PlanetPhysics/Quantum Symmetries From Group and Groupoid Representations

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Group and Groupoid representations

Whereas [[../AlgebraicCategoriesAndRepresentationsOfClassesOfAlgebras/|group representations]] of quantum unitary [[../QuantumOperatorAlgebra4/|operators]] are extensively employed in standard [[../QuantumParadox/|quantum mechanics]], the applications of groupoid representations

are still under development. For example, a description of stochastic quantum mechanics in curved [[../SR/|spacetime]] (Drechsler and Tuckey, 1996) involving a Hilbert bundle is possible in terms of [[../GroupRepresentations/|groupoid representations]] which can indeed be defined on such a Hilbert bundle $(X*ℍ,\pi )$, but cannot be expressed as the simpler [[../GroupRepresentations/|group representations]] on a [[../NormInducedByInnerProduct/|Hilbert space]] $ℍ$. On the other hand, as in the case of group representations, unitary groupoid representations induce associated [[../VonNeumannAlgebra2/|C*-algebra]] [[../CategoricalGroupRepresentation/|representations]]. In the next subsection we recall some of the basic results concerning groupoid representations and their associated groupoid *-algebra representations. For further details and recent results in the mathematical theory of groupoid representations one has also available (\htmladdnormallink{the succint monograph by Buneci (2003) and references cited therein}{www.utgjiu.ro/math/mbuneci/preprint.html}).

Let us consider first the relationships between these mainly [[../CoIntersections/|algebraic]] [[../PreciseIdea/|concepts]] and their [[../TopologicalOrder2/|extended quantum symmetries]], also including relevant [[../LQG2/|computation]] examples; then let us consider several further extensions of symmetry and [[../ModuleAlgebraic/|algebraic topology]] in the context of [[../MathematicalFoundationsOfQuantumTheories/|local quantum physics/]] [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theory]], symmetry breaking, quantum chromodynamics and the development of novel [[../Supersymmetry/|supersymmetry]] theories of [[../LQG2/|quantum gravity]]. In this respect one can also take spacetime `inhomogeneity' as a criterion for the comparisons between physical, partial or local, symmetries: on the one hand, the example of paracrystals reveals [[../Thermodynamics/|Thermodynamic]] disorder ([[../ThermodynamicLaws/|entropy]]) within its own spacetime framework, whereas in spacetime itself, whatever the selected model, the inhomogeneity arises through (super) gravitational effects. More specifically, in the former case one has the technique of the generalized Fourier--Stieltjes transform (along with [[../AssociatedGroupoidAlgebraRepresentations/|convolution]] and [[../HigherDimensionalQuantumAlgebroid/|Haar measure]]), and in view of the latter, we may compare the resulting `broken'/paracrystal--type symmetry with that of the supersymmetry predictions for weak gravitational [[../CosmologicalConstant/|fields]] (e.g., `ghost' [[../Particle/|particles]]) along with the broken supersymmetry in the presence of intense gravitational fields. Another significant extension of quantum symmetries may result from the superoperator algebra/algebroids of Prigogine's quantum superoperators which are defined only for irreversible, infinite-dimensional [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] (Prigogine, 1980).

Extended Quantum Groupoid and Algebroid Symmetries

[[../QuantumGroup4/|Quantum groups~]] $\to$ Representations ~ $\to$ [[../WeakHopfAlgebra/|weak Hopf algebras]] ~$\to$ [[../WeakHopfAlgebra/|~quantum groupoids]] and [[../Algebroids/|algebroids]] Our intention here is to view the latter scheme in terms of a weak Hopf C*--algebroid-- and/or other-- extended symmetries, which we propose to do, for example, by incorporating the concepts of [[../I3/|rigged Hilbert spaces]] and \emph{sectional [[../Bijective/|functions]] for a [[../Cod/|small category]]}. We note, however, that an alternative approach to quantum groupoids has already been reported (Maltsiniotis, 1992), (perhaps also related to [[../NoncommutativeGeometry4/|noncommutative geometry]]); this was later expressed in terms of deformation-quantization: the Hopf algebroid [[../CohomologicalProperties/|deformation]] of the universal enveloping algebras of [[../LieAlgebroids/|Lie algebroids]] (Xu, 1997) as the classical limit of a quantum `groupoid'; this also parallels the introduction of quantum `groups' as the deformation-quantization of Lie [[../QuantumOperatorAlgebra5/|bialgebras]]. Furthermore, such a Hopf algebroid approach (Lu, 1996) leads to [[../Cod/|categories]] of Hopf algebroid [[../RModule/|modules]] (Xu, 1997) which are monoidal, whereas the links between Hopf algebroids and monoidal bicategories were investigated by Day and Street (1997).

