In mathematics, compact quantum groups are generalisations of compact groups, where the commutative \mathrm{C}^*-algebra of continuous complex-valued functions on a compact group is generalised to an abstract structure-algebra, which plays the role of the "algebra of continuous complex-valued functions on the compact quantum group".

The basic motivation for this theory comes from the following analogy. The space of complex-valued functions on a compact Hausdorff topological space forms a commutative C*-algebra. On the other hand, by the Gelfand Theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism.

S. L. Woronowicz introduced the important concept of compact matrix quantum groups, which he initially called compact pseudogroups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry.

Formulation

For a compact topological group, , there exists a C*-algebra homomorphism

where is the minimal C*-algebra tensor product the completion of the algebraic tensor product of and such that

, where

. There also exists a linear multiplicative mapping

1. ,

such that

. Strictly speaking, this does not make into a Hopf algebra, unless is finite.

On the other hand, a finite-dimensional representation of can be used to generate a *-subalgebra of which is also a Hopf *-algebra. Specifically, if

is an -dimensional representation of , then

for all , and

for all . It follows that the *-algebra for all for all is a Hopf *-algebra: the counit is determined by

is the Kronecker delta), the antipode is , and the unit is given by

Compact matrix quantum groups

As a generalization, a compact matrix quantum group is defined as a pair , where is a C*-algebra and

is a matrix with entries in such that

• The *-subalgebra, , of , which is generated by the matrix elements of , is dense in ;

• There exists a C*-algebra homomorphism, called the comultiplication, (here is the C*-algebra tensor product - the completion of the algebraic tensor product of and ) such that

• There exists a linear antimultiplicative map, called the coinverse, where is the identity element of . Since is antimultiplicative, .

As a consequence of continuity, the comultiplication on is coassociative.

In general, is a bialgebra, and is a Hopf *-algebra.

Informally, can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and can be regarded as a finite-dimensional representation of the compact matrix quantum group.

Compact quantum groups

For C*-algebras and acting on the Hilbert spaces and respectively, their minimal tensor product is defined to be the norm completion of the algebraic tensor product in ; the norm completion is also denoted by .

A compact quantum group is defined as a pair , where is a unital C*-algebra and

• is a unital *-homomorphism satisfying ;

• the sets and are dense in .

Representations

A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra Furthermore, a representation, v, is called unitary if the matrix for v is unitary, or equivalently, if

Example

An example of a compact matrix quantum group is , where the parameter is a positive real number.

First definition

, where is the C*-algebra generated by and , subject to

and

. Note that is a representation, but not a unitary representation. is equivalent to the unitary representation

Second definition

, where is the C*-algebra generated by and , subject to

and

. Note that .

Limit case

If , then is equal to the concrete compact group .

References