Discrete quantum groups

Abstract

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 induces a comultiplication Δ on A by (Δf)(p, q) = f(pq) whenever f ∈ A and p, q ∈ G. If G is finite, one can identify the algebra of complex functions on G × G with A ⊗ A so that Δ: A → A ⊗ A. Then (A, Δ) is a Hopf algebra. If G is infinite, we still have Δ(f)(g ⊗ 1) and Δ(f)(1 ⊗ g) in A ⊗ A for all f and g. In this case (A, Δ) is a multiplier Hopf algebra. In fact, it is a multiplier Hopf * -algebra when A is given the natural involution defined by f*(p) = f(p) for all f ∈ A and p ∈ G. In this paper we call a multiplier Hopf *-algebra (A, Δ) a discrete quantum group if the underlying *-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. © 1996 Academic Press, Inc.