As defined under the following heading on [[../QuantumOperatorAlgebra5/|groupoids]], let Failed to parse (unknown function "\grp"): {\displaystyle (\grp_{lc},\tau)} be a [[../LocallyCompactGroupoid/|locally compact groupoid]] endowed with a (left) [[../QuantumOperatorAlgebra5/|Haar system]], and let Failed to parse (unknown function "\grp"): {\displaystyle A= C^*(\grp_{lc},\tau)} be the convolution $C^{*}$--algebra (we append $A$ with $\mathrm{𝟏}$ if necessary, so that $A$ is unital). Then consider such a \textit{groupoid representation} \\ Failed to parse (unknown function "\grp"): {\displaystyle \Lambda :(\grp_{lc}, \tau) \lra \{\mathcal H_x, \sigma_x \}_{x \in X}} that respects a compatible measure ${\sigma }_x$ on ${\mathscr{H}}_x$ (cf Buneci, 2003). On taking a state $\rho$ on $A$, we assume a parametrization

$({\mathscr{H}}_x,{\sigma }_x):=({\mathscr{H}}_{\rho },\sigma {)}_{x\in X}.$

Furthermore, each ${\mathscr{H}}_x$ is considered as a \emph{rigged Hilbert space} Bohm and Gadella (1989), that is, one also has the following nested inclusions:

${\mathrm{\Phi }}_x\subset ({\mathscr{H}}_x,{\sigma }_x)\subset {\mathrm{\Phi }}_x^{\times },$

in the usual manner, where ${\mathrm{\Phi }}_x$ is a dense subspace of ${\mathscr{H}}_x$ with the appropriate locally convex topology, and ${\mathrm{\Phi }}_x^{\times }$ is the space of continuous antilinear functionals of $\mathrm{\Phi }$~. For each $x\in X$, we require ${\mathrm{\Phi }}_x$ to be invariant under $\mathrm{\Lambda }$ and Failed to parse (unknown function "\IM"): {\displaystyle \IM~ \Lambda \vert \Phi_x} is a continuous representation of Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} on ${\mathrm{\Phi }}_x$~. With these conditions, representations of (proper) quantum groupoids that are derived for weak C*--Hopf algebras (or algebroids) modeled on rigged Hilbert spaces could be suitable generalizations in the framework of a [[../Hamiltonian2/|Hamiltonian]] generated [[../TrivialGroupoid/|semigroup]] of time evolution of a quantum system via integration of Schr\"odinger's equation Failed to parse (unknown function "\del"): {\displaystyle \iota \hslash \frac{\del \psi}{\del t} = H \psi} as studied in the case of [[../BilinearMap/|Lie groups]] (Wickramasekara and Bohm, 2006). The adoption of the rigged Hilbert spaces is also based on how the latter are recognized as reconciling the Dirac and von Neumann approaches to [[../QuantumOperatorAlgebra5/|quantum theories]] (Bohm and Gadella, 1989).

