# PlanetPhysics/Groupoid and Group Representations

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## Groupoid representations[edit | edit source]

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 (*www.utgjiu.ro/math/mbuneci/preprint.html* ).

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 symmetries[edit | edit source]

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).

### Groupoids[edit | edit source]

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
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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 groupoids[edit | edit source]

Let
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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*.

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## References[edit | edit source]

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^{1.0}^{1.1}M. R. Buneci.:*Groupoid Representations*, (orig. title "Reprezentari de Grupoizi"), Ed. Mirton: Timishoara (2003). - ↑
E. M. Alfsen and F. W. Schultz:
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