Compact quantum group

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In mathematics, a compact quantum group is an abstract structure which is determined by the noncommutative C*-algebra whose elements represent "continuous complex-valued functions" on the 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 $\Delta : C(G) \to C(G) \otimes C(G)$ (where $C(G) \otimes C(G)$ is the C*-algebra tensor product - the completion of the algebraic tensor product of $C (G)$ and $C (G)$ ), such that $Δ(f)(x, y) = f (x y)$ for all $f \in C(G)$ , and for all $x, y \in G$ (where $(f \otimes g)(x,y) = f(x) g(y)$ for all $f, g \in C(G)$ and all $x, y \in G$ ). There also exists a linear multiplicative mapping $\kappa : C(G) \to C(G)$ , such that $κ(f)(x) = f (x - 1)$ for all $f \in C(G)$ and all $x \in G$ . 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 $g \mapsto (u_{ij}(g))_{i,j}$ is an $n$ -dimensional representation of $G$ , then $u_{ij} \in C(G)$ for all $i, j$ , and $\Delta(u_{ij}) = \sum_k u_{ik} \otimes u_{kj}$ for all $i, j$ . It follows that the *-algebra generated by $u i j$ for all $i, j$ and $κ(u i j)$ for all $i, j$ is a Hopf *-algebra: the counit is determined by $ε(u i j) = δ i j$ for all $i, j$ (where $δ i j$ is the Kronecker delta), the antipode is $κ$ , and the unit is given by

1 =	∑	u_1kκ(u_k1) =	∑	κ(u_1k)u_k1.
	k		k

As a generalization, a compact matrix quantum group is defined as a pair $(C, u)$ , where $C$ is a C*-algebra and $u = (u_{ij})_{i,j = 1,\dots,n}$ is a matrix with entries in $C$ such that

The *-subalgebra, $C 0$ , of $C$ , which is generated by the matrix elements of $u$ , is dense in $C$ ;

There exists a C*-algebra homomorphism $\Delta : C \to C \otimes C$ (where $C \otimes C$ is the C*-algebra tensor product - the completion of the algebraic tensor product of $C$ and $C$ ) such that $\Delta(u_{ij}) = \sum_k u_{ik} \otimes u_{kj}$ for all $i, j$ ( $Δ$ is called the comultiplication);

There exists a linear antimultiplicative map $\kappa : C_0 \to C_0$ (the coinverse) such that $κ(κ(v * ) * ) = v$ for all $v \in C_0$ and

∑ κ(u_ik)u_kj = ∑ u_ikκ(u_kj) = δ_ijI,

k k

where $I$ is the identity element of $C$ . Since $κ$ is antimultiplicative, then $κ(v w) = κ(w)κ(v)$ for all $v, w \in C_0$ .

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

In general, $C$ is not a bialgebra, and $C 0$ 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 coassiative coalgebra $A$ is a square matrix $v = (v_{ij})_{i,j = 1,\dots,n}$ with entries in $A$ (so $v \in M_n(A)$ ) such that $\Delta(v_{ij}) = \sum_{k=1}^n v_{ik} \otimes v_{kj}$ for all $i, j$ and $ε(v i j) = δ i j$ for all $i, j$ ). Furthermore, a representation, v, is called unitary if the matrix for v is unitary (or equivalently, if $\kappa(v_{ij}) = v^*_{ji}$ for all i, j).

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

$\gamma \gamma^* = \gamma^* \gamma, \ \alpha \gamma = \mu \gamma \alpha, \ \alpha \gamma^* = \mu \gamma^* \alpha, \ \alpha \alpha^* + \mu \gamma^* \gamma = \alpha^* \alpha + \mu^{-1} \gamma^* \gamma = I,$

and $u = \left( \begin{matrix} \alpha & \gamma \\ - \gamma^* & \alpha^* \end{matrix} \right),$ so that the comultiplication is determined by $\Delta(\alpha) = \alpha \otimes \alpha - \gamma \otimes \gamma^*$ , $\Delta(\gamma) = \alpha \otimes \gamma + \gamma \otimes \alpha^*$ , and the coinverse is determined by $κ(α) = α *$ , $κ(γ) = - μ - 1 γ$ , $κ(γ *) = - μγ *$ , $κ(α *) = α$ . Note that $u$ is a representation, but not a unitary representation. $u$ is equivalent to the unitary representation $v = \left( \begin{matrix} \alpha & \sqrt{\mu} \gamma \\ - \frac{1}{\sqrt{\mu}} \gamma^* & \alpha^* \end{matrix} \right).$

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

$\beta \beta^* = \beta^* \beta, \ \alpha \beta = \mu \beta \alpha, \ \alpha \beta^* = \mu \beta^* \alpha, \ \alpha \alpha^* + \mu^2 \beta^* \beta = \alpha^* \alpha + \beta^* \beta = I,$

and $w = \left( \begin{matrix} \alpha & \mu \beta \\ - \beta^* & \alpha^* \end{matrix} \right),$ so that the comultiplication is determined by $\Delta(\alpha) = \alpha \otimes \alpha - \mu \beta \otimes \beta^*$ , $\Delta(\beta) = \alpha \otimes \beta + \beta \otimes \alpha^*$ , and the coinverse is determined by $κ(α) = α *$ , $κ(β) = - μ - 1 β$ , $κ(β *) = - μβ *$ , $κ(α *) = α$ . Note that $w$ is a unitary representation. The realizations can be identified by equating $\gamma = \sqrt{\mu} \beta$ .