NEW TURAEV BRAIDED GROUP CATEGORIES OVER ENTWINING STRUCTURES

作者：Wang Shuanhong^*

来源：Communications in Algebra, 2010, 38(3): 1019-1049.

DOI：10.1080/00927870902828736

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

Starting from the category (A)M(C)(psi) 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 psi : A circle times C -> A circle times C satisfying a mixed distributive law. Associated with any set map Psi : pi -> E(A, C), where E(A, C) denotes the set of all k-linear maps psi : A circle times C such that psi is an entwining map, and with the opposite group S(pi) of the semidirect product of the opposite group pi(op) of a group pi by pi, we introduce a Turaev S(pi)-category (A)M(S(pi))(C)(Psi) as a disjoint union of family of categories {(A)M(C)(psi(alpha, beta))}((alpha, beta)is an element of S(pi)) of entwined modules with a family of entwining maps {psi(alpha, beta)}((alpha, beta)is an element of S(pi)). We show that (A)M(S(pi))(C)(Psi) is a Turaev braided S(pi)-category if and only if there is a linear map @ : C circle times C -> A circle times A satisfying some conditions. For finite dimensional C (resp., A) there is a quasitriangular Turaev S(pi)-coalgebra C * A = {A#C*(alpha, beta)}((alpha, beta)is an element of S()pi) (resp., a coquasitriangular Turaev S(pi)-algebra A* x C = {A* x C(alpha, beta)}((alpha, beta)is an element of S(pi)) such that the category Rep (C * A) of representations of C * A is equivalent to (A)M(S(pi))(C)(Psi) (resp. the category Corep (A* x C) of corepresentations of A* x C is equivalent

• 出版日期2010
• 单位东南大学