Double Construction for Crossed Hopf Coalgebras

Recently, Turaev [19, 20] (see also Le and Turaev [8] and Virelizier [23]) generalized the notion of a TQFT and Reshetikhin-Turaev invariants to the case of 3-manifolds endowed with a homotopy classes of maps to K(π, 1), where π is a group. One of the key points in [19] is the notion of a crossed Hopf π-coalgebra, here called a Turaev coalgebra or, briefly, a T-coalgebra (see Section 1). As one can use categories of representations of modular Hopf algebras to construct Reshetikhin-Turaev invariants of 3-manifolds, one can use categories of representations of modular T-coalgebras to construct homotopy invariants of maps from 3-manifolds to K(π, 1). Similarly, to construct Virelizier Hennings-like homotopy invariants [23], we need a ribbon T-coalgebra H such that the neutral component H1 is unimodular [15]. Roughly speaking, a T-coalgebra H is a family {Hα}α∈π of algebras endowed with a comultiplication ∆α,β : Hαβ → Hα⊗Hβ , a counit ε : k → H1 (where 1 is the neutral element of π), and an antipode sα : Hα → Hα−1 . It is also required that H is endowed with a family of algebra isomorphisms φβ = φβ : Hα → Hβαβ−1 , the conjugation, compatible with the above structures and such that φβγ = φβ ◦ φγ . When π = {1}, we recover the definition of a Hopf algebra. A T-coalgebra H is of finite type when every Hα is finite-dimensional. H is totally-finite when the direct sum ⊕