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.
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
for all formula_3, and for all formula_4, where
for all formula_6 and all formula_7. There also exists a linear multiplicative mapping
such that
for all formula_3 and all formula_11. 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 generated by formula_15 for all and formula_16 for all is a Hopf *-algebra: the counit is determined by
for all formula_18 (where formula_19 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
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 separable C*-algebra and
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
so that the comultiplication is determined by formula_30, and the coinverse is determined by formula_31. 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 
so that the comultiplication is determined by formula_35, and the coinverse is determined by formula_36, formula_37. Note that is a unitary representation. The realizations can be identified by equating formula_38.
Limit Case.
If , then is equal to the concrete compact group .