Compact quantum group

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In mathematics, a compact quantum group is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of "continuous complex-valued functions" on a compact quantum group.

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 [1] 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[edit]

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

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

for all , and for all , where

for all and all . There also exists a linear multiplicative mapping

,

such that

for all and all . 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

is an n-dimensional representation of G, then

for all i, j, and

for all i, j. It follows that the *-algebra generated by for all i, j and for all i, j is a Hopf *-algebra: the counit is determined by

for all (where is the Kronecker delta), the antipode is κ, and the unit is given by

Compact Matrix Quantum Groups[edit]

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

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 : CCC (here CC is the C*-algebra tensor product - the completion of the algebraic tensor product of C and C) such that
  • There exists a linear antimultiplicative map, called the coinverse, κ : C0C0 such that for all and where I is the identity element of C. Since κ is antimultiplicative, κ(vw) = κ(w)κ(v) for all .

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.

Compact Quantum Groups[edit]

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 AB in B(HK); the norm completion is also denoted by AB.

A compact quantum group[2][3] is defined as a pair (C, Δ), where C is a unital separable C*-algebra and

  • Δ : CCC is a C*-algebra unital homomorphism satisfying (Δ ⊗ id) Δ = (id ⊗ Δ) Δ;
  • the sets {(C ⊗ 1) Δ(C)} and {(1 ⊗ C) Δ(C)} are dense in CC.

Representations[edit]

A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra[4] Furthermore, a representation, v, is called unitary if the matrix for v is unitary, or equivalently, if

Example[edit]

An example of a compact matrix quantum group is SUμ(2),[5] where the parameter μ is a positive real number.

First Definition[edit]

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

and

so that the comultiplication is determined by , and the coinverse is determined by . Note that u is a representation, but not a unitary representation. u is equivalent to the unitary representation

Second Definition[edit]

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

and

so that the comultiplication is determined by , and the coinverse is determined by , . Note that w is a unitary representation. The realizations can be identified by equating .

Limit Case[edit]

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

References[edit]

  1. ^ Woronowicz, S.L. "Compact Matrix Pseudogrooups", Commun. Math. Phys. 111 (1987), 613-665
  2. ^ Woronowicz, S.L. "Compact Quantum Groups". Notes from http://www.fuw.edu.pl/~slworono/PDF-y/CQG3.pdf
  3. ^ van Daele, A. and Maes, Ann. "Notes on compact quantum groups", arXiv:math/9803122
  4. ^ a corepresentation of a counital coassiative coalgebra A is a square matrix
    with entries in A (so that v ∈ M(n, A)) such that
  5. ^ van Daele, A. and Wang, S. "Universal quantum groups" Int. J. of Math. (1996), 255-263.