Discrete Quantum Groups Original Research Article

LetGbe any discrete group. Consider the algebraAof all complex functions with finite support onGwith pointwise operations. The multiplication onGinduces a comultiplication Δ onAby (Δf)(p, q)=f(pq) wheneverfset membership, variantAandp, qset membership, variantG. IfGis finite, one can identify the algebra of complex functions onG×GwithAcircle times operatorAso that Δ: A→Acircle times operatorA. Then (A, Δ) is a Hopf algebra. IfGis infinite, we still have Δ(f)(gcircle times operator1) and Δ(f)(1circle times operatorg) inAcircle times operatorAfor allfandg. In this case (A, Δ) is a multiplier Hopf algebra. In fact, it is a multiplier Hopf *-algebra whenAis given the natural involution defined by[formula]for allfset membership, variantAandpset membership, variantG. In this paper we call a multiplier Hopf *-algebra (A, Δ)adiscrete quantum group if the underlying *-algebraAis 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.