groupoid and group representations related to quantum symmetries

1 Groupoid representations

Whereas group representations (http://planetmath.org/GroupRepresentation) of quantum unitary operators areextensively employed in standard quantum mechanics, the applications ofgroupoid representations (http://planetmath.org/RepresentationsOfLocallyCompactGroupoids)are still under development. For example, a description of stochastic quantummechanics in curved spacetime (Drechsler and Tuckey, 1996)involving a Hilbert bundle is possible in terms ofgroupoid representations which can indeed be defined onsuch a Hilbert bundle $(X*\mathscr{H},\pi )$, but cannot be expressed asthe simpler group representations on a Hilbert space $\mathscr{H}$. On theother hand, as in the case of group representations, unitarygroupoid representations induce associated C*-algebrarepresentations. In the next subsection we recall some of thebasic results concerning groupoid representations and theirassociated groupoid *-algebra representations. For furtherdetails and recent results in the mathematical theory of groupoidrepresentations one has also available the succint monograph byBuneci (2003) and references cited therein (www.utgjiu.ro/math/mbuneci/preprint.html).

Let us consider first the relationships between these mainly algebraic concepts and their extendedquantum symmetries, also including relevant computation examples;then let us consider several further extensions of symmetryand 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 acriterion for the comparisons between physical, partial or local,symmetries: on the one hand, the example of paracrystalsreveals thermodynamic disorder (entropy) within its own spacetimeframework, whereas in spacetime itself, whatever the selectedmodel, the inhomogeneity arises through (super) gravitationaleffects. More specifically, in the former case one has thetechnique of the generalized Fourier–Stieltjes transform (alongwith convolution and Haar measure), and in view of the latter, wemay compare the resulting ‘broken’/paracrystal–type symmetry withthat of the supersymmetry predictions for weak gravitationalfields (e.g., ‘ghost’ particles) along with the brokensupersymmetry in the presence of intense gravitational fields.Another significant extension of quantum symmetries may resultfrom the superoperator algebra/algebroids of Prigogine’s quantumsuperoperators which are defined only for irreversible,infinite-dimensional systems (Prigogine, 1980).

1.1 Definition of extended quantum groupoid and algebroid symmetries

Quantum groups  $\to$ Representations   $\to$ Weak Hopf algebras  $\to$  Quantum groupoids and algebroids

Our intention here is to view the latter scheme in terms ofweak Hopf C*–algebroid– and/or other– extendedsymmetries, which we propose to do, for example, by incorporatingthe concepts of rigged Hilbert spaces and sectionalfunctions for a small category. We note, however, that analternative approach to quantum ‘groupoids’ has already beenreported (Maltsiniotis, 1992), (perhaps also related tononcommutative geometry); this was later expressed in terms ofdeformation-quantization: the Hopf algebroid deformation of theuniversal enveloping algebras of Lie algebroids (Xu, 1997) as theclassical limit of a quantum ‘groupoid’; this also parallels theintroduction of quantum ‘groups’ as the deformation-quantizationof Lie bialgebras. Furthermore, such a Hopf algebroid approach(Lu, 1996) leads to categories of Hopf algebroid modules (Xu,1997) which are monoidal, whereas the links between Hopfalgebroids and monoidal bicategories were investigated by Day andStreet (1997).

As defined under the following heading on groupoids, let$({𝖦}_{lc},\tau )$ be a locally compact groupoid endowed with a (left) Haar system,and let $A=C^{*}({𝖦}_{lc},\tau )$ be the convolution$C^{*}$–algebra (we append $A$ with $\mathrm{𝟏}$ if necessary, sothat $A$ is unital). Then consider such a groupoidrepresentation
$\mathrm{\Lambda }:({𝖦}_{lc},\tau )⟶{\{{\mathscr{H}}_x,{\sigma }_x\}}_{x\in X}$ 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

$$({\mathscr{H}}_x,{\sigma }_x):={({\mathscr{H}}_{\rho },\sigma )}_{x\in X}.$$(1.1)

