PlanetPhysics/Groupoid and Group Representations

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Groupoid representationsEdit

Whereas group representations of quantum unitary operators are extensively employed in standard quantum mechanics, the applications of groupoid representations 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  , but cannot be expressed as the simpler group representations on a Hilbert space  . 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 groupoid *-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. Let us consider first several further extensions of symmetry and algebraic topology in the context of local quantum physics/ 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 algebra/algebroids of Prigogine's quantum superoperators which are defined only for irreversible, infinite-dimensional systems (Prigogine, 1980).

Definition of extended quantum groupoid and algebroid symmetriesEdit

Quantum groups~   Representations ~   weak Hopf algebras ~  ~quantum groupoids 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 spaces and \emph{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 geometry); this was later expressed in terms of deformation-quantization: the Hopf algebroid deformation of the universal enveloping algebras of 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 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 Hopf algebroids and monoidal bicategories were investigated by Day and Street (1997).

As defined under the following heading on groupoids, let   be a locally compact groupoid endowed with a (left) Haar system, and let   be the convolution  --algebra (we append   with   if necessary, so that   is unital). Then consider such a groupoid representation

  that respects a compatible measure   on   (cf Buneci, 2003). On taking a state   on  , we assume a parametrization   Furthermore, each   is considered as a \emph{rigged Hilbert space} Bohm and Gadella (1989), that is, one also has the following nested inclusions:   in the usual manner, where   is a dense subspace of   with the appropriate locally convex topology, and   is the space of continuous antilinear functionals of  ~. For each  , we require   to be invariant under   and   is a continuous representation of   on  ~. 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 Hamiltonian generated semigroup of time evolution of a quantum system via integration of Schr\"odinger's equation   as studied in the case of 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 quantum theories (Bohm and Gadella, 1989).

Next, let   be a locally compact Hausdorff groupoid and   a locally compact Hausdorff space. (  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   be a continuous, open and surjective map. For each  , consider the fibre  , and set   equipped with a uniform norm  ~. Then we set  ~. We form a Banach bundle   as follows. Firstly, the projection is defined via the typical fibre  ~. Let   denote the continuous complex valued functions on   with compact support. We obtain a sectional function   defined via restriction as  ~. Commencing from the vector space  , the set   is dense in  ~. For each  , the function   is continuous on  , and each   is a continuous section of  ~. These facts follow from Seda (1982, theorem 1). Furthermore, under the convolution product  , \textit{the space   forms an associative algebra over Failed to parse (unknown function "\bC"): {\displaystyle \bC} } (cf. Seda, 1982, Theorem 3).


Recall that a groupoid   is, loosely speaking, a small category with inverses over its set of objects  ~. One often writes   for the set of morphisms in   from   to  ~. A topological groupoid consists of a space  , a distinguished subspace  , called {\it the space of objects} of  , together with maps Failed to parse (unknown function "\xymatrix"): {\displaystyle r,s~:~ \xymatrix{{\mathsf{G}} \ar@<1ex>[r]^r \ar[r]_s & {\mathsf{G}}^{(0)}} } called the {\it range} and {\it source maps} respectively, together with a law of composition   such that the following hold~:~

\item[(1)]  ~, for all  ~. \item[(2)]  ~, for all  ~. \item[(3)]  ~, for all  ~. \item[(4)]  ~. \item[(5)] Each   has a two--sided inverse   with  ~.

Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call   the set of objects of  ~. For  , the set of arrows   forms a group  , called the isotropy group of   at   . Thus, as is well kown, a topological groupoid is just a groupoid internal to the category of 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 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 tangent groupoid (e.g. [4])
  • (e) holonomy groupoids for foliations (e.g. [4])
  • (f) Poisson groupoids (e.g. [81])
  • (g) graph groupoids (e.g. [47, 64]).

As a simple example of a groupoid, consider (b) above. Thus, let R be an equivalence relation on a set X. Then R is a groupoid under the following operations:  . Here,  , (the diagonal of   ) and  .

Thus,   =  . When  , R is called a trivial groupoid. A special case of a trivial groupoid is  . (So every i is equivalent to every j ). Identify   with the matrix unit  . Then the groupoid   is just matrix multiplication except that we only multiply   when  , and  . We do not really lose anything by restricting the multiplication, since the pairs   excluded from groupoid multiplication just give the 0 product in normal algebra anyway.

For a groupoid   to be a locally compact groupoid means that   is required to be a (second countable) locally compact Hausdorff space , and the product and also inversion maps are required to be continuous. Each   as well as the unit space   is closed in  .

What replaces the left Haar measure on   is a system of measures   ( ), where   is a positive regular Borel measure on   with dense support. In addition, the   's are required to vary continuously (when integrated against   and to form an invariant family in the sense that for each x, the map   is a measure preserving homeomorphism from   onto  . Such a system   is called a left Haar system for the locally compact groupoid  .

This is defined more precisely next.

Haar systems for locally compact topological groupoidsEdit

Let Failed to parse (unknown function "\xymatrix"): {\displaystyle \xymatrix{ {\mathsf{G}} \ar@<1ex>[r]^r \ar[r]_s & {\mathsf{G}}^{(0)}}=X } be a locally compact, locally trivial topological groupoid with its transposition into transitive (connected) components. Recall that for  , the costar of   denoted   is defined as the closed set  , whereby  is a principal  --bundle relative to fixed base points  ~. Assuming all relevant sets are locally compact, then following Seda (1976), a (left) Haar system on   denoted   (for later purposes), is defined to comprise of i) a measure   on  , ii) a measure   on   and iii) a measure   on   such that for every Baire set   of  , the following hold on setting  ~:

 \item[(1)]   is measurable. \item[(2)]   ~. \item[(3)]  , for all   and  ~.

The presence of a left Haar system on   has important topological implications: it requires that the range map   is open. For such a   with a left Haar system, the vector space   is a convolution *--algebra , where for  :

  with  .

One has   to be the enveloping C*--algebra of   (and also representations are required to be continuous in the inductive limit topology). Equivalently, it is the completion of   where   is the universal representation of  . For example, if  , then   is just the finite dimensional algebra  , the span of the  s.

There exists (cf. [1]) a measurable Hilbert bundle   with   and a G-representation L on  . Then, for every pair   of square integrable sections of  , it is required that the function   be  --measurable. The representation   of   is then given by:\\  .

The triple   is called a measurable  --Hilbert bundle.

All SourcesEdit



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