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%%% Primary Title: quantum groupoids and C*-algebras
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\begin{document}
This is a topic entry on quantum groupoids, related mathematical \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} and their applications in modern quantum phyiscs.
\begin{definition} \emph{quantum groupoids}, $Q_{\grp}$' s, are currently defined either as quantized, \htmladdnormallink{locally compact groupoids}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} endowed with a left \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html}, $(\grp,\mu)$, or as \emph{\htmladdnormallink{weak Hopf algebras}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}} (WHA). This concept is also an extension of the notion of \htmladdnormallink{quantum group}{http://planetphysics.us/encyclopedia/QuantumGroup.html}, which is sometimes represented by a \htmladdnormallink{Hopf algebra}{http://planetphysics.us/encyclopedia/QuantumGroup.html}, $\mathsf{H}$. \emph{Quantum groupoid \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}} define \htmladdnormallink{extended quantum symmetries}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html} beyond the `Standard Model' (\htmladdnormallink{SUSY}{http://planetphysics.us/encyclopedia/SUSY4.html}) in \htmladdnormallink{mathematical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} or \htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}.
\end{definition}
\subsection{Mathematical Definitions and Related Physical Explanations}
\subsubsection{Quantum Groupoids and the Groupoid C*--Algebra}
Quantum groupoid (e.g., weak Hopf algebras) and \htmladdnormallink{algebroid}{http://planetphysics.us/encyclopedia/Algebroids.html} symmetries figure prominently both in the theory of dynamical \htmladdnormallink{deformations}{http://planetphysics.us/encyclopedia/CohomologicalProperties.html} of quantum `groups' (e.g., Hopf algebras) and the quantum Yang--Baxter equations (Etingof et al., 1999,2001). On the other hand, one can also consider the natural extension of \emph{locally compact} (quantum) \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} to locally compact (proper) \emph{\htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html}} equipped with a Haar measure and a corresponding \htmladdnormallink{groupoid representation}{http://planetphysics.us/encyclopedia/GroupRepresentations.html} theory (Buneci, 2003) as a major, potentially interesting source for locally compact (but
generally \emph{\htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html}}) \emph{quantum groupoids}. The corresponding quantum groupoid representations on bundles of
\htmladdnormallink{Hilbert spaces}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} extend \htmladdnormallink{quantum symmetries}{http://planetphysics.us/encyclopedia/HilbertBundle.html} well beyond those of quantum `groups'/Hopf algebras and simpler \htmladdnormallink{operator algebra}{http://planetphysics.us/encyclopedia/Groupoid.html} representations, and are also consistent with the \htmladdnormallink{locally compact quantum group}{http://planetphysics.us/encyclopedia/LCQG.html} representations that were recently studied in some detail by Kustermans and Vaes (2000, and references cited therein).
The latter quantum groups are neither Hopf algebras, nor are they equivalent to Hopf algebras
or their dual coalgebras. Quantum groupoid representations are, however, the next important
step towards unifying \htmladdnormallink{quantum field theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} with \htmladdnormallink{general relativity}{http://planetphysics.us/encyclopedia/SR.html} in a locally covariant
and quantized form. Such representations need not however be restricted to weak Hopf algebra representations, as
the latter have no known connection to any \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{GR}{http://planetphysics.us/encyclopedia/SR.html} theory and also appear to be inconsistent with GR.
One is also motivated by numerous, important quantum physics examples to introduce a framework for quantum symmetry breaking in terms of either \emph{\htmladdnormallink{locally compact quantum groupoid}{http://planetphysics.us/encyclopedia/UcLocallyCompactQuantumGroupoids.html}}, or related algebroid, representations, such as those of \emph{weak Hopf C*-algebroids with \htmladdnormallink{convolution}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html}}; the latter are usually realized in the context
of \emph{\htmladdnormallink{rigged Hilbert spaces}{http://planetphysics.us/encyclopedia/I3.html}} (Bohm and Gadella, 1989).
Furthermore, with regard to a unified and global framework for symmetry breaking,
as well as higher order quantum symmetries, one needs to look towards the \emph{\htmladdnormallink{double groupoid}{http://planetphysics.us/encyclopedia/WeakHomotopy.html}}
structures of Brown and Spencer (1976), to enable one to introduce the concepts
of \emph{quantum and graded Lie bi--algebroids} which are expected to carry a distinctive C*--algebroid convolution structure. The extension to \emph{\htmladdnormallink{supersymmetry}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}} leads then naturally to \htmladdnormallink{superalgebra}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html}, \htmladdnormallink{superfield}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html} symmetries and their involvement in \htmladdnormallink{supergravity}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html} or \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html} (\htmladdnormallink{QG}{http://planetphysics.us/encyclopedia/SUSY2.html}) theories for intense
gravitational \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} in fluctuating, quantized spacetimes. Their mathematical/quantum \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{classification}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} then involves \htmladdnormallink{superstructures}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} with such \htmladdnormallink{supersymmetries}{http://planetphysics.us/encyclopedia/Supersymmetry.html} that can only be
appropriately studied in (quantum) \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperdiagramsAsHeterofunctors.html}.
