PlanetPhysics/Weak Hopf Algebra

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Definition 0.1 :

In order to define a weak Hopf algebra , one `weakens' or relaxes certain axioms of a [[../QuantumOperatorAlgebra5/|Hopf algebra]] as follows~:

• [(1)] The comultiplication is not necessarily unit--preserving.
• [(2)] The counit Failed to parse (unknown function "\vep"): {\displaystyle \vep} is not necessarily a [[../TrivialGroupoid/|homomorphism]] of algebras.
• [(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 Template:'groupoidTemplate:' for a weak Hopf algebra.

Examples of weak Hopf algebras.

• [(1)] We refer here to Bais et al. (2002). Let $G$ be a [[../AbelianCategory3/|non-Abelian]] [[../TrivialGroupoid/|group]] and $H\subset G$ a discrete subgroup. Let $F(H)$ denote the space of [[../Bijective/|functions]] on $H$ and Failed to parse (unknown function "\bC"): {\displaystyle \bC H} the group algebra (which consists of the linear span of group elements with the group structure). The quantum double $D(H)$ (Drinfeld, 1987) is defined by Failed to parse (unknown function "\wti"): {\displaystyle D(H) = F(H)~ \wti{\otimes}~ \bC H~, } where, for $x\in H$, the `twisted [[../Tensor/|tensor]] product' is specified by Failed to parse (unknown function "\wti"): {\displaystyle \wti{\otimes} \mapsto ~(f_1 \otimes h_1) (f_2 \otimes h_2)(x) = f_1(x) f_2(h_1 x h_1^{-1}) \otimes h_1 h_2 ~. } The physical interpretation is often to take $H$ as the `electric [[../BoseEinsteinStatistics/|gauge group]]' and $F(H)$ as the `magnetic symmetry' generated by $\{f\otimes e\}$~. In terms of the counit Failed to parse (unknown function "\vep"): {\displaystyle \vep} , the double $D(H)$ has a trivial [[../CategoricalGroupRepresentation/|representation]] given by Failed to parse (unknown function "\vep"): {\displaystyle \vep(f \otimes h) = f(e)} ~. We next look at certain features of this construction. For the purpose of braiding [[../Bijective/|relations]] there is an $R$ [[../Matrix/|matrix]], Failed to parse (unknown function "\a"): {\displaystyle R \in D(H) \otimes D(H)<math>, leading to the [[../QuantumSpinNetworkFunctor2/|operator]] <math> \mathcal R \equiv \sigma \cdot (\Pi^A_{\a} \otimes \Pi^B_{\be}) (R)~, } in terms of the Clebsch--Gordan series Failed to parse (unknown function "\a"): {\displaystyle \Pi^A_{\a} \otimes \Pi^B_{\be} \cong N^{AB \gamma}_{\a \be C}~ \Pi^C_{\gamma}} , and where $\sigma$ denotes a flip operator. The operator ${ℛ}^2$ is sometimes called the monodromy or Aharanov--Bohm [[../PureState/|phase factor]]. In the case of a condensate in a state $|v⟩$ in the carrier space of some representation Failed to parse (unknown function "\a"): {\displaystyle \Pi^A_{\a}} ~. One considers the maximal Hopf subalgebra $T$ of a Hopf algebra $A$ for which $|v⟩$ is $T$--invariant; specifically ~: Failed to parse (unknown function "\a"): {\displaystyle \Pi^A_{\a} (P)~\vert v \rangle = \vep(P) \vert v \rangle~,~ \forall P \in T~. } \item[(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, $HK{H}^{-1}=K$~. Next, consider $A=D(H)$~. On breaking a purely electric condensate $|v⟩$, the magnetic symmetry remains unbroken, but the electric symmetry Failed to parse (unknown function "\bC"): {\displaystyle \bC H} is broken to Failed to parse (unknown function "\bC"): {\displaystyle \bC N_v} , with $N_v\subset H$, the stabilizer of $|v⟩<math>.Fromthisweobtain$T = F(H) \wti{\otimes} \bC N_v</math>~. \item[(3)] In Nikshych and Vainerman (2000) quantum groupoids (as weak C*--Hopf algebras, see below) 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 $B=B_1^Q<math>and$B_2 = B_1 \rtimes Q</math>~. Similarly, in Rehren (1997) `[[../Paragroups/|paragroups]]' (derived from weak C*--Hopf algebras) comprise (quantum) [[../GroupoidHomomorphism2/|groupoids]] of equivalence classes such as associated with 6j--symmetry groups (relative to a fusion rules algebra). They correspond to [[../Bijective/|type]] $II$ [[../CoordinateSpace/|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[(4)] 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.

