# PlanetPhysics/Quantum Symmetries From Group and Groupoid Representations

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### Group and 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 (\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 algebraic concepts and their extended quantum symmetries, also including relevant computation examples;
then let us consider 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).

### 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 a
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
**Failed to parse (unknown function "\grp"): {\displaystyle (\grp_{lc},\tau)}**
be a *locally compact groupoid* endowed with a (left) Haar system,
and let **Failed to parse (unknown function "\grp"): {\displaystyle A= C^*(\grp_{lc},\tau)}**
be the convolution
--algebra (we append with if necessary, so
that 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
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 **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 ~. 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
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \iota \hslash \frac{\del \psi}{\del t} = H \psi}**
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 **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
be a *locally compact Hausdorff groupoid* and a
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 surjective map.
For each , 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 norm ~. 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)\gamma = \{ \wti{\psi} : \psi \in C_c(\grp) \}\{
\wti{\psi}(z) : \wti{\psi} \in \gamma \}\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\wti{\psi}</math> is a
continuous section of **Failed to parse (unknown function "\A"): {\displaystyle p : \A \lra X \times X}**
~. These facts
follow from Seda (1982, theorem 1). Furthermore, under the convolution
product , \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).

### GroupoidsEdit

Recall that a groupoid **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
is, loosely speaking, a small
category with inverses over its set of 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 morphisms in **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \grp}**
from
to ~. *A 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 source maps} respectively,

together with a law of 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)] (\gamma_1, \gamma_2) \in \grp^{(2)}</math>~.

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

\item[(3)] \gamma \in \grp</math>~.

\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 **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
group **Failed to parse (unknown function "\grp"): {\displaystyle \grp_u}**
, called the *isotropy group of Failed to parse (unknown function "\grp"): {\displaystyle \grp}
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, **Failed to parse (unknown function "\grp"): {\displaystyle \grp^0 = X }**
, (the diagonal of ) and .

So = .
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 **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 (**Failed to parse (unknown function "\grp"): {\displaystyle u \in \grp_{lc}^0}**
), where is a positive regular Borel measure on **Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u}**
with dense support. In addition, the '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 is a measure preserving 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
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 groupoidsEdit

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 , the *costar of * denoted
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 ~. 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 on **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
, ii) a
measure on and iii) a measure on
such that for every Baire set of **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
, the following hold on
setting ~:

\item[(1)] is measurable. \item[(2)] ~. \item[(3)] , for allFailed to parse (unknown function "\grp"): {\displaystyle t \in \grp(x,z)}andFailed 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})}**
:

with

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
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
s.

There exists (cf. ^{[1]}) 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 of square integrable sections of ,
it is required that the function \nu\Phi</math> of
**Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})}**
is then given by:\\ .

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

## All SourcesEdit

^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[1]}^{[8]}^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}^{[14]}^{[15]}^{[16]}^{[17]}^{[18]}^{[19]}^{[20]}^{[21]}^{[22]}^{[23]}^{[24]}^{[25]}^{[26]}^{[27]}

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