Turaev group coalgebras and twisted Drinfeld double

Abstract

A new method of constructing Turaev group coalgebras with quasitriangular structure is introduced. Starting from a Hopf dual pairing (A, B, σ) with appropriate homomorphisms: φ: π → Aut(A) and (ψ: π → Aut(B), we construct a twisted Drinfeld double D(A,B,σ;φ,ψ) which is a Turaev S(π)-coalgebra, where the group S(π) is a twisted semi-direct square of π. Moreover, when A or B is finite-dimensional, we define a non-trivial quasitriangular structure on D (A, B, σ ; φ, ψ). This approach allows us to produce new examples of Turaev π-coalgebras for many infinite groups π such as GL n (k) and ( C * ) l with l ≥ 1.