There are already several, significant examples that motivate the consideration of weak C*-Hopf algebras which also deserve mentioning in the context of standard quantum theories. Furthermore, notions such as (proper) weak C*-algebroids can provide the main framework for symmetry breaking and 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 Hopf algebra by weakening the definining 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 C*-algebra.

(1)

In Nikshych and Vainerman (2000) quantum groupoids were considered as weak C*-Hopf algebras and 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).

(2)

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.

Von Neumann Algebras (or $W^*$ -algebras).

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

(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 the 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).

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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