Next, let Failed to parse (unknown function "\grp"): {\displaystyle \grp} be a locally compact Hausdorff groupoid and $X$ a [[../LocallyCompactHausdorffSpaces/|locally compact Hausdorff space]]. (Failed to parse (unknown function "\grp"): {\displaystyle \grp} will be called a \emph{locally compact groupoid, or lc- groupoid} for short). In order to achieve a small C*--category we follow a suggestion of A. Seda (private communication) by using a general principle in the context of Banach bundles (Seda, 1976, 982)). Let Failed to parse (unknown function "\grp"): {\displaystyle q= (q_1, q_2) : \grp \lra X \times X} be a continuous, open and [[../BCConjecture/|surjective]] map. For each $z=(x,y)\in X\times X$, consider the fibre Failed to parse (unknown function "\grp"): {\displaystyle \grp_z = \grp (x,y) = q^{-1}(z)} , and set Failed to parse (unknown function "\A"): {\displaystyle \A_z = C_0(\grp_z) = C_0(\grp(x,y))} equipped with a uniform [[../NormInducedByInnerProduct/|norm]] $\Vert {\Vert }_z$~. Then we set Failed to parse (unknown function "\A"): {\displaystyle \A = \bigcup_z \A_z<math>~. We form a Banach bundle } p : \A \lra X \times X</math> as follows. Firstly, the projection is defined via the typical fibre Failed to parse (unknown function "\A"): {\displaystyle p^{-1}(z) = \A_z = \A_{(x,y)}} ~. Let Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp)} denote the continuous complex valued functions on Failed to parse (unknown function "\grp"): {\displaystyle \grp} with compact support. We obtain a sectional function Failed to parse (unknown function "\wti"): {\displaystyle \wti{\psi} : X \times X \lra \A<math> defined via restriction as } \wti{\psi}(z) = \psi \vert \grp_z = \psi \vert \grp (x,y)$.Commencingfromthe[[../NormInducedByInnerProduct/|vectorspace]]$\gamma = \{ \wti{\psi} : \psi \in C_c(\grp) \}$,theset$\{ \wti{\psi}(z) : \wti{\psi} \in \gamma \}$isdensein$\A_z</math>~. For each Failed to parse (unknown function "\wti"): {\displaystyle \wti{\psi} \in \gamma} , the function Failed to parse (unknown function "\wti"): {\displaystyle \Vert \wti{\psi} (z) \Vert_z<math> is continuous on } X$,andeach$\wti{\psi}</math> is a continuous [[../IsomorphicObjectsUnderAnIsomorphism/|section]] of Failed to parse (unknown function "\A"): {\displaystyle p : \A \lra X \times X} ~. These facts follow from Seda (1982, [[../Formula/|theorem]] 1). Furthermore, under the convolution product $f*g$, \textit{the space Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp)} forms an associative algebra over Failed to parse (unknown function "\bC"): {\displaystyle \bC} } (cf. Seda, 1982, Theorem 3).

Groupoids

Recall that a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} is, loosely speaking, a small category with inverses over its set of [[../TrivialGroupoid/|objects]] Failed to parse (unknown function "\grp"): {\displaystyle X = Ob(\grp)} ~. One often writes Failed to parse (unknown function "\grp"): {\displaystyle \grp^y_x} for the set of [[../TrivialGroupoid/|morphisms]] in Failed to parse (unknown function "\grp"): {\displaystyle \grp} from $x$ to $y$~. A [[../GroupoidHomomorphism2/|topological groupoid]] consists of a space Failed to parse (unknown function "\grp"): {\displaystyle \grp} , a distinguished subspace Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(0)} = \obg \subset \grp} , called {\it the space of objects} of Failed to parse (unknown function "\grp"): {\displaystyle \grp} , together with maps

Failed to parse (unknown function "\xymatrix"): {\displaystyle r,s~:~ \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)} } }

called the {\it range} and {\it [[../SmallCategory/|source maps]]} respectively,

together with a law of [[../Cod/|composition]]

Failed to parse (unknown function "\grp"): {\displaystyle \circ~:~ \grp^{(2)}: = \grp \times_{\grp^{(0)}} \grp = \{ ~(\gamma_1, \gamma_2) \in \grp \times \grp ~:~ s(\gamma_1) = r(\gamma_2)~ \}~ \lra ~\grp~, }

such that the following hold~:~

\item[(1)] $s({\gamma }_1\circ {\gamma }_2)=r({\gamma }_2),r({\gamma }_1\circ {\gamma }_2)=r({\gamma }_1)<math>,forall$(\gamma_1, \gamma_2) \in \grp^{(2)}</math>~.

\item[(2)] $s(x)=r(x)=x$~, for all Failed to parse (unknown function "\grp"): {\displaystyle x \in \grp^{(0)}} ~.

\item[(3)] $\gamma \circ s(\gamma )=\gamma ,r(\gamma )\circ \gamma =\gamma <math>,forall$\gamma \in \grp</math>~.

\item[(4)] $({\gamma }_1\circ {\gamma }_2)\circ {\gamma }_3={\gamma }_1\circ ({\gamma }_2\circ {\gamma }_3)$~.

