Quantum Group - Compact Matrix Quantum Groups

Compact Matrix Quantum Groups

See also compact quantum group.

S.L. Woronowicz introduced compact matrix quantum groups. 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.

The continuous complex-valued functions on a compact Hausdorff topological space form a commutative C*-algebra. 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.

For a compact topological group, G, there exists a C*-algebra homomorphism Δ: C(G) → C(G) ⊗ C(G) (where C(G) ⊗ C(G) is the C*-algebra tensor product - the completion of the algebraic tensor product of C(G) and C(G)), such that Δ(f)(x, y) = f(xy) for all f ∈ C(G), and for all x, y ∈ G (where (f ⊗ g)(x, y) = f(x)g(y) for all f, g ∈ C(G) and all x, y ∈ G). There also exists a linear multiplicative mapping κ: C(G) → C(G), such that κ(f)(x) = f(x−1) for all f ∈ C(G) and all x ∈ G. Strictly, this does not make C(G) 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 u_ij ∈ C(G) and

It follows that the *-algebra generated by u_ij for all i, j and κ(u_ij) for all i, j is a Hopf *-algebra: the counit is determined by ε(u_ij) = δ_ij for all i, j (where δ_ij 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, fu), where C is a C*-algebra and is a matrix with entries in C such that

• The *-subalgebra, C₀, of C, which is generated by the matrix elements of u, is dense in C;

• There exists a C*-algebra homomorphism called the comultiplication Δ: C → C ⊗ C (where C ⊗ C is the C*-algebra tensor product - the completion of the algebraic tensor product of C and C) such that for all i, j we have:

• There exists a linear antimultiplicative map κ: C₀ → C₀ (the coinverse) such that κ(κ(v*)*) = v for all v ∈ C₀ and

where I is the identity element of C. Since κ is antimultiplicative, then κ(vw) = κ(w) κ(v) for all v, w in C₀.

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

In general, C is not 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.

A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebra A is a square matrix with entries in A (so v belongs to M(n, A)) such that

for all i, j and ε(v_ij) = δ_ij for all i, j). Furthermore, a representation v, is called unitary if the matrix for v is unitary (or equivalently, if κ(v_ij) = v*_ij for all i, j).

An example of a compact matrix quantum group is SU_μ(2), where the parameter μ is a positive real number. So SU_μ(2) = (C(SU_μ(2)), u), where C(SU_μ(2)) is the C*-algebra generated by α and γ, subject to

and

so that the comultiplication is determined by ∆(α) = α ⊗ α − γ ⊗ γ*, ∆(γ) = α ⊗ γ + γ ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(γ) = −μ−1γ, κ(γ*) = −μγ*, κ(α*) = α. Note that u is a representation, but not a unitary representation. u is equivalent to the unitary representation

Equivalently, SU_μ(2) = (C(SU_μ(2)), w), where C(SU_μ(2)) is the C*-algebra generated by α and β, subject to

and

so that the comultiplication is determined by ∆(α) = α ⊗ α − μβ ⊗ β*, Δ(β) = α ⊗ β + β ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(β) = −μ−1β, κ(β*) = −μβ*, κ(α*) = α. Note that w is a unitary representation. The realizations can be identified by equating .

When μ = 1, then SU_μ(2) is equal to the algebra C(SU(2)) of functions on the concrete compact group SU(2).