In mathematics, compact quantum groups are generalisations of compact groups, where the commutative
-algebra of continuous complex-valued functions on a compact group is generalised to an abstract structure on a not-necessarily commutative unital
-algebra, which plays the role of the "algebra of continuous complex-valued functions on the compact quantum group".
[1] 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[2] 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.
For a compact topological group, G, there exists a C*-algebra homomorphism where C(G) ⊗ C(G) is the minimal C*-algebra tensor product — the completion of the algebraic tensor product of C(G) and C(G)) — such that for all
There also exists a linear multiplicative mapping such that for all
Strictly speaking, this does not make C(G) into a Hopf algebra, unless G is finite.
On the other hand, a finite-dimensional representation of G can be used to generate a *-subalgebra of C(G) which is also a Hopf *-algebra.
Specifically, if is an n-dimensional representation of G, then for all i, j, and for all i, j.
for all i, j is a Hopf *-algebra: the counit is determined by for all
is the Kronecker delta), the antipode is κ, and the unit is given by As a generalization, a compact matrix quantum group is defined as a pair (C, u), where C is a C*-algebra and is a matrix with entries in C such that As a consequence of continuity, the comultiplication on C is coassociative.
In general, C is a bialgebra, and C0 is a Hopf *-algebra.
Informally, C can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and u can be regarded as a finite-dimensional representation of the compact matrix quantum group.
For C*-algebras A and B acting on the Hilbert spaces H and K respectively, their minimal tensor product is defined to be the norm completion of the algebraic tensor product A ⊗ B in B(H ⊗ K); the norm completion is also denoted by A ⊗ B.
A compact quantum group[3][4] is defined as a pair (C, Δ), where C is a unital C*-algebra and A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra[5] Furthermore, a representation, v, is called unitary if the matrix for v is unitary, or equivalently, if An example of a compact matrix quantum group is SUμ(2),[6] where the parameter μ is a positive real number.
SUμ(2) = (C(SUμ(2)), u), where C(SUμ(2)) is the C*-algebra generated by α and γ, subject to and so that the comultiplication is determined by
κ ( α ) =
u is equivalent to the unitary representation SUμ(2) = (C(SUμ(2)), w), where C(SUμ(2)) is the C*-algebra generated by α and β, subject to and so that the comultiplication is determined by
Note that w is a unitary representation.
The realizations can be identified by equating
If μ = 1, then SUμ(2) is equal to the concrete compact group SU(2).