\item[(5)] Each $\gamma$ has a two--sided inverse ${\gamma }^{-1}$ with $\gamma {\gamma }^{-1}=r(\gamma ),{\gamma }^{-1}\gamma =s(\gamma )$~.

Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(0)} = Ob(\grp)} {\it the set of objects} of Failed to parse (unknown function "\grp"): {\displaystyle \grp} ~. For Failed to parse (unknown function "\grp"): {\displaystyle u \in Ob(\grp)} , the set of arrows Failed to parse (unknown function "\lra"): {\displaystyle u \lra u} forms a [[../TrivialGroupoid/|group]] Failed to parse (unknown function "\grp"): {\displaystyle \grp_u} , called the isotropy group of Failed to parse (unknown function "\grp"): {\displaystyle \grp} at $u$ .

Thus, as is well kown, a topological groupoid is just a groupoid internal to the category of [[../CoIntersections/|topological]] spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalise bundles of groups, group actions, and [[../TrivialGroupoid/|equivalence relations]]. For a further study of groupoids we refer the reader to Brown (2006).

Several examples of groupoids are:

• (a) locally compact groups, transformation groups, and any group in general (e.g. [59]
• (b) equivalence relations
• (c) tangent bundles
• (d) the [[../MoyalDeformation/|tangent groupoid]] (e.g. [4])
• (e) holonomy groupoids for foliations (e.g. [4])
• (f) Poisson groupoids (e.g. [81])
• (g) [[../Cod/|graph]] groupoids (e.g. [47, 64]).

As a simple example of a groupoid, consider (b) above. Thus, let R be an equivalence [[../Bijective/|relation]] on a set X. Then R is a groupoid under the following [[../Cod/|operations]]: $(x,y)(y,z)=(x,z),(x,y{)}^{-1}=(y,x)$. Here, Failed to parse (unknown function "\grp"): {\displaystyle \grp^0 = X } , (the diagonal of $X\times X$ ) and $r((x,y))=x,s((x,y))=y$.

So $R^2$ = $\{((x,y),(y,z)):(x,y),(y,z)\in R\}$. When $R=X\times X$, R is called a trivial groupoid. A special case of a [[../TrivialGroupoid/|trivial groupoid]] is $R=R_n=\{1,2,...,n\}$ $\times$ $\{1,2,...,n\}$. (So every i is equivalent to every j ). Identify $(i,j)\in R_n$ with the [[../Matrix/|matrix]] unit $e_{ij}$. Then the groupoid $R_n$ is just [[../Matrix/|matrix multiplication]] except that we only multiply $e_{ij},e_{kl}$ when $k=j$, and $(e_{ij}{)}^{-1}=e_{ji}$. We do not really lose anything by restricting the multiplication, since the pairs $e_{ij},e_{kl}$ excluded from groupoid multiplication just give the 0 product in normal algebra anyway.

For a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} to be a locally compact groupoid means that Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is required to be a (second countable) locally compact Hausdorff space , and the product and also inversion maps are required to be continuous. Each Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u} as well as the unit space Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^0} is closed in Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} .

What replaces the left Haar measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is a system of measures ${\lambda }^u$ (Failed to parse (unknown function "\grp"): {\displaystyle u \in \grp_{lc}^0} ), where ${\lambda }^u$ is a positive [[../CoIntersections/|regular]] Borel measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u} with dense support. In addition, the ${\lambda }^u$ 's are required to vary continuously (when integrated against Failed to parse (unknown function "\grp"): {\displaystyle f \in C_c(\grp_{lc}))} and to form an invariant family in the sense that for each x, the map $y\mapsto xy$ is a measure preserving [[../TrivialGroupoid/|homeomorphism]] from Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^s(x)} onto Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^r(x)} . Such a system $\left\{{\lambda }^u\right\}$ is called a left Haar system for the locally compact groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} .

This is defined more precisely next.

