Locally compact quantum group

The locally compact quantum group is a relatively new C*-algebraic formalism for quantum group

Quantum group

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...

s, generalizing the Kac algebra, compact quantum group

Compact quantum group

In mathematics, a compact quantum group is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of "continuous complex-valued functions" on a compact quantum group....

 and Hopf algebra

Hopf algebra

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...

 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

Noncommutative topology

Noncommutative topology in mathematics is a term applied to the strictly C*-algebraic part of the noncommutative geometry program. The program has its origins in the Gel'fand duality between the topology of locally compact spaces and the algebraic structure of commutative C*-algebras.Several...

 analogue of the Haar measure

Haar measure

In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....

.

The category of l.c. quantum groups allow for a dual construction, generalizing the Pontryagin duality

Pontryagin duality

In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...

 of abelian groups.

The theory has an equivalent formulation in terms of von Neumann algebra

Von Neumann algebra

In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...

s.