Editing Quantum group (section)

===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.