Haar systems for locally compact topological groupoids

Let

Failed to parse (unknown function "\xymatrix"): {\displaystyle \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)}}=X }

be a locally compact, locally trivial topological groupoid with its transposition into transitive (connected) components. Recall that for $x\in X$, the costar of $x$ denoted ${\mathrm{C}\mathrm{O}}^{\mathrm{*}}\mathrm{(}\mathrm{x}\mathrm{)}$ is defined as the closed set Failed to parse (unknown function "\grp"): {\displaystyle \bigcup\{ \grp(y,x) : y \in \grp \}} , whereby

Failed to parse (unknown function "\grp"): {\displaystyle \grp(x_0, y_0) \hookrightarrow \rm{CO}^*(x) \lra X~, }

is a principal Failed to parse (unknown function "\grp"): {\displaystyle \grp(x_0, y_0)} --bundle relative to fixed base points $(x_0,y_0)$~. Assuming all relevant sets are locally compact, then following Seda (1976), a \emph{(left) Haar system on Failed to parse (unknown function "\grp"): {\displaystyle \grp} } denoted Failed to parse (unknown function "\grp"): {\displaystyle (\grp, \tau)} (for later purposes), is defined to comprise of i) a measure $\kappa$ on Failed to parse (unknown function "\grp"): {\displaystyle \grp} , ii) a measure $\mu$ on $X$ and iii) a measure ${\mu }_x$ on ${\mathrm{C}\mathrm{O}}^{\mathrm{*}}\mathrm{(}\mathrm{x}\mathrm{)}$ such that for every Baire set $E$ of Failed to parse (unknown function "\grp"): {\displaystyle \grp} , the following hold on setting $E_x=E\cap {\mathrm{C}\mathrm{O}}^{\mathrm{*}}\mathrm{(}\mathrm{x}\mathrm{)}$~:

 \item[(1)] $x\mapsto {\mu }_x(E_x)$ is measurable. \item[(2)] $\kappa (E)=\underset{x}{\int }{\mu }_x(E_x)d{\mu }_x$ ~. \item[(3)] ${\mu }_z(t{E}_x)={\mu }_x(E_x)$, for all Failed to parse (unknown function "\grp"): {\displaystyle t \in \grp(x,z)}
 and Failed to parse (unknown function "\grp"): {\displaystyle x, z \in \grp}
~.

The presence of a left Haar system on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} has important topological implications: it requires that the range map Failed to parse (unknown function "\grp"): {\displaystyle r: \grp_{lc} \rightarrow \grp_{lc}^0} is open. For such a Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} with a left Haar system, the vector space Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})} is a convolution *--algebra , where for Failed to parse (unknown function "\grp"): {\displaystyle f, g \in C_c(\grp_{lc})} :

$f*g(x)=\int f(t)g(t^{-1}x)d{\lambda }^{r(x)}(t),$ with $f*(x)=\overline{f(x^{-1})}.$

One has Failed to parse (unknown function "\grp"): {\displaystyle C^*(\grp_{lc})} to be the enveloping C*--algebra of Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})} (and also representations are required to be continuous in the inductive limit topology). Equivalently, it is the completion of Failed to parse (unknown function "\grp"): {\displaystyle \pi_{univ}(C_c(\grp_{lc}))} where ${\pi }_{univ}$ is the universal representation of Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} . For example, if Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc} = R_n} , then Failed to parse (unknown function "\grp"): {\displaystyle C^*(\grp_{lc})} is just the finite dimensional algebra Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc}) = M_n} , the span of the $e_{i^{\prime }j}$s.

There exists (cf. ) a measurable Hilbert bundle Failed to parse (unknown function "\grp"): {\displaystyle (\grp_{lc}^0, \mathbb{H}, \mu)} with Failed to parse (unknown function "\grp"): {\displaystyle \mathbb{H} = \left\{ \mathbb{H}^u_{u \in \grp_{lc}^0} \right\}<math> and a G-representation L on } \H</math>. Then, for every pair $\xi ,\eta$ of [[../PiecewiseLinear/|square]] integrable sections of $ℍ$, it is required that the function $x\mapsto (L(x)\xi (s(x)),\eta (r(x)))<math>be$\nu$--measurable.Therepresentation$\Phi</math> of Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})} is then given by:\\ $⟨\mathrm{\Phi }(f)\xi |,\eta ⟩=\int f(x)(L(x)\xi (s(x)),\eta (r(x)))d{\nu }_0(x)$.

The triple $(\mu ,ℍ,L)$ is called a \textit{measurable Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} --Hilbert bundle}.