In mathematics and theoretical physics, a locally compact quantum group is a relatively new 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.
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).
denote the set of positive elements of
We use the following notation: Types of weights.
Definition (one-parameter group).
A one-parameter group on
is continuous (surely this should be called strongly continuous?).
Definition (analytic extension of a one-parameter group).
Given a norm-continuous one-parameter group
, we are going to define an analytic extension of
, let which is a horizontal strip in the complex plane.
We call a function
norm-regular if and only if the following conditions hold: Suppose now that
is uniquely determined (by the theory of complex-analytic functions), so
is then called the analytic extension of
, called the set of analytic elements of
and there exists a norm-continuous one-parameter group
the multiplier algebra of
is a state (i.e., a positive linear functional of norm
Definition (Locally compact quantum group).
A (C*-algebraic) locally compact quantum group is an ordered pair
is a non-degenerate *-homomorphism called the co-multiplication, that satisfies the following four conditions: From the definition of a locally compact quantum group, it can be shown that the right-invariant K.M.S.
is a redundant condition and does not need to be postulated.
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.
The theory has an equivalent formulation in terms of von Neumann algebras.