For a compact topological group, G, there exists a C*-algebra homomorphism

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

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

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

for all ${\displaystyle f\in C(G)}$ , and for all ${\displaystyle x,y\in G}$ , where

${\displaystyle (f\otimes g)(x,y)=f(x)g(y)}$ 

for all ${\displaystyle f,g\in C(G)}$  and all ${\displaystyle x,y\in G}$ . There also exists a linear multiplicative mapping

${\displaystyle \kappa :C(G)\to C(G)}$ ,

such that

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

for all ${\displaystyle f\in C(G)}$  and all ${\displaystyle x\in G}$ . 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

${\displaystyle g\mapsto (u_{ij}(g))_{i,j}}$ 

is an n-dimensional representation of G, then

${\displaystyle u_{ij}\in C(G)}$ 

for all i, j, and

${\displaystyle \Delta (u_{ij})=\sum _{k}u_{ik}\otimes u_{kj}}$ 

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

${\displaystyle \epsilon (u_{ij})=\delta _{ij}}$ 

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

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

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

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

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 (here C ⊗ C is the C*-algebra tensor product - the completion of the algebraic tensor product of C and C) such that

${\displaystyle \forall i,j:\qquad \Delta (u_{ij})=\sum _{k}u_{ik}\otimes u_{kj};}$ 

• There exists a linear antimultiplicative map, called the coinverse, κ : C₀ → C₀ such that ${\displaystyle \kappa (\kappa (v*)*)=v}$  for all ${\displaystyle v\in C_{0}}$  and ${\displaystyle \sum _{k}\kappa (u_{ik})u_{kj}=\sum _{k}u_{ik}\kappa (u_{kj})=\delta _{ij}I,}$  where I is the identity element of C. Since κ is antimultiplicative, κ(vw) = κ(w)κ(v) for all ${\displaystyle v,w\in C_{0}}$ .

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.

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); the norm completion is also denoted by A ⊗ B.

A compact quantum group^[3][4] 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;
• the sets {(C ⊗ 1) Δ(C)} and {(1 ⊗ C) Δ(C)} are dense in C ⊗ C.

A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra^[5] Furthermore, a representation, v, is called unitary if the matrix for v is unitary, or equivalently, if

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

An example of a compact matrix quantum group is SU_μ(2),^[6] where the parameter μ is a positive real number.

First definition edit

SU_μ(2) = (C(SU_μ(2)), u), where C(SU_μ(2)) is the C*-algebra generated by α and γ, subject to

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

and

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

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

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

Second definition edit

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

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

and

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

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

Limit case edit

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