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Discrete Quantum Groups

✍ Scribed by A. Van Daele

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
162 KB
Volume
180
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let G be any discrete group. Consider the algebra A of all complex functions with finite support on G with pointwise operations. The multiplication on G Ž .Ž . Ž . induces a comultiplication ⌬ on A by ⌬ f p, q sf pq whenever f g A and p, q g G. If G is finite, one can identify the algebra of complex functions on

A for all f and g. In Ž . this case A, ⌬ is a multiplier Hopf algebra. In fact, it is a multiplier Hopf U U Ž .

-algebra when A is given the natural involution defined by f p s f p for all Ž .

U

Ž

. fgA and p g G. In this paper we call a multiplier Hopf -algebra A, ⌬ a discrete quantum group if the underlying U -algebra A is a direct sum of full matrix algebras. We study these discrete quantum groups and we give a simple proof of the existence and uniqueness of a left and a right invariant Haar measure.

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