weak Hopf C*-algebra

Definition 0.1.

A weak Hopf C*-algebraMathworldPlanetmathPlanetmathPlanetmath is defined as a weak Hopf algebraPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/WeakHopfCAlgebra) which admits a faithfulPlanetmathPlanetmathPlanetmath *–representation on a Hilbert spaceMathworldPlanetmath. The weak C*–Hopf algebraMathworldPlanetmathPlanetmathPlanetmath is therefore much more likely to be closely related to a quantum groupoidPlanetmathPlanetmath than the weak Hopf algebra. However, one can argue that locally compact groupoidsPlanetmathPlanetmath equipped with a Haar measure are even closer to defining quantum groupoids (http://planetmath.org/QuantumGroupoids2).

There are already several, significant examples that motivate the consideration of weak C*-Hopf algebras which also deserve mentioning in the context of standard quantum theoriesPlanetmathPlanetmath. Furthermore, notions such as (proper) weak C*-algebroids can provide the main framework for symmetry breaking and quantum gravity that we are considering here. Thus, one may consider the quasi-group symmetriesPlanetmathPlanetmath constructed by means of special transformationsPlanetmathPlanetmath of the coordinate space M.

Remark: Recall that the weak Hopf algebra is defined as the extensionPlanetmathPlanetmathPlanetmath of a Hopf algebra by weakening the definining axioms of a Hopf algebra as follows:

  • (1)

    The comultiplication is not necessarily unit-preserving.

  • (2)

    The counit ε is not necessarily a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of algebras.

  • (3)

    The axioms for the antipode map S:AA with respect to the counit are as follows. For all hH,

    m(idS)Δ(h) =(εid)(Δ(1)(h1)) (0.1)
    m(Sid)Δ(h) =(idε)((1h)Δ(1))
    S(h) =S(h(1))h(2)S(h(3)).

These axioms may be appended by the following commutative diagramsMathworldPlanetmath

AASidAAΔmA@ >uεA  AAidSAAΔmA@ >uεA (0.2)

along with the counit axiom:

\xymatrix@C=3pc@R=3pcAA\ar[d]ε1&A\ar[l]Δ\ar[dl]idA\ar[d]ΔA&AA\ar[l]1ε (0.3)

Some authors substitute the term quantum groupoid for a weak Hopf algebra.

0.1 Examples of weak Hopf C*-algebra.

  • (1)

    In Nikshych and Vainerman (2000) quantum groupoids were considered as weak C*–Hopf algebras and were studied in relationship to the noncommutative symmetries of depth 2 von Neumann subfactors. If

    ABB1B2 (0.4)

    is the Jones extension induced by a finite index depth 2 inclusion AB of II1 factors, then Q=AB2 admits a quantum groupoid structureMathworldPlanetmath and acts on B1, so that B=B1Q and B2=B1Q . Similarly, in Rehren (1997) ‘paragroups’ (derived from weak C*–Hopf algebras) comprise (quantum) groupoidsPlanetmathPlanetmathPlanetmathPlanetmath of equivalence classesMathworldPlanetmathPlanetmath such as associated with 6j–symmetry groups (relative to a fusion rules algebra). They correspond to type II von Neumann algebrasMathworldPlanetmath in quantum mechanics, and arise as symmetries where the local subfactors (in the sense of containment of observables within fields) have depth 2 in the Jones extension. Related is how a von Neumann algebra N, such as of finite index depth 2, sits inside a weak Hopf algebra formed as the crossed product NA (Böhm et al. 1999).

  • (2)

    In Mack and Schomerus (1992) using a more general notion of the Drinfeld construction, develop the notion of a quasi triangular quasi–Hopf algebra (QTQHA) is developed with the aim of studying a range of essential symmetries with special properties, such the quantum groupPlanetmathPlanetmathPlanetmathPlanetmath algebra Uq(sl2) with |q|=1 . If qp=1, then it is shown that a QTQHA is canonically associated with Uq(sl2). Such QTQHAs are claimed as the true symmetries of minimalPlanetmathPlanetmath conformal field theories.

0.2 Von Neumann Algebras (or W*-algebras).

Let denote a complex (separablePlanetmathPlanetmath) Hilbert space. A von Neumann algebra 𝒜 acting on is a subset of the *–algebra of all bounded operatorsMathworldPlanetmathPlanetmath () such that:

  • (1)

    𝒜 is closed underPlanetmathPlanetmath the adjoint operation (with the adjoint of an element T denoted by T*).

  • (2)

    𝒜 equals its bicommutant, namely:

    𝒜={A():B(),C𝒜,(BC=CB)(AB=BA)}. (0.5)

If one calls a commutant of a set 𝒜 the special set of bounded operators on () which commute with all elements in 𝒜, then this second condition implies that the commutant of the commutant of 𝒜 is again the set 𝒜.

On the other hand, a von Neumann algebra 𝒜 inherits a unital subalgebraMathworldPlanetmathPlanetmath from (), and according to the first condition in its definition 𝒜 does indeed inherit a *-subalgebra structure, as further explained in the next sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on C*-algebras. Furthermore, we have the notable Bicommutant Theorem which states that 𝒜 is a von Neumann algebra if and only if A is a *-subalgebra of L(H), closed for the smallest topologyMathworldPlanetmath defined by continuous maps (ξ,η)(Aξ,η) for all <Aξ,η)> where <.,.> denotes the inner product defined on H . For further instruction on this subject, see e.g. Aflsen and Schultz (2003), Connes (1994).

CommutativePlanetmathPlanetmathPlanetmathPlanetmath and noncommutative Hopf algebras form the backbone of quantum ‘groups’ and are essential to the generalizationsPlanetmathPlanetmath of symmetry. Indeed, in most respects a quantum ‘group’ is identifiable with a Hopf algebra. When such algebras are actually associated with proper groups of matrices there is considerable scope for their representations on both finite and infinite dimensional Hilbert spaces.


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Title weak Hopf C*-algebra
Canonical name WeakHopfCalgebra
Date of creation 2013-03-22 18:12:47
Last modified on 2013-03-22 18:12:47
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 62
Author bci1 (20947)
Entry type Topic
Classification msc 08C99
Classification msc 16S40
Classification msc 81R15
Classification msc 81R50
Classification msc 16W30
Classification msc 57T05
Synonym quantum groupoids
Related topic WeakHopfAlgebra
Related topic VonNeumannAlgebra
Related topic TopologicalAlgebra
Related topic QuantumGroupoids2
Related topic LocallyCompactQuantumGroup
Related topic HopfAlgebra
Related topic LocallyCompactGroupoids
Related topic QuantumGroupoids2
Related topic GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries
Related topic GrassmanHopfAlgebrasAndTheirDualCoAlge
Defines weak Hopf algebra
Defines weak Hopf C*-algebra
Defines weak bialgebra
Defines quantum group
Defines quantum groupoid
Defines von Neumann algebra
Defines W*–algebra