The Size of the Intrinsic Group of a Multiplier Hopf Algebra

Abstract

In the theory of Hopf algebras, grouplike elements are well studied. We argue that this notion is too restrictive when dealing with multiplier Hopf algebras. We introduce the “intrinsic” group, situated in the multiplier algebra of the multiplier Hopf algebra. When dealing with an algebraic quantum group A, the intrinsic group of the dual  characterizes a special class of automorphisms on A. For multiplier Hopf algebras which are paired in the sense of Drabant and Van Daele. (Drabant, B., Van Daele, A. Pairing and quantum double for multiplier Hopf algebras. (2001). Algebras and Representation Theory 4:109–132), we prove that the intrinsic group characterizes the semi-invariants of the action associated to the pairing.

Key Words:

• Grouplike elements
• Semi-invariants

Mathematics Subject Classification (1991):

• Primary 16W30

Acknowledgment

The author would like to thank A. Van Daele for many helpful discussions during the preparation of this paper.