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# weak Hopf C*-algebra

###### Definition 0.1.

A weak Hopf $C^{*}$-algebra is defined as a weak Hopf algebra which admits a faithful $*$–representation on a Hilbert space. The weak C*–Hopf algebra is therefore much more likely to be closely related to a quantum groupoid than the weak Hopf algebra. However, one can argue that locally compact groupoids equipped with a Haar measure are even closer to defining quantum groupoids.

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 $\varepsilon$ 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 $h\in H$,

$\displaystyle m({\rm id}\otimes S)\Delta(h)$ $\displaystyle=(\varepsilon\otimes{\rm id})(\Delta(1)(h\otimes 1))$ (0.1) $\displaystyle m(S\otimes{\rm id})\Delta(h)$ $\displaystyle=({\rm id}\otimes\varepsilon)((1\otimes h)\Delta(1))$ $\displaystyle S(h)$ $\displaystyle=S(h_{{(1)}})h_{{(2)}}S(h_{{(3)}})~{}.$

These axioms may be appended by the following commutative diagrams

${\begin{matrix}A\otimes A&\cd@stack{\rightarrowfill@}{S\otimes{\rm id}}{}&A% \otimes A\\ {\Delta}{\Big\uparrow}&&{}{\Big\downarrow}{m}&&\\ A@ >u\circ\varepsilon>>A\end{matrix}}\qquad{\begin{matrix}A\otimes A&\cd@stack% {\rightarrowfill@}{{\rm id}\otimes S}{}&A\otimes A\\ {\Delta}{\Big\uparrow}&&{}{\Big\downarrow}{m}&&\\ A@ >u\circ\varepsilon>>A\end{matrix}}$ | (0.2) |

along with the counit axiom:

$\xymatrix@C=3pc@R=3pc{A\otimes A\ar[d]_{{\varepsilon\otimes 1}}&A\ar[l]_{{% \Delta}}\ar[dl]_{{{\rm id}_{A}}}\ar[d]^{{\Delta}}\\ A&A\otimes A\ar[l]^{{1\otimes\varepsilon}}}$ | (0.3) |

Some authors substitute the term *quantum groupoid* for a weak Hopf algebra.

# 0.1 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

$A\subset B\subset B_{1}\subset B_{2}\subset\ldots$ (0.4) is the Jones extension induced by a finite index depth $2$ inclusion $A\subset B$ of $II_{1}$ factors, then $Q=A^{{\prime}}\cap B_{2}$ admits a quantum groupoid structure and acts on $B_{1}$, so that $B=B_{1}^{{Q}}$ 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 ${\rm U}_{q}(\rm{sl}_{2})$ with $|q|=1$ . 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.

# 0.2 Von Neumann Algebras (or $W^{*}$-algebras).

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

$\mathcal{A}=\{A\in\mathcal{L}(\mathcal{H}):\forall B\in\mathcal{L}(\mathcal{H}% ),\forall C\in\mathcal{A},~{}(BC=CB)\Rightarrow(AB=BA)\}~{}.$ (0.5)

If one calls a *commutant* of a set $\mathcal{A}$ the special set of
bounded operators on $\mathcal{L}(\mathcal{H})$ 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 $\mathcal{L}(\mathcal{H})$, 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 the notable
*Bicommutant Theorem* which states that $\mathcal{A}$ *is a von
Neumann algebra if and only if $\mathcal{A}$ is a *-subalgebra of
$\mathcal{L}(\mathcal{H})$, 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 $\mathcal{H}$* . 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.

# References

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## Mathematics Subject Classification

08C99*no label found*16S40

*no label found*81R15

*no label found*81R50

*no label found*16W30

*no label found*57T05

*no label found*

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