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

$${\mathrm{\Phi }}_x\subset ({\mathscr{H}}_x,{\sigma }_x)\subset {\mathrm{\Phi }}_x^{\times },$$(1.2)

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 antilinearfunctionals of $\mathrm{\Phi }$ . For each $x\in X$, we require ${\mathrm{\Phi }}_x$ tobe invariant under $\mathrm{\Lambda }$ and $\mathrm{Im}\mathrm{\Lambda }|{\mathrm{\Phi }}_x$ is acontinuous representation of ${𝖦}_{lc}$ on ${\mathrm{\Phi }}_x$ . With theseconditions, representations of (proper) quantum groupoids that arederived for weak C*–Hopf algebras (or algebroids) modeled onrigged Hilbert spaces could be suitable generalizations in theframework of a Hamiltonian generated semigroup of time evolutionof a quantum system via integration of Schrödinger’s equation$\iota \mathrm{\hslash }\frac{\partial \psi }{\partial t}=H\psi$ as studied inthe case of Lie groups (Wickramasekara and Bohm, 2006). Theadoption of the rigged Hilbert spaces is also based on how thelatter are recognized as reconciling the Dirac and von Neumannapproaches to quantum theories (Bohm and Gadella, 1989).

Next, let $𝖦$ be a locally compact Hausdorff groupoid and $X$ alocally compact Hausdorff space. ($𝖦$ will be called a locally compact groupoid,or lc- groupoid for short). In order to achieve a small C*–categorywe follow a suggestion of A. Seda (private communication) by using ageneral principle in the context of Banach bundles (Seda, 1976, 982)).Let $q=(q_1,q_2):𝖦⟶X\times X$ be a continuous, open and surjective map.For each $z=(x,y)\in X\times X$, consider the fibre${𝖦}_z=𝖦(x,y)=q^{-1}(z)$, and set ${𝒜}_z=C_0({𝖦}_z)=C_0(𝖦(x,y))$ equippedwith a uniform norm $\parallel {\parallel }_z$ . Then we set $𝒜={\bigcup }_z{𝒜}_z$ . We form a Banach bundle $p:𝒜⟶X\times X$as follows. Firstly, the projection is defined via the typicalfibre $p^{-1}(z)={𝒜}_z={𝒜}_{(x,y)}$ . Let $C_c(𝖦)$ denote thecontinuous complex valued functions on $𝖦$ with compactsupport. We obtain a sectional function $\tilde{\psi }:X\times X⟶𝒜$ defined via restriction as $\tilde{\psi }(z)=\psi |{𝖦}_z=\psi |𝖦(x,y)$ . Commencing from the vector space$\gamma =\{\tilde{\psi }:\psi \in C_c(𝖦)\}$, the set $\{\tilde{\psi }(z):\tilde{\psi }\in \gamma \}$ is dense in ${𝒜}_z$ . Foreach $\tilde{\psi }\in \gamma$, the function ${\parallel \tilde{\psi }(z)\parallel }_z$ is continuous on $X$, and each $\tilde{\psi }$ is acontinuous section of $p:𝒜⟶X\times X$ . These factsfollow from Seda (1982, Theorem 1). Furthermore, under the convolutionproduct $f*g$, the space $C_c\mathit{}\mathrm{(}\mathrm{G}\mathrm{)}$ forms an associative algebraover $\mathrm{C}$ (cf. Seda, 1982, Theorem 3).

1.2 Groupoids

Recall that a groupoid $𝖦$ is, loosely speaking, a smallcategory with inverses over its set of objects $X=Ob(𝖦)$ . Oneoften writes ${𝖦}_x^y$ for the set of morphisms in $𝖦$ from$x$ to $y$ . A topological groupoid consists of a space$𝖦$, a distinguished subspace ${𝖦}^{(0)}=\mathrm{Ob}(𝖦)\subset 𝖦$,called the space of objects of $𝖦$, together with maps

$$r,s:$$(1.3)