# Locally compact quantum group

The

**locally compact (l.c.) quantum group**is a relatively newC*-algebra ic formalism forquantum group s, generalizing theKac algebra ,compact quantum group andHopf algebra approaches. Earlier attempts of a unifying definition of quantum groups using e.g. multiplicative unitaries have had some success, but have also ran into several technical problems.One of the main features distinguishing it from other approaches is the axiomatic existence of an

invariant weight , giving a noncommutative analogue of theHaar measure .The category of l.c. quantum groups allow for a dual construction, generalizing the

Pontryagin duality of abelian groups.The theory has an equivalent formulation in terms of

von Neumann algebra s.

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