Compact quantum group explained

In mathematics, compact quantum groups are generalisations of compact groups, where the commutative

C ^{*} -algebra of continuous complex-valued functions on a compact group is generalised to an abstract structure on a not-necessarily commutative unital

C ^{*} -algebra, which plays the role of the "algebra of continuous complex-valued functions on the compact quantum group".^[1]

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^[2] 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,, there exists a C*-algebra homomorphism

\Delta : C(G) \to C(G) ⊗ C(G)

where is the minimal C*-algebra tensor product - the completion of the algebraic tensor product of and) - such that

\Delta(f)(x,y) = f(xy)

for all

f \in C(G) , and for all x, y \in G , where(f ⊗ 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 into a Hopf algebra, unless is finite.

On the other hand, a finite-dimensional representation of can be used to generate a

• -subalgebra

of which is also a Hopf *-algebra. Specifically, if

g \mapsto (u _{ij} (g)) _{i,j}

is an -dimensional representation of, then

u _{ij} \in C(G)

for all, and

\Delta(u _{ij} ) = \sum _{k} u _{ik} ⊗ u _{kj}

for all . It follows that the

• -algebra

generated by

u _{ij} for all and \kappa(u _{ij} ) for all is a Hopf *-algebra: the counit is determined by\epsilon(u _{ij} ) = \delta _{ij}

for all

i, j (where \delta _{ij} is the Kronecker delta), the antipode is, and the unit is given by1 = \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, where is a C*-algebra and

u = (u _{ij} ) _{i,j}

is a matrix with entries in such that

• The *-subalgebra,, of, which is generated by the matrix elements of, is dense in ;
• There exists a C*-algebra homomorphism, called the comultiplication, (here is the C*-algebra tensor product - the completion of the algebraic tensor product of and) such that

\forall i, j: \Delta(u _{ij} ) = \sum _{k} u _{ik} ⊗ u _{kj} ;

• There exists a linear antimultiplicative map, called 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 is the identity element of . Since is antimultiplicative, for all v, w \in C _{0} .

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

In general, is a bialgebra, and is a Hopf *-algebra.

Informally, can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and can be regarded as a finite-dimensional representation of the compact matrix quantum group.

Compact quantum groups

For C*-algebras and acting on the Hilbert spaces and respectively, their minimal tensor product is defined to be the norm completion of the algebraic tensor product in ; the norm completion is also denoted by .

A compact quantum group^{[3] [4]} is defined as a pair, where is a unital C*-algebra and

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

Representations

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

\forall i, j: \kappa(v _{ij} ) = v ^{*} _{ji} .

Example

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

First definition

, where is the C*-algebra generated by and, 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{matrix} \alpha & \gamma \ - \gamma ^{*} & \alpha ^{*} \end{matrix} \right),

so that the comultiplication is determined by

\Delta(\alpha) = \alpha ⊗ \alpha - \gamma ⊗ \gamma ^{*,} \Delta(\gamma) = \alpha ⊗ \gamma + \gamma ⊗ \alpha ^{*} , and the coinverse is determined by \kappa(\alpha) = \alpha ^{*,} \kappa(\gamma) = - \mu ^{-1} \gamma, \kappa(\gamma ^{*)} = - \mu \gamma ^{*,} \kappa(\alpha ^{*)} = \alpha . Note that is a representation, but not a unitary representation. is equivalent to the unitary representation v = \left( \begin{matrix} \alpha & \sqrt{\mu} \gamma \ - (1)/(\sqrt{\mu) } \gamma^* & \alpha^* \end \right).

Second definition

, where is the C*-algebra generated by and, 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{matrix} \alpha & \mu \beta \ - \beta ^{*} & \alpha ^{*} \end{matrix} \right),

so that the comultiplication is determined by

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

Limit case

If, then is equal to the concrete compact group .

Notes and References

1. Book: Banica . Teo . Introduction to Quantum Groups . 2023 . Springer . 978-3-031-23816-1.
2. Woronowicz, S.L. "Compact Matrix Pseudogrooups", Commun. Math. Phys. 111 (1987), 613-665
3. Woronowicz, S.L. "Compact Quantum Groups". Notes from http://www.fuw.edu.pl/~slworono/PDF-y/CQG3.pdf
4. van Daele, A. and Maes, Ann. "Notes on compact quantum groups", arXiv:math/9803122
5. a corepresentation of a counital coassiative coalgebra is a square matrixv = (v _{ij} ) _{i,j}

   with entries in (so that) such that

   \forall i, j: \Delta(v _{ij} ) = \sum ^{n} _{k =1} v _{ik} ⊗ v _{kj} \forall i, j: \epsilon(v _{ij} ) = \delta _{ij} .
6. van Daele, A. and Wang, S. "Universal quantum groups" Int. J. Math. (1996), 255-263.