08C99-WeakHopfCalgebra.tex

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%\pmkeywords{bialgebras}
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\pmrelated{VonNeumannAlgebra}
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\pmrelated{QuantumGroupoids2}
\pmrelated{LocallyCompactQuantumGroup}
\pmrelated{HopfAlgebra}
\pmrelated{LocallyCompactGroupoids}
\pmrelated{QuantumGroupoids2}
\pmrelated{GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries}
\pmrelated{GrassmanHopfAlgebrasAndTheirDualCoAlge}
\pmdefines{weak Hopf algebra}
\pmdefines{weak Hopf C*-algebra}
\pmdefines{weak bialgebra}
\pmdefines{quantum group}
\pmdefines{quantum groupoid}
\pmdefines{von Neumann algebra}
\pmdefines{$W^*$--algebra}

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\begin{document}
\begin{definition}  A {\em weak Hopf $C^*$-algebra} is defined as a \PMlinkname{weak Hopf algebra}{WeakHopfCAlgebra} 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 \PMlinkname{quantum groupoids}{QuantumGroupoids2}. \end{definition}



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) \emph{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 $M$.


\textbf{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:

\begin{itemize}
\item[(1)] The comultiplication is not necessarily unit-preserving.

\med
\item[(2)] The counit $\vep$ is not necessarily a homomorphism of algebras.

\med
\item[(3)] The axioms for the antipode map $S : A \lra A$ with respect to the
counit are as follows. For all $h \in H$,
\begin{equation}
\begin{aligned} 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)}) ~.
\end{aligned}
\end{equation}
\end{itemize}

These axioms may be appended by the following commutative diagrams
\begin{equation}
{\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}}
\end{equation}
along with the counit axiom:
\begin{equation}
\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}}
\end{equation}

Some authors substitute the term \emph{quantum groupoid} for a weak Hopf algebra.
 
 \med

\subsection{Examples of weak Hopf C*-algebra.}
\begin{itemize}

\item[(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 $N \rtimes A$ (B\"ohm et al. 1999).

\med
\item[(2)]
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 $\U_q (\rm{sl}_2)$ 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.
\end{itemize}


\subsection {Von Neumann Algebras (or $W^*$-algebras).}

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

\begin{itemize}

\item[(1)] $\A$ is closed under the adjoint operation (with the
adjoint of an element $T$ denoted by $T^*$).


\item[(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}
\end{itemize}

If one calls a \emph{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
\emph{unital} subalgebra from $\cL(\H)$, and according to the
first condition in its definition $\A$ does indeed inherit a
\emph{*-subalgebra} structure, as further explained in the next
section on C*-algebras. Furthermore, we have the notable
\emph{Bicommutant Theorem} which states that $\A$ \emph{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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