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; 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).
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 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
![]() |
(0.1) |
Furthermore, each
is considered as a rigged Hilbert
space Bohm and Gadella (1989), that is, one also has the following nested inclusions:
![]() |
(0.2) |
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 Schrödinger'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
, the space
forms an associative algebra
over
(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 the space of objects of
, together with maps
![]() |
(0.3) |
called the range and source maps respectively, together with a law of composition
![]() |
(0.4) |
such that the following hold :
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:
As a simple example of a groupoid, consider (b) above. Thus, let R be an equivalence relationhttp://planetphysics.org/encyclopedia/Bijective.html on a set X. Then R is a groupoid under the following operations:
. Here,
, (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.
This is defined more precisely next.
Let
![]() |
(0.5) |
![]() |
(0.6) |
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. [7]) 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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