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\begin{document}
\textbf{Definition 0.1}:
In order to define a \emph{weak Hopf algebra}, one `weakens' or relaxes certain axioms of a \htmladdnormallink{Hopf algebra}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} as follows~:
\begin{itemize}
\item [(1)] The comultiplication is not necessarily unit--preserving.
\item [(2)] The counit $\vep$ is not necessarily a \htmladdnormallink{homomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of algebras.
\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 \htmladdnormallink{commutative diagrams}{http://planetphysics.us/encyclopedia/Commutativity.html}
\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.
\subsection{Examples of weak Hopf algebras.}
\begin{itemize}
\item [(1)]
We refer here to Bais et al. (2002). Let $G$ be a \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and $H \subset G$ a discrete subgroup. Let $F(H)$ denote the space
of \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} on $H$ and $\bC H$ the group algebra (which consists
of the linear span of group elements with the group structure).
\emph{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 \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} 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 \htmladdnormallink{gauge group}{http://planetphysics.us/encyclopedia/BoseEinsteinStatistics.html}' 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 \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} given by $\vep(f \otimes h) =
f(e)$~. We next look at certain features of this construction.
For the purpose of braiding \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html} there is an $R$ \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html}, $R
\in D(H) \otimes D(H)$, leading to the \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} \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 \emph{monodromy} or
\emph{Aharanov--Bohm \htmladdnormallink{phase factor}{http://planetphysics.us/encyclopedia/PureState.html}}. 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}
\item[(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 $T = F(H) \wti{\otimes} \bC N_v$~.
\item[(3)]
In Nikshych and Vainerman (2000) quantum groupoids (as weak
C*--Hopf algebras, see below) were studied in relationship to the
\htmladdnormallink{noncommutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} 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)
`\htmladdnormallink{paragroups}{http://planetphysics.us/encyclopedia/Paragroups.html}' (derived from weak C*--Hopf algebras) comprise
(quantum) \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} of equivalence classes such as associated with
6j--symmetry groups (relative to a fusion rules algebra). They
correspond to \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} $II$ \htmladdnormallink{von Neumann algebras}{http://planetphysics.us/encyclopedia/CoordinateSpace.html} in \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html},
and arise as symmetries where the local subfactors (in the sense
of containment of \htmladdnormallink{observables}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} within \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}) 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).
\item[(4)]
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 \htmladdnormallink{quantum group}{http://planetphysics.us/encyclopedia/QuantumGroup4.html} 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}
\section{Definitions of Related Concepts}
Let us recall two basic \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of \htmladdnormallink{quantum operator algebra}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} that are essential to \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{quantum theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html}. \\
\subsection {Definition of a Von Neumann Algebra.}
Let $\H$ denote a complex (separable) \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html}. A \emph{von
Neumann algebra} $\A$ acting on $\H$ is a subset of the algebra of
all bounded \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} $\cL(\H)$ such that:
\begin{itemize}
\item[(1)] $\A$ is closed under the adjoint \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} (with the
adjoint of an element $T$ denoted by $T^*$).
\item[(2)]
$\A$ equals its \htmladdnormallink{bicommutant}{http://planetphysics.us/encyclopedia/CoordinateSpace.html}, 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{\htmladdnormallink{commutant}{http://planetphysics.us/encyclopedia/CoordinateSpace.html}} of a set $\A$ the special set of
bounded operators on $\cL(\H)$ which \htmladdnormallink{commute}{http://planetphysics.us/encyclopedia/Commutator.html} 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
\htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} on \htmladdnormallink{C*-algebras}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html}. Furthermore, we have notable
\emph{Bicommutant \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}} 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 \htmladdnormallink{inner product}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} defined on $\H$}~. For
further instruction on this subject, see e.g. Aflsen and Schultz
(2003), Connes (1994).
\subsection{Definition of a Hopf algebra}
Firstly, a {\em unital associative algebra} consists of a linear space
$A$ together with two linear maps
\begin{equation}
\begin{aligned} m &: A \otimes A \lra A~,~(multiplication)
\eta &: \bC \lra A~,~ (unity)
\end{aligned}
\end{equation}
satisfying the conditions
\begin{equation}
\begin{aligned}
m(m \otimes \mathbf 1) &= m (\mathbf 1 \otimes m) \\ m(\mathbf 1
\otimes \eta) &= m (\eta \otimes \mathbf 1) = \ID~.
\end{aligned}
\end{equation}
This first condition can be seen in terms of a commuting \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}~:
\begin{equation}
\begin{CD}
A \otimes A \otimes A @> m \otimes \ID>> A \otimes A
\\ @V \ID \otimes mVV @VV m V
\\ A \otimes A @ > m >> A
\end{CD}
\end{equation}
Next suppose we consider `reversing the arrows', and take an
algebra $A$ equipped with a linear homorphisms $\Delta : A \lra A
\otimes A$, satisfying, for $a,b \in A$ :
\begin{equation}
\begin{aligned} \Delta(ab) &= \Delta(a) \Delta(b)
\\ (\Delta \otimes \ID) \Delta &= (\ID \otimes \Delta) \Delta~.
\end{aligned}
\end{equation}
We call $\Delta$ a \emph{comultiplication}, which is said to be
\emph{coasociative} in so far that the following diagram commutes
\begin{equation}
\begin{CD}
A \otimes A \otimes A @< \Delta\otimes \ID<< A \otimes A
\\ @A \ID \otimes \Delta AA @AA \Delta A
\\ A \otimes A @ < \Delta << A
\end{CD}
\end{equation}
There is also a counterpart to $\eta$, the \emph{counity} map
$\vep : A \lra \bC$ satisfying
\begin{equation}
(\ID \otimes \vep) \circ \Delta = (\vep \otimes \ID) \circ \Delta
= \ID~.
\end{equation}
A \emph{\htmladdnormallink{bialgebra}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}} $(A, m, \Delta, \eta,
\vep)$ is a linear space $A$ with maps $m, \Delta, \eta, \vep$
satisfying the above properties.
Now to recover anything resembling a group structure, we must
append such a bialgebra with an \htmladdnormallink{antihomomorphism}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} $S : A \lra A$,
satisfying $S(ab) = S(b) S(a)$, for $a,b \in A$~. This map is
defined implicitly via the property~:
\begin{equation} m(S \otimes
\ID) \circ \Delta = m(\ID \otimes S) \circ \Delta = \eta \circ
\vep~~.
\end{equation}
We call $S$ the \emph{antipode map}. A \emph{Hopf algebra} is then
a bialgebra $(A,m, \eta, \Delta, \vep)$ 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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