PlanetPhysics/Weak Hopf C Algebra 2

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 A weak Hopf $C^{*}$-algebra  is defined as a weak Hopf algebra which admits a

faithful $*$--representation on a [[../NormInducedByInnerProduct/|Hilbert space]]. The weak C*--Hopf algebra is therefore much more likely to be closely related to a `[[../WeakHopfAlgebra/|quantum groupoid]]' than the weak Hopf algebra. However, one can argue that [[../LocallyCompactGroupoid/|locally compact groupoids]] equipped with a [[../HigherDimensionalQuantumAlgebroid/|Haar measure]] are even closer to defining quantum groupoids. There are already several, significant examples that motivate the consideration of weak C*-Hopf algebras which also deserve mentioning in the context of `standard' [[../QuantumOperatorAlgebra5/|quantum theories]]. Furthermore, notions such as (proper) weak C*-algebroids can provide the main framework for symmetry breaking and [[../LQG2/|quantum gravity]] that we are considering here. Thus, one may consider the quasi-group symmetries constructed by means of special transformations of the "coordinate space" $M$.

Remark : Recall that the weak Hopf algebra is defined as the extension of a [[../QuantumOperatorAlgebra5/|Hopf algebra]] by weakening the definining axioms of a Hopf algebra as follows~:

\item[(1)] The comultiplication is not necessarily unit-preserving. \item[(2)] The counit Failed to parse (unknown function "\vep"): {\displaystyle \vep} is not necessarily a [[../TrivialGroupoid/|homomorphism]] of algebras. \item[(3)] The axioms for the antipode map Failed to parse (unknown function "\lra"): {\displaystyle S : A \lra A} with respect to the counit are as follows. For all

$h\in H$

,

Failed to parse (unknown function "\ID"): {\displaystyle m(\ID \otimes S) \Delta (h) &= (\vep \otimes \ID)(\Delta (1) (h \otimes 1)) \\ m(S \otimes \ID) \Delta (h) &= (\ID \otimes \vep)((1 \otimes h) \Delta(1)) \\ S(h) &= S(h_{(1)}) h_{(2)} S(h_{(3)}) ~. }

These axioms may be appended by the following [[../Commutativity/|commutative diagrams]]

Failed to parse (unknown function "\begin{CD}"): {\displaystyle {\begin{CD} A \otimes A @> S\otimes \ID >> A \otimes A \\ @A \Delta AA @VV m V \\ A @ > u \circ \vep >> A \end{CD}} \qquad {\begin{CD} A \otimes A @> \ID\otimes S >> A \otimes A \\ @A \Delta AA @VV m V \\ A @ > u \circ \vep >> A \end{CD}} }

along with the counit axiom:

Failed to parse (unknown function "\xymatrix"): {\displaystyle \xymatrix@C=3pc@R=3pc{ A \otimes A \ar[d]_{\vep \otimes 1} & A \ar[l]_{\Delta} \ar[dl]_{\ID_A} \ar[d]^{\Delta} \\ A & A \otimes A \ar[l]^{1 \otimes \vep}} }

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

Examples of weak Hopf C*-algebra.

\item[(1)] In Nikshych and Vainerman (2000) quantum groupoids were considered as weak C*--Hopf algebras and were studied in relationship to the [[../AbelianCategory3/|noncommutative]] symmetries of depth 2 von Neumann subfactors. If

$A\subset B\subset B_1\subset B_2\subset \dots$

is the Jones extension induced by a finite index depth

$2$

inclusion

$A\subset B$

of

$I{I}_1$

factors, then

$Q=A^{\prime }\cap B_2$

admits a quantum groupoid structure and acts on

$B_1$

, so that </math>B = B_1^{Q}

$and$

B_2 = B_1 \rtimes QFailed to parse (syntax error): {\displaystyle ~. Similarly, in Rehren (1997) `[[../Paragroups/|paragroups]]' (derived from weak C*--Hopf algebras) comprise (quantum) [[../QuantumOperatorAlgebra5/|groupoids]] of equivalence classes such as associated with 6j--symmetry [[../TrivialGroupoid/|groups]] (relative to a fusion rules algebra). They correspond to [[../Bijective/|type]] <math>II} von Neumann algebras in [[../QuantumParadox/|quantum mechanics]], and arise as symmetries where the local subfactors (in the sense of containment of [[../QuantumSpinNetworkFunctor2/|observables]] within [[../CosmologicalConstant/|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

