In mathematics and theoretical physics, a locally compact quantum group is a C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unitaries have enjoyed some success but have also encountered several technical problems.

One of the main features distinguishing this new approach from its predecessors is the axiomatic existence of left and right invariant weights. This gives a noncommutative analogue of left and right Haar measures on a locally compact Hausdorff group.

Definitions

Before we can even begin to properly define a locally compact quantum group, we first need to define a number of preliminary concepts and also state a few theorems.

Definition (weight). be a C*-algebra denote the set of positive elements. A weight such that

• , and

• .

Some notation for weights.. We use the following notation:

• , which is called the set of all -integrable elements.

• , which is called the set of all -square-integrable elements.

• , which is called the set of all -integrable.

Types of weights..

• is faithful.

• is lower semi-continuous.

• is densely defined is a dense subset.

• is proper if and only if it is non-zero, lower semi-continuous and densely defined.

Definition (one-parameter group). be a C*-algebra. A one-parameter group is norm-continuous is continuous.

Definition (analytic extension of a one-parameter group)., we are going to define an analytic extension, let

1. ,

norm-regular if and only if the following conditions hold:

• .

• .

• .

, and let

is then called the analytic extension.

Theorem 1., called the set of analytic elements.

Definition (K.M.S. weight). is a K.M.S. weight is a proper weight such that

• , and

• .

the multiplier algebra.

Theorem 2..

Theorem 3. is a state (i.e., a positive linear functional.

Definition (Locally compact quantum group). A (C*-algebraic) locally compact quantum group is a non-degenerate *-homomorphism called the co-multiplication, that satisfies the following four conditions:

• .

• .

• .

• .

is a redundant condition and does not need to be postulated.

Duality

The category of locally compact quantum groups allows for a dual construction with which one can prove that the bi-dual of a locally compact quantum group is isomorphic to the original one. This result gives a far-reaching generalization of Pontryagin duality for locally compact Hausdorff abelian groups.

Alternative formulations

The theory has an equivalent formulation in terms of von Neumann algebras.

See also

• Locally compact space

• Locally compact field

• Locally compact group

References

• Johan Kustermans & Stefaan Vaes. "Locally Compact Quantum Groups." Annales Scientifiques de l’École Normale Supérieure. Vol. 33, No. 6 (2000), pp.837–934.

• Thomas Timmermann. "An Invitation to Quantum Groups and Duality – From Hopf Algebras to Multiplicative Unitaries and Beyond." EMS Textbooks in Mathematics, European Mathematical Society (2008).