Compact quantum group

In mathematics, compact quantum groups are generalisations of compact groups, where the commutative ${\mathrm{C}}^{*}$-algebra of continuous complex-valued functions on a compact group is generalised to an abstract structure on a not-necessarily commutative unital ${\mathrm{C}}^{*}$-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, Template:Mvar, there exists a C*-algebra homomorphism

$\mathrm{\Delta }:C(G)\to C(G)\otimes C(G)$

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

$\mathrm{\Delta }(f)(x,y)=f(xy)$

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

$\kappa (f)(x)=f(x^{-1})$

for all $f\in C(G)$ and all $x\in G$. Strictly speaking, this does not make C(G) into a Hopf algebra, unless Template:Mvar is finite.

On the other hand, a finite-dimensional representation of Template:Mvar 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 Template:Mvar-dimensional representation of Template:Mvar, then

$u_{ij}\in C(G)$

for all i, j, and

$\mathrm{\Delta }(u_{ij})=\sum _k{u}_{ik}\otimes u_{kj}$

for all i, j. It follows that the *-algebra generated by $u_{ij}$ for all i, j and $\kappa (u_{ij})$ for all i, j is a Hopf *-algebra: the counit is determined by

$ϵ(u_{ij})={\delta }_{ij}$

for all $i,j$ (where ${\delta }_{ij}$ is the Kronecker delta), the antipode is Template:Mvar, and the unit is given by

$1=\sum _k{u}_{1k}\kappa (u_{k1})=\sum _k\kappa (u_{1k})u_{k1}.$

Compact matrix quantum groups

As a generalization, a compact matrix quantum group is defined as a pair (C, u), where Template:Mvar is a C*-algebra and

$u=(u_{ij}{)}_{i,j=1,\dots ,n}$

is a matrix with entries in Template:Mvar such that

• The *-subalgebra, C₀, of Template:Mvar, which is generated by the matrix elements of Template:Mvar, is dense in Template:Mvar;
• There exists a C*-algebra homomorphism, called the comultiplication, Δ : C → C ⊗ C (here C ⊗ C is the C*-algebra tensor product - the completion of the algebraic tensor product of Template:Mvar and Template:Mvar) such that

$\mathrm{\forall }i,j:\mathrm{\Delta }(u_{ij})=\sum _k{u}_{ik}\otimes u_{kj};$

• There exists a linear antimultiplicative map, called the coinverse, κ : C₀ → C₀ such that $\kappa (\kappa (v*)*)=v$ for all $v\in C_0$ and $\sum _k\kappa (u_{ik})u_{kj}=\sum _k{u}_{ik}\kappa (u_{kj})={\delta }_{ij}I,$ where Template:Mvar is the identity element of Template:Mvar. Since Template:Mvar is antimultiplicative, κ(vw) = κ(w)κ(v) for all $v,w\in C_0$.

As a consequence of continuity, the comultiplication on Template:Mvar is coassociative.

In general, Template:Mvar is a bialgebra, and C₀ is a Hopf *-algebra.

Informally, Template:Mvar can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and Template:Mvar can be regarded as a finite-dimensional representation of the compact matrix quantum group.

Compact quantum groups

For C*-algebras Template:Mvar and Template:Mvar acting on the Hilbert spaces Template:Mvar and Template:Mvar 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 is defined as a pair (C, Δ), where Template:Mvar is a unital C*-algebra and

• Δ : C → C ⊗ C is a unital *-homomorphism satisfying (Δ ⊗ id) Δ = (id ⊗ Δ) Δ;
• the sets {(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

$\mathrm{\forall }i,j:\kappa (v_{ij})=v_{ji}^{*}.$

Example

An example of a compact matrix quantum group is SU_μ(2), where the parameter Template:Mvar is a positive real number.

First definition

SU_μ(2) = (C(SU_μ(2)), u), where C(SU_μ(2)) is the C*-algebra generated by Template:Mvar and Template:Mvar, 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

$u=\left(\begin{array}{cc}\alpha & \gamma \\ -{\gamma }^{*}& {\alpha }^{*}\end{array}\right),$

so that the comultiplication is determined by $\mathrm{\Delta }(\alpha )=\alpha \otimes \alpha -\gamma \otimes {\gamma }^{*},\mathrm{\Delta }(\gamma )=\alpha \otimes \gamma +\gamma \otimes {\alpha }^{*}$, and the coinverse is determined by $\kappa (\alpha )={\alpha }^{*},\kappa (\gamma )=-{\mu }^{-1}\gamma ,\kappa ({\gamma }^{*})=-\mu {\gamma }^{*},\kappa ({\alpha }^{*})=\alpha$. Note that Template:Mvar is a representation, but not a unitary representation. Template:Mvar is equivalent to the unitary representation

$v=\left(\begin{array}{cc}\alpha & \sqrt{\mu }\gamma \\ -\frac{1}{\sqrt{\mu }}{\gamma }^{*}& {\alpha }^{*}\end{array}\right).$

Second definition

SU_μ(2) = (C(SU_μ(2)), w), where C(SU_μ(2)) is the C*-algebra generated by Template:Mvar and Template:Mvar, 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

$w=\left(\begin{array}{cc}\alpha & \mu \beta \\ -{\beta }^{*}& {\alpha }^{*}\end{array}\right),$

so that the comultiplication is determined by $\mathrm{\Delta }(\alpha )=\alpha \otimes \alpha -\mu \beta \otimes {\beta }^{*},\mathrm{\Delta }(\beta )=\alpha \otimes \beta +\beta \otimes {\alpha }^{*}$, and the coinverse is determined by $\kappa (\alpha )={\alpha }^{*},\kappa (\beta )=-{\mu }^{-1}\beta ,\kappa ({\beta }^{*})=-\mu {\beta }^{*}$, $\kappa ({\alpha }^{*})=\alpha$. Note that Template:Mvar is a unitary representation. The realizations can be identified by equating $\gamma =\sqrt{\mu }\beta$.

Limit case

If μ = 1, then SU_μ(2) is equal to the concrete compact group SU(2).