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 Template:Math is the minimal C*-algebra tensor product — the completion of the algebraic tensor product of Template:Math and Template:Math) — 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 Template:Math 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 Template:Math 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 Template:Math, and

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

for all Template:Math. It follows that the *-algebra generated by $u_{ij}$ for all Template:Math and $\kappa (u_{ij})$ for all Template:Math 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 Template:Math, 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, Template:Math, 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, Template:Math (here Template:Math 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, Template:Math 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, Template:Math 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 Template:Math 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 Template:Math in Template:Math; the norm completion is also denoted by Template:Math.

A compact quantum group is defined as a pair Template:Math, where Template:Mvar is a unital C*-algebra and

• Template:Math is a unital *-homomorphism satisfying Template:Math;
• the sets Template:Math and Template:Math are dense in Template:Math.

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 Template:Math, where the parameter Template:Mvar is a positive real number.

First definition

Template:Math, where Template:Math 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

Template:Math, where Template:Math 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 Template:Math, then Template:Math is equal to the concrete compact group Template:Math.