1998/03/02 by B. Drabant, Bernhard Drabant, Alfons Van Daele +5 · 1 citation
Mathematics · #16D10 #16W30 #17B37 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #math.OA #math.QA #msc:16D10 #msc:16W30 #msc:17B37
paper · pdf · doi:10.48550/arxiv.math/9803005
55 pages, AMS-TeX
arxiv created 1998/03/02 · openalex publication_date 1998/03/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an action α of a group G on an algebra R (over \Bbb C), the crossed product R×αG is the vector space of R-valued functions with finite support in G, together with the twisted convolution product given by (ξη)(p) = ∑q ∈ G ξ(q) αq (η(q-1p)) where p∈ G. This construction has been extended to the theory of Hopf algebras. Given an action of a Hopf algebra A on an algebra R, it is possible to make the tensor product R\ot A into an algebra by using a twisted product, involving the action. In this case, the algebra is called the smash product and denoted by R# A. In the group case, the action α of G on R yields an action of the group algebra \Bbb C G as a Hopf algebra on R and the crossed R×αG coincides with the smash product R# \Bbb C G. In this paper we extend the theory of actions of Hopf algebras to actions of multiplier Hopf algebras. We also construct the smash product and we obtain results very similar as in the original situation for Hopf algebras. The main result in the paper is a duality theorem for such actions. We consider dual pairs of multiplier Hopf algebras to formulate this duality theorem. We prove a result in the case of an algebraic quantum group and its dual. The more general case is only stated and will be proven in a separate paper on coactions. These duality theorems for actions are substantial generalizations of the corresponding theorem for Hopf algebras. Also the techniques that are used here to prove this result are slightly different and simpler.