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 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 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 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 algebras.

• [(1)] We refer here to Bais et al. (2002). Let $G$ be a non-Abelian group and $H\subset G$ a discrete subgroup. Let $F(H)$ denote the space of 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 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 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 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 relations there is an $R$ 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 ${\mathcal {R}}^{2}$ is sometimes called the monodromy or Aharanov--Bohm phase factor. In the case of a condensate in a state $\vert v\rangle$ 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 $\vert v\rangle$ 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, $HKH^{-1}=K$~. Next, consider $A=D(H)$~. On breaking a purely electric condensate $\vert v\rangle$, the magnetic symmetry remains unbroken, but the electric symmetry Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \bC H} is broken to Failed to parse (unknown function "\bC"): {\displaystyle \bC N_v} , with $N_{v}\subset H$, the stabilizer of $\vert v\rangle <math>~.Fromthisweobtain$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 noncommutative symmetries of depth 2 von Neumann subfactors. If $A\subset B\subset B_{1}\subset B_{2}\subset \ldots$ is the Jones extension induced by a finite index depth $2$ inclusion $A\subset B$ of $II_{1}$ factors, then $Q=A'\cap B_{2}$ 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' (derived from weak C*--Hopf algebras) comprise (quantum) groupoids of equivalence classes such as associated with 6j--symmetry groups (relative to a fusion rules algebra). They correspond to type $II$ von Neumann algebras 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 $N\rtimes A$ (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 quantum group algebra Failed to parse (unknown function "\U"): {\displaystyle \U_q (\rm{sl}_2)} with $\vert q\vert =1$~. 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 concepts of quantum operator algebra that are essential to algebraic quantum theories. \\

\subsection {Definition of a Von Neumann Algebra.}

Let $\mathbb {H}$ denote a complex (separable) Hilbert space. A \emph{von Neumann algebra} Failed to parse (unknown function "\A"): {\displaystyle \A} acting on $\mathbb {H}$ is a subset of the algebra of all bounded 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 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 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 section on C*-algebras. Furthermore, we have notable Bicommutant 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 $(\xi ,\eta )\longmapsto (A\xi ,\eta )$ for all $<A\xi ,\eta )>$ where $<.,.>$ denotes the inner product defined on $\mathbb {H}$}~. 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 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 $\Delta$ 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 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 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.