$N⋊A$

(B\"ohm et al. 1999). \item[(2)] In Mack and Schomerus (1992) using a more general notion of the Drinfeld construction, develop the notion of a \emph{quasi triangular quasi--Hopf algebra} (QTQHA) is developed with the aim of studying a range of essential symmetries with special properties, such the [[../QuantumGroup4/|quantum group]] algebra Failed to parse (unknown function "\U"): {\displaystyle \U_q (\rm{sl}_2)} with

$|q|=1$

~. If

$q^p=1$

, then it is shown that a QTQHA is canonically associated with Failed to parse (unknown function "\U"): {\displaystyle \U_q (\rm{sl}_2)} . Such QTQHAs are claimed as the true symmetries of minimal conformal field theories.

\subsection {Von Neumann Algebras (or $W^{*}$-algebras).}

Let $ℍ$ denote a complex (separable) Hilbert space. A \emph{von Neumann algebra} Failed to parse (unknown function "\A"): {\displaystyle \A} acting on $ℍ$ is a subset of the $*$--algebra of all bounded [[../QuantumOperatorAlgebra4/|operators]] Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} such that:

\item[(1)] Failed to parse (unknown function "\A"): {\displaystyle \A} is closed under the adjoint [[../Cod/|operation]] (with the adjoint of an element

$T$

denoted by

$T^{*}$

). \item[(2)] Failed to parse (unknown function "\A"): {\displaystyle \A} equals its bicommutant, namely:

Failed to parse (unknown function "\A"): {\displaystyle \A= \{A \in \cL(\mathbb{H}) : \forall B \in \cL(\mathbb{H}), \forall C\in \A,~ (BC=CB)\Rightarrow (AB=BA)\}~. }

If one calls a commutant of a set Failed to parse (unknown function "\A"): {\displaystyle \A} the special set of bounded operators on Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} which [[../Commutator/|commute]] with all elements in Failed to parse (unknown function "\A"): {\displaystyle \A} , then this second condition implies that the commutant of the commutant of Failed to parse (unknown function "\A"): {\displaystyle \A} is again the set Failed to parse (unknown function "\A"): {\displaystyle \A} .

On the other hand, a von Neumann algebra Failed to parse (unknown function "\A"): {\displaystyle \A} inherits a unital subalgebra from Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} , and according to the first condition in its definition Failed to parse (unknown function "\A"): {\displaystyle \A} does indeed inherit a *-subalgebra structure, as further explained in the next [[../IsomorphicObjectsUnderAnIsomorphism/|section]] on [[../VonNeumannAlgebra2/|C*-algebras]]. Furthermore, we have the notable Bicommutant [[../Formula/|theorem]] which states that Failed to parse (unknown function "\A"): {\displaystyle \A} \emph{is a von Neumann algebra if and only if Failed to parse (unknown function "\A"): {\displaystyle \A} is a *-subalgebra of Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} , closed for the smallest topology defined by continuous maps $(\xi ,\eta )⟼(A\xi ,\eta )$ for all $<A\xi ,\eta )>$ where $<.,.>$ denotes the [[../NormInducedByInnerProduct/|inner product]] defined on $ℍ$}~. For further instruction on this subject, see e.g. Aflsen and Schultz (2003), Connes (1994). \\

Commutative and noncommutative Hopf algebras form the backbone of quantum `groups' and are essential to the generalizations 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 [[../Matrix/|matrices]] there is considerable scope for their [[../CategoricalGroupRepresentation/|representations]] on both finite and infinite dimensional Hilbert spaces.