Pure Appl. Algebra, 1989, 59(1): 1–41

5 Moerdijk I. Monads on tensor categories. J. Pure Appl. Algebra, 2002, 168(2–3): 189–208

6 Bruguières A, Virelizier A. Hopf monads. Adv. Math., 2007, 215(2): 679–733

7 Bruguières A, Virelizier A. Hopf monads on monoidal categories. Adv. Math., 2011, 227(2): 745–800

8 Mesablishvili B, Wisbauer R. Bimonads and Hopf Monads on Categories. J. K-theory, 2011, 7(2): 349–388

9 Böhm G, Brzezi`nski T, Wisbauer R. Monads and comands on module categories. J. Algebra, 2009, 322(5): 1719–1747

10 Power J, Watanabe H. Combining a monad and a comonad. Theor. Comput. Sci., 2002, 280(1–2): 137–162

11 Wang D G, Dai R X. Entwining structures of monads and comonads. Acta Mathematica Sinica, Chinese Series, 2008, 51(5): 927–932 (in Chinese)

12 Caenepeel S, Goyvaerts I. Monoidal Hom-Hopf algebras. Comm Algebra, 2011, 39: 2216–2240

13 Hartwig J, Larsson D, Silvestrov S. Deformations of Lie algebras using q-derivations. J. Algebra, 2006, 295: 314–361

14 Hu N H. q-Witt algebras, q-Lie algebras, q-holomorph structure and representations. Algebra Colloq., 1999, 6(1): 51–70

15 Makhlouf A, Silvestrov S. Hom-algebras structures. J. Gen. Lie Theory Appl., 2008, 2: 51–64

16 Yau D. Hom-quantum groups I: Quasitriangular Hom-bialgebras. J. Phys. A, 2012, 45(6): 065203

17 Makhlouf A, Silvestrov S. Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras. Generalized Lie theory in Mathematics. Physics and Beyond. Springer-Verlag, Berlin, 2008, Chp 17, 189–206

18 Wang Z W, Chen Y Y, Zhang L Y. The antipode and Drinfel’d double of Hom-Hopf algebras (in Chinese). Sci Sin Math, 2012, 42(11): 1079–1093

19 Frégier Y, Gohr A. On Hom-type algebras. J. Gen. Lie Theory Appl., 2010, 4: 1–16

20 Rafel M. Separable functors revisited. Comm. Algebra, 1990, 18: 1445–1459