New Turaev Braided Group Categories Over Entwining Structures

Abstract

Starting from the category _A ℳ ^C (ψ) of entwined modules, that is, A-modules and C-comodules over Hopf algebras A and C, where the structures are only related by an entwining map ψ: A ⊗ C → A ⊗ C satisfying a mixed distributive law. Associated with any set map Ψ: π → 𝔼(A, C), where 𝔼(A, C) denotes the set of all k-linear maps ψ: A ⊗ C → A ⊗ C such that ψ is an entwining map, and with the opposite group 𝒮(π) of the semidirect product of the opposite group π ^op of a group π by π, we introduce a Turaev 𝒮(π)-category as a disjoint union of family of categories { _A ℳ ^C (ψ(α, β))}_{(α, β)∈𝒮(π)} of entwined modules with a family of entwining maps {ψ(α, β)}_{(α, β)∈𝒮(π)}. We show that is a Turaev braided 𝒮(π)-category if and only if there is a linear map 𝒬: C ⊗ C → A ⊗ A satisfying some conditions. For finite dimensional C (resp., A) there is a quasitriangular Turaev 𝒮(π)-coalgebra C*A = {A#C*(α, β)}_{(α, β)∈𝒮(π)} (resp., a coquasitriangular Turaev 𝒮(π)-algebra A* × C = {A* × C(α, β)}_{(α, β)∈𝒮(π)}) such that the category Rep(C*A) of representations of C*A is equivalent to (resp. the category Corep(A* × C) of corepresentations of A* × C is equivalent to ). Finally, we study dualities of any object in the category and the dual of the category itself.

Key Words:

• (Co)Quasitriangular structure
• Entwining structure
• Hopf algebra
• Smash (co)product
• Turaev braided group category

2000 Mathematics Subject Classification:

• 16W30

ACKNOWLEDGMENTS

The author would like to thank Vladimir Turaev for his constant encouragement and useful suggestions. He would like to thank the Indiana University of Bloomington for its warm hospitality and thanks M. D. Staic for his giving article [11] when he visited the University in January 2008. This work was partially supported by the NNSF of Jiangsu Province (No. BK2009258), the FNS of CHINA (10871042) and the EMSTRKF of China (108154).

Notes

Communicated by V. A. Artamonov.