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weak Hopf algebra

(Definition)

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 $\varepsilon$ is not necessarily a homomorphism of algebras.

(3)

The axioms for the antipode map $S : A {\longrightarrow}A$ with respect to the counit are as follows. For all $h \in H$ ,

$\begin{equation*}\begin{aligned}m({\rm id}\otimes S) \Delta (h) &= (\varepsilon ... ...lta(1)) \\ S(h) &= S(h_{(1)}) h_{(2)} S(h_{(3)}) ~. \end{aligned}\end{equation*}$

These axioms may be appended by the following commutative diagrams

$\displaystyle {\begin{CD}A \otimes A @> S\otimes {\rm id}>> A \otimes A \\ @A \... ...A \otimes A \\ @A \Delta AA @VV m V \\ A @ > u \circ \varepsilon >> A \end{CD}}$

(0.2)

along with the counit axiom:

$\displaystyle \xymatrix@C=3pc@R=3pc{ A \otimes A \ar[d]_{\varepsilon \otimes 1}... ...{\rm id}_A} \ar[d]^{\Delta} \\ A & A \otimes A \ar[l]^{1 \otimes \varepsilon }}$

(0.3)

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 $\mathbb{C}H$ 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

$\displaystyle D(H) = F(H)~ \widetilde {\otimes}~ \mathbb{C}H~,$

(0.4)

where, for $x \in H$ , the `twisted tensor product' is specified by

$\displaystyle \widetilde {\otimes} \mapsto ~(f_1 \otimes h_1) (f_2 \otimes h_2)(x) = f_1(x) f_2(h_1 x h_1^{-1}) \otimes h_1 h_2 ~.$

(0.5)

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 $\varepsilon$ , the double $D(H)$

has a trivial representation given by $\varepsilon (f \otimes h) = f(e)$ . We next look at certain features of this construction.

For the purpose of braiding relations there is an $R$ matrix, $R \in D(H) \otimes D(H)$ , leading to the operator

$\displaystyle \mathcal R \equiv \sigma \cdot (\Pi^A_{\alpha } \otimes \Pi^B_{\beta }) (R)~,$

(0.6)

in terms of the Clebsch–Gordan series $\Pi^A_{\alpha } \otimes \Pi^B_{\beta } \cong N^{AB \gamma}_{\alpha \beta C}~ \Pi^C_{\gamma}$ , 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 $\Pi^A_{\alpha }$ . One considers the maximal Hopf subalgebra $T$

of a Hopf algebra $A$

for which $\vert v \rangle$ is $T$

–invariant; specifically :

$\displaystyle \Pi^A_{\alpha } (P)~\vert v \rangle = \varepsilon (P) \vert v \rangle~,~ \forall P \in T~.$

(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

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 $\mathbb{C}H$ is broken to $\mathbb{C}N_v$ , with $N_v \subset H$ , the stabilizer of $\vert v \rangle$ . From this we obtain $T = F(H) \widetilde {\otimes} \mathbb{C}N_v$ .

(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

$\displaystyle A \subset B \subset B_1 \subset B_2 \subset \ldots$

(0.8)

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 $B = B_1^{Q}$ and $B_2 = B_1 \rtimes Q$ . 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 $N \rtimes A$ (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 group algebra ${\rm U}_q (\rm {sl}_2)$ with $\vert q \vert =1$ . If $q^p=1$

, then it is shown that a QTQHA is canonically associated with ${\rm U}_q (\rm {sl}_2)$ . 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.

Definition of a Von Neumann Algebra.

Let $\H$ denote a complex (separable) Hilbert space. A von Neumann algebra $\mathcal A$ acting on $\H$ is a subset of the algebra of all bounded operators $\mathcal L(\H )$ such that:

(1)

$\mathcal A$ is closed under the adjoint operation (with the adjoint of an element $T$

denoted by $T^*$

(2)

$\mathcal A$ equals its bicommutant, namely:

$\displaystyle \mathcal A= \{A \in \mathcal L(\H ) : \forall B \in \mathcal L(\H ), \forall C\in \mathcal A,~ (BC=CB)\Rightarrow (AB=BA)\}~.$

(1.1)

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

On the other hand, a von Neumann algebra $\mathcal A$ inherits a unital subalgebra from $\mathcal L(\H )$ , and according to the first condition in its definition $\mathcal A$ 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 $\mathcal A$ is a von Neumann algebra if and only if $\mathcal A$ is a *-subalgebra of $\mathcal L(\H )$ , 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 $\H$ . 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

$\begin{equation*}\begin{aligned}m &: A \otimes A {\longrightarrow}A~,~(multiplication) \eta &: \mathbb{C}{\longrightarrow}A~,~ (unity) \end{aligned}\end{equation*}$

satisfying the conditions

$\begin{equation*}\begin{aligned} m(m \otimes \mathbf 1) &= m (\mathbf 1 \otimes ... ...es \eta) &= m (\eta \otimes \mathbf 1) = {\rm id}~. \end{aligned}\end{equation*}$

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

$\displaystyle \begin{CD} A \otimes A \otimes A @> m \otimes {\rm id}>> A \otimes A \\ @V {\rm id}\otimes mVV @VV m V \\ A \otimes A @ > m >> A \end{CD}$

(1.4)

Next suppose we consider `reversing the arrows', and take an algebra $A$

equipped with a linear homorphisms $\Delta : A {\longrightarrow}A \otimes A$ , satisfying, for $a,b \in A$ :

$\begin{equation*}\begin{aligned}\Delta(ab) &= \Delta(a) \Delta(b) \\ (\Delta \ot... ...rm id}) \Delta &= ({\rm id}\otimes \Delta) \Delta~. \end{aligned}\end{equation*}$

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

$\displaystyle \begin{CD} A \otimes A \otimes A @< \Delta\otimes {\rm id}<< A \o... ... {\rm id}\otimes \Delta AA @AA \Delta A \\ A \otimes A @ < \Delta << A \end{CD}$

(1.6)

There is also a counterpart to $\eta$ , the counity map $\varepsilon : A {\longrightarrow}\mathbb{C}$ satisfying

$\displaystyle ({\rm id}\otimes \varepsilon ) \circ \Delta = (\varepsilon \otimes {\rm id}) \circ \Delta = {\rm id}~.$

(1.7)

A bialgebra $(A, m, \Delta, \eta, \varepsilon )$ is a linear space $A$

with maps $m, \Delta, \eta, \varepsilon$ satisfying the above properties.

Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism $S : A {\longrightarrow}A$ , satisfying $S(ab) = S(b) S(a)$ , for $a,b \in A$ . This map is defined implicitly via the property :

$\displaystyle m(S \otimes {\rm id}) \circ \Delta = m({\rm id}\otimes S) \circ \Delta = \eta \circ \varepsilon ~~.$

(1.8)

We call

the antipode map. A Hopf algebra is then a bialgebra $(A,m, \eta, \Delta, \varepsilon )$ 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.

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