Compact Quantum Group

-algebra" class="copylinks">C
algebra . The geometry of a compact matrix quantum group is a special case of a Noncommutative Geometry .

-algebra. 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 .

For a compact ), the antipode is

\kappa

, and the unit is given by

1 = \sum_k u_{1k} \kappa(u_{k1}) = \sum_k \kappa(u_{1k}) u_{k1}.

-algebra and $u = (u_{ij})_{i,j = 1,\dots,n}$ is a matrix with entries in $C$ such that

The
subalgebra, $C_0$ , of $C$ , which is generated by the matrix elements of $u$ , is dense in $C$ ;

There exists a C
algebra homomorphism $\Delta : C o C \otimes C$ (where $C \otimes C$ is the C
algebra tensor product - the completion of the algebraic tensor product of $C$ and $C$ ) such that $\Delta(u_{ij}) = \sum_k u_{ik} \otimes u_{kj}$ for all $i, j$ ( $\Delta$ is called the comultiplication);

There exists a linear antimultiplicative map $\kappa : C_0 o C_0$ (the coinverse) 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 $I$ is the identity element of $C$ . Since $\kappa$ is antimultiplicative, then $\kappa(vw) = \kappa(w) \kappa(v)$ for all $v, w \in C_0$ .

As a consequence of continuity, the comultiplication on

C

is coassociative.

-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.

-algebra (a corepresentation of a counital coassiative coalgebra $A$ is a square matrix $v = (v_{ij})_{i,j = 1,\dots,n}$ with entries in $A$ (so $v \in M_n(A)$ ) such that $\Delta(v_{ij}) = \sum_{k=1}^n v_{ik} \otimes v_{kj}$ for all $i, j$ and $\epsilon(v_{ij}) = \delta_{ij}$ for all $i, j$ ). Furthermore, a representation, ''v'', is called unitary if the matrix for ''v'' is unitary (or equivalently, if $\kappa(v_{ij}) = v^---_{ji}$ for all ''i'', ''j'').

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

& \alpha^--- \end{matrix} ight), so that the comultiplication is determined by $\Delta(\alpha) = \alpha \otimes \alpha - \gamma \otimes \gamma^---$ , $\Delta(\gamma) = \alpha \otimes \gamma + \gamma \otimes \alpha^---$ , and the coinverse is determined by $\kappa(\alpha) = \alpha^---$ , $\kappa(\gamma) = - \mu^{-1} \gamma$ , $\kappa(\gamma^---) = - \mu \gamma^---$ , $\kappa(\alpha^---) = \alpha$ . Note that $u$ is a representation, but not a unitary representation. $u$ is equivalent to the unitary representation $v = \left( \begin{matrix} \alpha & \sqrt{\mu} \gamma \ - rac{1}{\sqrt{\mu}} \gamma^--- & \alpha^--- \end{matrix}
ight).$

= \beta^--- \beta, \ \alpha \beta = \mu \beta \alpha, \ \alpha \beta^--- = \mu \beta^--- \alpha, \ \alpha \alpha^--- + \mu^2 \beta^--- \beta = \alpha^--- \alpha + \beta^--- \beta = I,

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

When

\mu = 1

, then

SU_{\mu}(2)

is equal to the concrete compact group

SU(2)

	CATEGORIES ABOUT COMPACT QUANTUM GROUP

	quantum groups

	c-algebras

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