Thus, a natural extension of quantum symmetries leads one to \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/HigherDimensionalAlgebra2.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html})
and may involve, for example, both \textit{`quantum' double groupoids} defined as `locally compact'
double groupoids equipped with Haar measures via convolution, and an extension of superalgebra
to double (super) algebroids, (that are naturally much more general than the Lie \htmladdnormallink{double algebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html} defined in Mackenzie,
2004).
One can now proceed to formally define several \htmladdnormallink{Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} concepts that are
needed to express the extended quantum symmetries in terms of proper quantum groupoid and \htmladdnormallink{algebroid representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}. `Hidden', higher dimensional quantum symmetries will then also emerge either
\emph{via} generalized \htmladdnormallink{quantization procedures}{http://planetphysics.us/encyclopedia/MoyalDeformation.html} from higher dimensional algebra representations or be
determined as global or local invariants obtainable-- at least in principle-- through
\htmladdnormallink{non-Abelian algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} (\htmladdnormallink{NAAT}{http://planetphysics.us/encyclopedia/NonabelianAlgebraicTopology3.html}) methods.
\subsubsection{Weak Hopf Algebras}
Let us begin by recalling the notion of a quantum group in \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} to a Hopf algebra where the former is often realized as an automorphism group for a quantum space, that is, an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} in a suitable \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of generally \htmladdnormallink{noncommutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} algebras. One of the most common guises of a quantum `group' is as the dual of a
\htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}, non-associative Hopf algebra. The Hopf algebras (cf. Chaician and Demichev 1996;
Majid,1996), and their generalizations (Karaali, 2007), are some of the fundamental building blocks
of \htmladdnormallink{quantum operator algebra}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} (see the former's definition in the Appendix), even though they cannot be `integrated' to groups like the `integration' of \htmladdnormallink{Lie algebras}{http://planetphysics.us/encyclopedia/BilinearMap.html} to \htmladdnormallink{Lie groups}{http://planetphysics.us/encyclopedia/BilinearMap.html}; furthermore, the connection of Hopf algebras to quantum symmetries seems to be only indirect.
In order to define a \emph{weak Hopf algebra}, one can relax certain axioms of a Hopf algebra as follows~:
\begin{itemize}
\item[(1)]
The comultiplication is not necessarily unit--preserving.
\end{itemize}
\begin{itemize}
\item[(2)]
The counit $\vep$ is not necessarily a \htmladdnormallink{homomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of algebras.
\end{itemize}
\begin{itemize}
\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}
Several mathematicians substitute the term \emph{quantum
groupoid} for a weak Hopf algebra, although this algebra in
itself is not a proper groupoid, but it may have a component
\emph{group} algebra as in the example of the quantum double
discussed next; nevertheless, weak Hopf algebras generalize Hopf
algebras --that with additional properties-- were previously
introduced as quantum `groups' by mathematical physicists. (The
latter are defined in the Appendix and, as already discussed, are
not mathematical groups but algebras). As it will be shown in the
next subsection, quasi--triangular quasi--Hopf algebras are
directly related to quantum symmetries in conformal (quantum)
field theories. Furthermore, weak C*--Hopf \htmladdnormallink{quantum algebras}{http://planetphysics.us/encyclopedia/QuantumAlgebra.html} lead
to weak C*--Hopf algebroids that are linked to quasi--group
quantum symmetries, and also to certain \htmladdnormallink{Lie algebroids}{http://planetphysics.us/encyclopedia/LieAlgebroids.html} (and their
associated Lie--Weinstein groupoids) used to define \htmladdnormallink{Hamiltonian}{http://planetphysics.us/encyclopedia/Hamiltonian2.html} (quantum) algebroids over the phase space of (quantum)
$W_N$--gravity.
\subsubsection{Two Examples of Weak Hopf Algebras}
One can refer here to the example given by Bais et al. (2002). Let G be a non--Abelian group
and $H \subset G$ a discrete subgroup. Let {\em 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} \emph{D(H)} (Drinfel'd, 1987) is defined by the eqn :
$D(H) = F(H)~ \wti{\otimes}~ \bC H~$, where, for $x \in H$, the `twisted \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} product'
is specified by the next eqn:
$\wti{\otimes} \mapsto ~(f_1 \otimes h_1) (f_2 \otimes h_2)(x) =
f_1(x) f_2(h_1 x h_1^{-1}) \otimes h_1 h_2 ~$.
