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". 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, 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 f \in C(G), and for all x, y \in G, where for all and all x, y \in G. There also exists a linear multiplicative mapping such that for all f \in C(G) and all x \in G. 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 . It follows that the *-algebra generated by u_{ij} for all i, j and for all i, j is a Hopf *-algebra: the counit is determined by for all i, j (where \delta_{ij} is the Kronecker delta), the antipode is κ, and the unit is given by

Compact matrix quantum groups

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 C0 , of C, which is generated by the matrix elements of u, is dense in C; Δ : C → C ⊗ C (here C ⊗ C is the C*-algebra tensor product - the completion of the algebraic tensor product of C and C) such that κ : C0 → C0 such that for all v \in C_0 and where I is the identity element of C. Since κ is antimultiplicative, for all. 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.

Compact quantum groups

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) A ⊗ B . A compact quantum group is defined as a pair (C, Δ) , where C is a unital C*-algebra and Δ : C → C ⊗ C is a unital *-homomorphism satisfying (Δ ⊗ id) Δ = (id ⊗ Δ) Δ {(C ⊗ 1) Δ(C)} and {(1 ⊗ C) Δ(C)} are dense in C ⊗ C .

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 SUμ(2) , where the parameter μ is a positive real number.

First definition

, where C(SUμ(2)) is the C*-algebra generated by α and γ, subject to and so that the comultiplication is determined by, and the coinverse is determined by. Note that u is a representation, but not a unitary representation. u is equivalent to the unitary representation

Second definition

, where C(SUμ(2)) is the C*-algebra generated by α and β, subject to and so that the comultiplication is determined by, and the coinverse is determined by ,. Note that w is a unitary representation. The realizations can be identified by equating.

Limit case

If , then SUμ(2) is equal to the concrete compact group SU(2) .