New BraidedT-Categories over Weak Crossed Hopf Group Coalgebras

Xuan Zhou,Tao Yang

doi:10.1155/2013/626394

Abstract

LetHbe a weak crossed Hopf group coalgebra over groupπ; we first introduce a kind of newα-Yetter-Drinfel’d module categories𝒲𝒴𝒟α(H)forα∈πand use it to construct a braidedT-category𝒲𝒴𝒟(H). As an application, we give the concept of a Long dimodule categoryH𝒲ℒHfor a weak crossed Hopf group coalgebraHwith quasitriangular and coquasitriangular structures and obtain thatH𝒲ℒHis a braidedT-category by translating it into a weak Yetter-Drinfel'd module subcategory𝒲𝒴𝒟(H⊗H).

Highlights

Braided crossed categories over a group π, introduced by Turaev [1] in the study of 3-dimensional homotopy quantum field theories, are braided monoidal categories in Freyd-Yetter categories of crossed π-sets [2]
Recall from Van Daele and Wang that a weak semi-Hopf group coalgebra H = {Hα, mα, 1α, Δ, ε}α∈π is a family of algebras {Hα, mα, 1α}α∈π and at the same time a group coalgebra {Hα, Δ = {Δ α,β}, ε}α,β∈π, such that the following conditions hold
A weak Hopf group coalgebra over π is a weak semi-Hopf group coalgebra H = {Hα, mα, 1α, Δ, ε}α∈π endowed with a family of k-linear maps S = {Sα : Hα → Hα−1 }α∈π satisfying the following equations: mα (Sα−1 ⊗ idHα ) Δ α−1,α (h) = 1(1,α)ε (h1(2,1)), mα Δ α,α−1 (h) = ε (1(1,1)h) 1(2,α), (5)

Summary

Introduction

Braided crossed categories over a group π (i.e., braided T-categories), introduced by Turaev [1] in the study of 3-dimensional homotopy quantum field theories, are braided monoidal categories in Freyd-Yetter categories of crossed π-sets [2] Such categories play an important role in the construction of homotopy invariants. We mainly generalize the above constructions shown in [6, 8], replacing their Hopf group coalgebras (or Hopf group algebras) by weak crossed Hopf group coalgebras [11] and provide new examples of braided T-categories. We obtain a new category HWLH = ∐α∈π HWLHα and show that as H is a quasitriangular and coquasitriangular weak crossed Hopf group coalgebra, HWLH is a braided T-subcategory of Yetter-Drinfel’d category WYD(H ⊗ H)

Preliminary

Yetter-Drinfel’d Categories for Weak Crossed Hopf Group Coalgebras

Braided T-Categories over Weak Long Dimodule Categories