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The Drinfel’d double for group-cograded multiplier Hopf algebras. (English) Zbl 1161.16028
Summary: 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 $$M(K(G)\otimes K(G))$$ 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 $$\mathbb{C}$$) such that there is an embedding $$I\colon K(G)\to M(B)$$. 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 $$\pi$$ 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 $$D^\pi$$ where the coproduct and the product depend on the action $$\pi$$. We also treat the *-algebra case. If $$\pi$$ 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.

MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16S40 Smash products of general Hopf actions
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References:
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