The locally compact (l.c.) quantum group is a relatively new C*-algebraic formalism for quantum groups, generalizing the Kac algebra, compact quantum group and Hopf 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 the Haar 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 algebras.
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