Editing Quantum group (section)

==Compact matrix quantum groups==
{{Main|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 representation|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''<sup>−1</sup>) 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 [[group representation|representation]] of ''G'' can be used to generate a *-subalgebra of ''C''(''G'') which is also a Hopf *-algebra.  Specifically, if <math>g \mapsto (u_{ij}(g))_{i,j}</math> is an ''n''-dimensional representation of ''G'', then for all ''i'', ''j'' ''u<sub>ij</sub>'' ∈ ''C''(''G'') and

:<math>\Delta(u_{ij}) = \sum_k u_{ik} \otimes u_{kj}.</math>

It follows that the *-algebra generated by ''u<sub>ij</sub>'' for all ''i, j'' and ''κ''(''u<sub>ij</sub>'') for all ''i, j'' is a Hopf *-algebra: the counit is determined by ε(''u<sub>ij</sub>'') = δ<sub>''ij''</sub> for all ''i, j'' (where ''δ''<sub>''ij''</sub> is the [[Kronecker delta]]), the antipode is ''κ'', and the unit is given by

:<math>1 = \sum_k u_{1k} \kappa(u_{k1}) = \sum_k \kappa(u_{1k}) u_{k1}.</math>

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

:*The *-subalgebra, ''C''<sub>0</sub>, 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:
:::<math>\Delta(u_{ij}) = \sum_k u_{ik} \otimes u_{kj}</math>

:*There exists a linear antimultiplicative map κ: ''C''<sub>0</sub> → ''C''<sub>0</sub> (the coinverse) such that ''κ''(''κ''(''v''*)*) = ''v'' for all ''v'' ∈ ''C''<sub>0</sub> and
:::<math>\sum_k \kappa(u_{ik}) u_{kj} = \sum_k u_{ik} \kappa(u_{kj}) = \delta_{ij} I,</math>
where ''I'' is the identity element of ''C''.  Since κ is antimultiplicative, then ''κ''(''vw'') = ''κ''(''w'') ''κ''(''v'') for all ''v'', ''w'' in ''C''<sub>0</sub>.

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

In general, ''C'' is not a bialgebra, and ''C''<sub>0</sub> 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.

===Representations===
A representation of the compact matrix quantum group is given by a [[coalgebra|corepresentation]] of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebra ''A'' is a square matrix <math>v = (v_{ij})_{i,j = 1,\dots,n}</math> with entries in ''A'' (so ''v'' belongs to M(''n'', ''A'')) such that

:<math>\Delta(v_{ij}) = \sum_{k=1}^n v_{ik} \otimes v_{kj}</math>

for all ''i'', ''j'' and ''ε''(''v<sub>ij</sub>'') = δ<sub>''ij''</sub> for all ''i, j'').  Furthermore, a representation ''v'', is called unitary if the matrix for ''v'' is unitary (or equivalently, if κ(''v<sub>ij</sub>'') = ''v*<sub>ij</sub>'' for all ''i'', ''j'').

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

:<math>\gamma \gamma^* = \gamma^* \gamma, </math>
:<math>\alpha \gamma = \mu \gamma \alpha, </math>
:<math>\alpha \gamma^* = \mu \gamma^* \alpha, </math>
:<math>\alpha \alpha^* + \mu \gamma^* \gamma = \alpha^* \alpha + \mu^{-1} \gamma^* \gamma = I,</math>

and

:<math>u = \left( \begin{matrix} \alpha & \gamma \\ - \gamma^* & \alpha^* \end{matrix} \right),</math>

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<sup>−1</sup>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

:<math>v = \left( \begin{matrix} \alpha & \sqrt{\mu} \gamma \\ - \frac{1}{\sqrt{\mu}} \gamma^* & \alpha^* \end{matrix} \right).</math>

Equivalently, SU<sub>μ</sub>(2) = (C(SU<sub>μ</sub>(2)), ''w''), where C(SU<sub>μ</sub>(2)) is the C*-algebra generated by α and β, subject to

:<math>\beta \beta^* = \beta^* \beta,</math>
:<math>\alpha \beta = \mu \beta \alpha,</math>
:<math>\alpha \beta^* = \mu \beta^* \alpha,</math>
:<math>\alpha \alpha^* + \mu^2 \beta^* \beta = \alpha^* \alpha + \beta^* \beta = I,</math>

and

:<math>w = \left( \begin{matrix} \alpha & \mu \beta \\ - \beta^* & \alpha^* \end{matrix} \right),</math>

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<sup>−1</sup>β, κ(β*) = −μβ*, κ(α*) = α.  Note that ''w'' is a unitary representation.  The realizations can be identified by equating <math>\gamma = \sqrt{\mu} \beta</math>.

When μ = 1, then SU<sub>μ</sub>(2) is equal to the algebra ''C''(SU(2)) of functions on the concrete compact group SU(2).