The Drinfel’d Double for Group-cograded Multiplier Hopf Algebras

Abstract

Let G be any group and let K(G) denote the multiplier Hopf algebra of complex functions with finite support in G. The product in K(G) is pointwise. The comultiplication on K(G) is defined with values in the multiplier algebra M(K(G) ⊗K(G )) by the formula \((\Delta(f\,)) (p,q)\!\! =\!\! f(pq)\) for all \(f\! \in\! K(G)\) and \(p, q \!\in\! G\). In this paper we consider multiplier Hopf algebras B (over \(\Bbb C\)) such that there is an embedding I: K(G) →M(B). This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps K(G) into the center of M(B). These multiplier Hopf algebras are called G-cograded multiplier Hopf algebras. They are a generalization of the Hopf group-coalgebras as studied by Turaev and Virelizier. In this paper, we also consider an admissible action π of the group G on a G-cograded multiplier Hopf algebra B. When B is paired with a multiplier Hopf algebra A, we construct the Drinfel’d double Dπ where the coproduct and the product depend on the action π. We also treat the *-algebra case. If π is the trivial action, we recover the usual Drinfel’d double associated with the pair \(\langle A, B \rangle\). On the other hand, also the Drinfel’d double, as constructed by Zunino for a finite-type Hopf group-coalgebra, is an example of the construction above. In this case, the action is non-trivial but related with the adjoint action of the group on itself. Now, the double is again a G-cograded multiplier Hopf algebra.