Definitions of Related Concepts

Let us recall two basic [[../PreciseIdea/|concepts]] of [[../QuantumOperatorAlgebra5/|quantum operator algebra]] that are essential to [[../CoIntersections/|algebraic]] [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum theories]]. \\

\subsection {Definition of a Von Neumann Algebra.}

Let $ℍ$ denote a complex (separable) [[../NormInducedByInnerProduct/|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 [[../CoordinateSpace/|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 [[../CoordinateSpace/|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 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).

Definition of a Hopf algebra

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

Failed to parse (syntax error): {\displaystyle m &: A \otimes A \lra A~,~(multiplication) \eta &: \bC \lra A~,~ (unity) } satisfying the conditions Failed to parse (syntax error): {\displaystyle m(m \otimes \mathbf 1) &= m (\mathbf 1 \otimes m) \\ m(\mathbf 1 \otimes \eta) &= m (\eta \otimes \mathbf 1) = \ID~. } This first condition can be seen in terms of a commuting [[../TrivialGroupoid/|diagram]]~: Failed to parse (unknown function "\begin{CD}"): {\displaystyle \begin{CD} A \otimes A \otimes A @> m \otimes \ID>> A \otimes A \\ @V \ID \otimes mVV @VV m V \\ A \otimes A @ > m >> A \end{CD} } Next suppose we consider `reversing the arrows', and take an algebra $A$ equipped with a linear homorphisms Failed to parse (unknown function "\lra"): {\displaystyle \Delta : A \lra A \otimes A<math>, satisfying, for } a,b \in A</math> :

Failed to parse (syntax error): {\displaystyle \Delta(ab) &= \Delta(a) \Delta(b) \\ (\Delta \otimes \ID) \Delta &= (\ID \otimes \Delta) \Delta~. } We call $\mathrm{\Delta }$ a comultiplication , which is said to be coasociative in so far that the following diagram commutes Failed to parse (unknown function "\begin{CD}"): {\displaystyle \begin{CD} A \otimes A \otimes A @< \Delta\otimes \ID<< A \otimes A \\ @A \ID \otimes \Delta AA @AA \Delta A \\ A \otimes A @ < \Delta << A \end{CD} } There is also a counterpart to $\eta$, the counity map Failed to parse (unknown function "\vep"): {\displaystyle \vep : A \lra \bC} satisfying Failed to parse (unknown function "\ID"): {\displaystyle (\ID \otimes \vep) \circ \Delta = (\vep \otimes \ID) \circ \Delta = \ID~. } A [[../QuantumOperatorAlgebra5/|bialgebra]] Failed to parse (unknown function "\vep"): {\displaystyle (A, m, \Delta, \eta, \vep)<math> is a linear space } A$withmaps$m, \Delta, \eta, \vep</math> satisfying the above properties.

Now to recover anything resembling a group structure, we must append such a bialgebra with an [[../QuantumOperatorAlgebra5/|antihomomorphism]] Failed to parse (unknown function "\lra"): {\displaystyle S : A \lra A} , satisfying $S(ab)=S(b)S(a)$, for $a,b\in A$~. This map is defined implicitly via the property~: Failed to parse (unknown function "\ID"): {\displaystyle m(S \otimes \ID) \circ \Delta = m(\ID \otimes S) \circ \Delta = \eta \circ \vep~~. } We call $S$ the antipode map . A Hopf algebra is then a bialgebra Failed to parse (unknown function "\vep"): {\displaystyle (A,m, \eta, \Delta, \vep)} equipped with an antipode map $S$~.

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 matrices there is considerable scope for their representations on both finite and infinite dimensional Hilbert spaces.