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

(3)

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

These axioms may be appended by the following commutative diagrams

(0.2)

(0.3)

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 $\bC H$ 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 \begin{equation} D(H) = F(H)~ \wti{\otimes}~ \bC H~, \end{equation}where, for $x \in H$ , the twisted tensor product is specified by \begin{equation} \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 ~. \end{equation}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 $\vep$ , the double $D(H)$ has a trivial representation given by $\vep(f \otimes h) = f(e)$ . We next look at certain features of this construction.

For the purpose of braiding relations there is an $R$ matrix, $R \in D(H) \otimes D(H)$ , leading to the operator \begin{equation} \mathcal R \equiv \sigma \cdot (\Pi^A_{\a} \otimes \Pi^B_{\be}) (R)~, \end{equation}in terms of the Clebsch-Gordan series $\Pi^A_{\a} \otimes \Pi^B_{\be} \cong N^{AB \gamma}_{\a \be C}~ \Pi^C_{\gamma}$ , 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 $\Pi^A_{\a}$ . One considers the maximal Hopf subalgebra $T$ of a Hopf algebra $A$ for which $\vert v \rangle$ is $T$ -invariant; specifically : \begin{equation} \Pi^A_{\a} (P)~\vert v \rangle = \vep(P) \vert v \rangle~,~ \forall P \in T~. \end{equation}

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

(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 \begin{equation} A \subset B \subset B_1 \subset B_2 \subset \ldots \end{equation}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 $\U_q (\rm{sl}_2)$ with $\vert q \vert =1$ . If $q^p=1$ , then it is shown that a QTQHA is canonically associated with $\U_q (\rm{sl}_2)$ . Such QTQHAs are claimed as the true symmetries of minimal conformal field theories.

Definitions of Related Concepts

Definition of a Von Neumann Algebra.

(1)

$\A$ is closed under the adjoint operation (with the adjoint of an element $T$ denoted by $T^*$ ).

(2)

$\A$ equals its bicommutant, namely:

\begin{equation} \A= \{A \in \cL(\H) : \forall B \in \cL(\H), \forall C\in \A,~ (BC=CB)\Rightarrow (AB=BA)\}~. \end{equation}

If one calls a commutant of a set $\A$ the special set of bounded operators on $\cL(\H)$ 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 $\cL(\H)$ , and according to the first condition in its definition $\A$ 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 $\A$ is a von Neumann algebra if and only if $\A$ is a *-subalgebra of $\cL(\H)$ , 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

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

(1.4)

(1.5)

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

(1.6)

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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