weak Hopf algebra

Definition 0.1: In order to define a weak Hopf algebraPlanetmathPlanetmathPlanetmathPlanetmath, one weakens, or relaxes certain axioms of a Hopf algebraPlanetmathPlanetmathPlanetmath as follows :

  • (1)

    The comultiplication is not necessarily unit–preserving.

  • (2)

    The counit ε is not necessarily a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of algebrasMathworldPlanetmathPlanetmathPlanetmath.

  • (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 groupoidPlanetmathPlanetmath for a weak Hopf algebra. Therefore, the weak Hopf algebra is considered by some authors as an important concept in quantum operator algebraPlanetmathPlanetmathPlanetmath (QOA).

0.1 Examples of weak Hopf algebras

  • (1)

    We refer here to Bais et al. (2002). Let G be a non-Abelian groupMathworldPlanetmath and HG a discrete subgroup. Let F(H) denote the space of functions on H and H the group algebraPlanetmathPlanetmath (which consists of the linear span of group elements with the group structureMathworldPlanetmath).

    The quantum double D(H) (Drinfeld, 1987) is defined by

    D(H)=F(H)~H, (0.4)

    where, for xH, the twisted tensor product is specified by

    ~(f1h1)(f2h2)(x)=f1(x)f2(h1xh1-1)h1h2. (0.5)

    The physical interpretationMathworldPlanetmath is often to take H as the ‘electric gauge group’ and F(H) as the ‘magnetic symmetryMathworldPlanetmathPlanetmathPlanetmath’ generated by {fe} . In terms of the counit ε, the double D(H) has a trivial representation given by ε(fh)=f(e) . We next look at certain features of this construction.

    For the purpose of braiding relationsMathworldPlanetmathPlanetmathPlanetmath there is an R matrix, RD(H)D(H), leading to the operator

    σ(ΠαAΠβB)(R), (0.6)

    in terms of the Clebsch–Gordan series ΠαAΠβBNαβCABγΠγC, and where σ denotes a flip operator. The operator 2 is sometimes called the monodromyMathworldPlanetmath or Aharanov–Bohm phase factor. In the case of a condensate in a state |v in the carrierPlanetmathPlanetmath space of some representationPlanetmathPlanetmath ΠαA . One considers the maximal Hopf subalgebraPlanetmathPlanetmath T of a Hopf algebra A for which |v is TinvariantMathworldPlanetmath; specifically  :

    ΠαA(P)|v=ε(P)|v,PT. (0.7)
  • (2)

    For the second example, consider A=F(H) . The algebra of functions on H can be broken to the algebra of functions on H/K, that is, to F(H/K), where K is normal in H, that is, HKH-1=K . Next, consider A=D(H) . On breaking a purely electric condensate |v, the magnetic symmetry remains unbroken, but the electric symmetry H is broken to Nv, with NvH, the stabilizerMathworldPlanetmath of |v . From this we obtain T=F(H)~Nv .

  • (3)

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

    ABB1B2 (0.8)

    is the Jones extensionPlanetmathPlanetmathPlanetmath induced by a finite index depth 2 inclusion AB of II1 factors, then Q=AB2 admits a quantum groupoid structure and acts on B1, so that B=B1Q and B2=B1Q . Similarly, in Rehren (1997) ‘paragroups’ (derived from weak C*–Hopf algebras) comprise (quantum) groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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).

  • (4)

    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.

1 Definitions of Related Concepts

Let us recall two basic concepts of quantum operator algebra that are essential to Algebraic Quantum TheoriesPlanetmathPlanetmath.

1.1 Definition of a Von Neumann Algebra.

Let denote a complex (separable) Hilbert spaceMathworldPlanetmath. 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 adjointPlanetmathPlanetmathPlanetmathPlanetmath 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)}. (1.1)

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 subalgebra from (), and according to the first condition in its definition 𝒜 does indeed inherit a *-subalgebra structure, as further explained in the next sectionMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on C*-algebras. Furthermore, we have 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 productMathworldPlanetmath defined on H . For further instruction on this subject, see e.g. Aflsen and Schultz (2003), Connes (1994).

1.2 Definition of a Hopf algebra

Firstly, a unital associative algebra consists of a linear spacePlanetmathPlanetmath A together with two linear maps

m :AAA,(multiplication) (1.2)
η :A,(unity)

satisfying the conditions

m(m𝟏) =m(𝟏m) (1.3)
m(𝟏η) =m(η𝟏)=id.

This first condition can be seen in terms of a commuting diagram :

AAAmidAAidmmAA@ >mA (1.4)

Next suppose we consider ‘reversing the arrows’, and take an algebra A equipped with a linear homorphisms Δ:AAA, satisfying, for a,bA :

Δ(ab) =Δ(a)Δ(b) (1.5)
(Δid)Δ =(idΔ)Δ.

We call Δ a comultiplication, which is said to be coasociative in so far that the following diagram commutes

AAAΔidAAidΔΔAA@ <ΔA (1.6)

There is also a counterpart to η, the counity map ε:A satisfying

(idε)Δ=(εid)Δ=id. (1.7)

A bialgebraPlanetmathPlanetmathPlanetmath (A,m,Δ,η,ε) is a linear space A with maps m,Δ,η,ε satisfying the above properties.

Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism S:AA, satisfying S(ab)=S(b)S(a), for a,bA . This map is defined implicitly via the property :

m(Sid)Δ=m(idS)Δ=ηε. (1.8)

We call S the antipode map. A Hopf algebra is then a bialgebra (A,m,η,Δ,ε) equipped with an antipode map S .

CommutativePlanetmathPlanetmathPlanetmath 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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    arXiv:0709.4364v2 [quant–ph]
Title weak Hopf algebra
Canonical name WeakHopfAlgebra
Date of creation 2013-03-22 18:12:43
Last modified on 2013-03-22 18:12:43
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 35
Author bci1 (20947)
Entry type Definition
Classification msc 08C99
Classification msc 81R15
Classification msc 57T05
Classification msc 81R50
Classification msc 16W30
Synonym quantum groupoids v.1
Related topic HopfAlgebra
Related topic WeakHopfCAlgebra2
Related topic WeakHopfCAlgebra2
Related topic CommutativeDiagram
Related topic GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries
Related topic GrassmanHopfAlgebrasAndTheirDualCoAlgebras
Related topic WeakHopfCAlgebra2
Defines weak bialgebra
Defines commutant of a set
Defines counit axiom
Defines antipode map
Defines counity
Defines twisted tensor product
Defines quantum double
Defines QOA