Locally compact quantum group
The locally compact (l.c.) quantum group is a relatively new
C*-algebraic formalism for quantum groups, generalizing the Kac algebra, compact quantum groupand Hopf algebraapproaches. 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 dualityof abelian groups.
The theory has an equivalent formulation in terms of
von Neumann algebras.
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