Locally compact quantum group


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. Let be a C*-algebra, and let denote the set of positive elements of. A weight on is a function such that
Some notation for weights. Let be a weight on a C*-algebra. We use the following notation:
  • , which is called the set of all positive -integrable elements of.
  • , which is called the set of all -square-integrable elements of.
  • , which is called the set of all -integrable elements of.
Types of weights. Let be a weight on a C*-algebra.
Definition. Let be a C*-algebra. A one-parameter group on is a family of *-automorphisms of that satisfies for all. We say that is norm-continuous if and only if for every, the mapping defined by is continuous.
Definition. Given a norm-continuous one-parameter group on a C*-algebra, we are going to define an analytic extension of. For each, let
which is a horizontal strip in the complex plane. We call a function norm-regular if and only if the following conditions hold:
  • It is analytic on the interior of, i.e., for each in the interior of, the limit exists with respect to the norm topology on.
  • It is norm-bounded on.
  • It is norm-continuous on.
Suppose now that, and let
Define by. The function is uniquely determined, so is well-defined indeed. The family is then called the analytic extension of.
Theorem 1. The set, called the set of analytic elements of, is a dense subset of.
Definition. Let be a C*-algebra and a weight on. We say that is a K.M.S. weight on if and only if is a proper weight on and there exists a norm-continuous one-parameter group on such that
  • is invariant under, i.e., for all, and
  • for every, we have.
We denote by the multiplier algebra of.
Theorem 2. If and are C*-algebras and is a non-degenerate *-homomorphism, then we can uniquely extend to a *-homomorphism.
Theorem 3. If is a state on, then we can uniquely extend to a state on.
Definition. A locally compact quantum group is an ordered pair, where is a C*-algebra and is a non-degenerate *-homomorphism called the co-multiplication, that satisfies the following four conditions:
  • The co-multiplication is co-associative, i.e.,.
  • The sets and are linearly dense subsets of.
  • There exists a faithful K.M.S. weight on that is left-invariant, i.e., for all and.
  • There exists a K.M.S. weight on that is right-invariant, i.e., for all and.
From the definition of a locally compact quantum group, it can be shown that the right-invariant K.M.S. weight is automatically faithful. Therefore, the faithfulness of 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.