# The Drinfel'd double for group-cograded multiplier Hopf algebras

Research paper by **L. Delvaux, A. Van Daele**

Indexed on: **02 Apr '04**Published on: **02 Apr '04**Published in: **Mathematics - Quantum Algebra**

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#### 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 $M(K(G) \otimes K(G))$ 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 $\Bbb C$) such that there is an embedding
$I: K(G) \to M(B)$. 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 {\it $G$-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 {\it 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 $<A, B>$. 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.