groupoid and group representations related to quantum symmetries

1 Groupoid representations

Whereas group representationsMathworldPlanetmathPlanetmath ( of quantum unitary operators are extensively employed in standard quantum mechanics, the applications of groupoid representationsPlanetmathPlanetmathPlanetmathPlanetmath ( are still under development. For example, a description of stochastic quantum mechanics in curved spacetime (Drechsler and Tuckey, 1996) involving a Hilbert bundle is possible in terms of groupoid representations which can indeed be defined on such a Hilbert bundle (X*,π), but cannot be expressed as the simpler group representations on a Hilbert spaceMathworldPlanetmath . On the other hand, as in the case of group representations, unitary groupoid representations induce associated C*-algebra representations. In the next subsection we recall some of the basic results concerning groupoid representations and their associated groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath *-algebra representations. For further details and recent results in the mathematical theory of groupoid representations one has also available the succint monograph by Buneci (2003) and references cited therein (

Let us consider first the relationships between these mainly algebraic concepts and their extended quantum symmetries, also including relevant computation examples; then let us consider several further extensionsPlanetmathPlanetmathPlanetmath of symmetryPlanetmathPlanetmathPlanetmath and algebraic topology in the context of local quantum physics/algebraic quantum field theory, symmetry breaking, quantum chromodynamics and the development of novel supersymmetry theories of 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 thermodynamic disorder (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 convolution and 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 fields (e.g., ‘ghost’ particles) along with the broken supersymmetry in the presence of intense gravitational fields. Another significant extension of quantum symmetries may result from the superoperator algebraPlanetmathPlanetmathPlanetmath/algebroids of Prigogine’s quantum superoperators which are defined only for irreversible, infinite-dimensionalPlanetmathPlanetmath systems (Prigogine, 1980).

1.1 Definition of extended quantum groupoid and algebroid symmetries

Quantum groupsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath  Representations   Weak Hopf algebras    Quantum groupoidsPlanetmathPlanetmath and algebroids

Our intention here is to view the latter scheme in terms of weak Hopf C*–algebroid– and/or other– extended symmetries, which we propose to do, for example, by incorporating the concepts of rigged Hilbert spacesMathworldPlanetmathPlanetmath and sectional functions for a small category. We note, however, that an alternative approach to quantum ‘groupoids’ has already been reported (Maltsiniotis, 1992), (perhaps also related to noncommutative geometryPlanetmathPlanetmath); this was later expressed in terms of deformation-quantization: the Hopf algebroid deformation of the universal enveloping algebras of Lie algebroidsMathworldPlanetmath (Xu, 1997) as the classical limit of a quantum ‘groupoid’; this also parallels the introduction of quantum ‘groups’ as the deformation-quantization of Lie bialgebras. Furthermore, such a Hopf algebroid approach (Lu, 1996) leads to categoriesMathworldPlanetmath of Hopf algebroid 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 groupoids, let (𝖦lc,τ) be a locally compact groupoidPlanetmathPlanetmath endowed with a (left) Haar systemPlanetmathPlanetmath, and let A=C*(𝖦lc,τ) be the convolution C*–algebra (we append A with 𝟏 if necessary, so that A is unital). Then consider such a groupoid representation
Λ:(𝖦lc,τ){x,σx}xX that respects a compatibleMathworldPlanetmath measure σx on x (cf Buneci, 2003). On taking a state ρ on A, we assume a parametrization

(x,σx):=(ρ,σ)xX. (1.1)

Furthermore, each x is considered as a rigged Hilbert space Bohm and Gadella (1989), that is, one also has the following nested inclusions:

Φx(x,σx)Φx×, (1.2)

in the usual manner, where Φx is a dense subspace of x with the appropriate locally convex topologyMathworldPlanetmath, and Φx× is the space of continuousPlanetmathPlanetmath antilinear functionalsMathworldPlanetmathPlanetmathPlanetmath of Φ . For each xX, we require Φx to be invariantMathworldPlanetmath under Λ and ImΛ|Φx is a continuous representation of 𝖦lc on Φx . With these conditions, representations of (proper) quantum groupoids that are derived for weak C*–Hopf algebrasPlanetmathPlanetmath (or algebroids) modeled on rigged Hilbert spaces could be suitable generalizationsPlanetmathPlanetmath in the framework of a HamiltonianPlanetmathPlanetmath generated semigroup of time evolution of a quantum system via integration of Schrödinger’s equation ιψt=Hψ as studied in the case of Lie groupsMathworldPlanetmath (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 quantum theoriesPlanetmathPlanetmath (Bohm and Gadella, 1989).

Next, let 𝖦 be a locally compact HausdorffPlanetmathPlanetmath groupoid and X a locally compact Hausdorff spacePlanetmathPlanetmath. (𝖦 will be called a 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 q=(q1,q2):𝖦X×X be a continuous, open and surjective map. For each z=(x,y)X×X, consider the fibre 𝖦z=𝖦(x,y)=q-1(z), and set 𝒜z=C0(𝖦z)=C0(𝖦(x,y)) equipped with a uniform norm z . Then we set 𝒜=z𝒜z . We form a Banach bundle p:𝒜X×X as follows. Firstly, the projectionPlanetmathPlanetmath is defined via the typical fibre p-1(z)=𝒜z=𝒜(x,y) . Let Cc(𝖦) denote the continuous complex valued functions on 𝖦 with compact support. We obtain a sectional function ψ~:X×X𝒜 defined via restrictionPlanetmathPlanetmath as ψ~(z)=ψ|𝖦z=ψ|𝖦(x,y) . Commencing from the vector spaceMathworldPlanetmath γ={ψ~:ψCc(𝖦)}, the set {ψ~(z):ψ~γ} is dense in 𝒜z . For each ψ~γ, the function ψ~(z)z is continuous on X, and each ψ~ is a continuous section of p:𝒜X×X . These facts follow from Seda (1982, Theorem 1). Furthermore, under the convolution productPlanetmathPlanetmath f*g, the space Cc(G) forms an associative algebra over C (cf. Seda, 1982, Theorem 3).

1.2 Groupoids

Recall that a groupoid 𝖦 is, loosely speaking, a small category with inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath over its set of objects X=Ob(𝖦) . One often writes 𝖦xy for the set of morphismsMathworldPlanetmath in 𝖦 from x to y . A topological groupoid consists of a space 𝖦, a distinguished subspaceMathworldPlanetmathPlanetmath 𝖦(0)=Ob(𝖦)𝖦, called the space of objects of 𝖦, together with maps

r,s: (1.3)