\
The physical interpretation given to this construction usually proceeds by considering $H$
as the `electric \htmladdnormallink{gauge group',}{http://planetphysics.us/encyclopedia/BoseEinsteinStatistics.html} and {\em F(H)} as the `magnetic symmetry' generated by
$\{f \otimes e\}$~. In terms of the counit $\vep$, the double {\em D(H)} has a trivial representation
given by $\vep(f \otimes h) = f(e)$~. there are several very interesting features of this construction.
For the purpose of braiding relations there is available an $R$ \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html}, $R \in D(H) \otimes D(H)$,
leading to the following \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}:
$\mathcal R \equiv \sigma \cdot (\Pi^A_{\a} \otimes \Pi^B_{\be})$ to be defined 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 $\R^2$ 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 ~:
$\Pi^A_{\a} (P)~\vert v \rangle = \vep(P) \vert v \rangle~,~
\forall P \in T~.$
For the second example, consider the example provided by Mack and Schomerus (1992)
using a more general notion of the Drinfel'd construction--the notion of a \emph{quasi
triangular quasi--Hopf algebra} (QTQHA) which was developed with the aim
of studying a range of essential symmetries with special properties, such as the
\emph{quantum group algebra} $\U_q (\rm{sl}_2)$ with $\vert q \vert =1$~. If $q^p=1$,
then it was 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.
\subsubsection{The Weak Hopf C*--Algebra in Relation to Quantum Symmetry Breaking}
In our setting,a \emph{Weak C*--Hopf algebra} is a weak *--Hopf
algebra which admits a faithful \htmladdnormallink{*--representation on a Hilbert space}{http://planetphysics.us/encyclopedia/CoordinateSpace.html}. The weak C*--Hopf algebra is therefore much more likely to
be closely related to a `quantum groupoid' representation than any
weak Hopf algebra. However, one can argue that locally compact
groupoids equipped with a Haar measure (after \htmladdnormallink{quantization}{http://planetphysics.us/encyclopedia/MoyalDeformation.html}) come
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) \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
`\htmladdnormallink{coordinate space}{http://planetphysics.us/encyclopedia/CoordinateSpace.html}' $M$. These transformations along with the
coordinate space $M$ define certain \htmladdnormallink{Lie groupoids}{http://planetphysics.us/encyclopedia/LieAlgebroids.html}, and also their
infinitesimal version - the Lie algebroids $\mathbf{A}$, when the
former are Weinstein groupoids. If one then lifts the algebroid
action from $M$ to the principal homogeneous space $\R$ over the
cotangent bundle $T^*M \lra M$, one obtains a physically
significant algebroid structure. The latter was called the
\htmladdnormallink{Hamiltonian algebroid}{http://planetphysics.us/encyclopedia/Algebroids.html}, ${\mathcal A}^H$, related to the Lie
algebroid, $\mathbf{A}$. The Hamiltonian algebroid is an analog of
the \htmladdnormallink{Lie Algebra}{http://planetphysics.us/encyclopedia/TopologicalOrder2.html} of symplectic \htmladdnormallink{vector fields}{http://planetphysics.us/encyclopedia/NeutrinoRestMass.html} with respect to the
canonical symplectic structure on $\R$ or $T^*M$. In this recent
example, the Hamiltonian algebroid, $\mathcal{A}^H$ over $\R$, was
defined over the phase space of $W_N$--gravity, with the anchor
map to Hamiltonians of canonical transformations (Levin and
Olshanetsky, 2003,2008). Hamiltonian algebroids thus generalize
Lie algebras of canonical transformations; canonical
transformations of the Poisson sigma model phase space define a
\emph{Hamiltonian algebroid} with the Lie brackets related to such
a Poisson structure on the target space. The Hamiltonian algebroid
approach was utilized to analyze the symmetries of generalized
deformations of complex structures on \htmladdnormallink{Riemann surfaces}{http://planetphysics.us/encyclopedia/RiemannSurface.html}
$\sum_{g,n}$ of genus $g $ with $n$ marked points. However, its
implicit algebraic connections to von Neumann *--algebras and/or
\emph{weak C*--algebroid representations} have not yet been
investigated. This example suggests that algebroid (quantum)
symmetries are implicated in the foundation of relativistic
quantum gravity theories and supergravity.
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\end{document}
