groupoid and group representations related to quantum symmetries

groupoid and group representations related to quantum symmetries

1

Groupoid representations

Whereas

group representations

(

http://planetmath.org/GroupRepresentation

) of quantum

unitary operators

are
extensively employed in standard quantum mechanics, the applications of

groupoid representations

(

http://planetmath.org/RepresentationsOfLocallyCompactGroupoids

)
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

(

X

*

ℋ

,

π

)

, but cannot be expressed as
the simpler group representations on a

Hilbert space

ℋ

. 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;
then let us consider several further

extensions

of

symmetry

and

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 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).

1.1

Definition of extended quantum groupoid and algebroid symmetries

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

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

(

𝖦

l

⁢

c

,

τ

)

be a

locally compact groupoid

endowed with a (left)

Haar system

,
and let

A

=

C

*

⁢

(

𝖦

l

⁢

c

,

τ

)

be the convolution

C

*

–algebra (we append

A

with

𝟏

if necessary, so
that

A

is unital). Then consider such a

groupoid
representation

Λ

:

(

𝖦

l

⁢

c

,

τ

)

⟶

{

ℋ

x

,

σ

x

}

x

∈

X

that respects a

compatible

measure

σ

x

on

ℋ

x

(cf Buneci, 2003). On taking a state

ρ

on

A

, we assume a parametrization

(

ℋ

x

,

σ

x

)

:=

(

ℋ

ρ

,

σ

)

x

∈

X

.

(1.1)

Furthermore, each

ℋ

x

is considered as a

rigged Hilbert
space

Bohm and Gadella (1989), that is, one also has the following nested

inclusions

:

Φ

x

⊂

(

ℋ

x

,

σ

x

)

⊂

Φ

x

×

,

(1.2)

in the usual manner, where

Φ

x

is a dense subspace of

ℋ

x

with the appropriate locally convex

topology

, and

Φ

x

×

is the space of

continuous

antilinear

functionals

of

Φ

. For each

x

∈

X

, we require

Φ

x

to
be

invariant

under

Λ

and

Im

Λ

|

Φ

x

is a
continuous representation of

𝖦

l

⁢

c

on

Φ

x

. 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

ι

⁢

ℏ

⁢

∂

⁡

ψ

∂

⁡

t

=

H

⁢

ψ

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

𝖦

be a

locally compact

Hausdorff

groupoid

and

X

a

locally compact Hausdorff space

. (

𝖦

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

q

=

(

q

1

,

q

2

)

:

𝖦

⟶

X

×

X

be a continuous, open and surjective map.
For each

z

=

(

x

,

y

)

∈

X

×

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

)

)

equipped
with a

uniform norm

∥

∥

z

. Then we set

𝒜

=

⋃

z

𝒜

z

. We form a Banach bundle

p

:

𝒜

⟶

X

×

X

as follows. Firstly, the

projection

is defined via the typical
fibre

p

-

1

⁢

(

z

)

=

𝒜

z

=

𝒜

(

x

,

y

)

. Let

C

c

⁢

(

𝖦

)

denote the
continuous complex valued functions on

𝖦

with compact
support. We obtain a sectional function

ψ

~

:

X

×

X

⟶

𝒜

defined via

restriction

as

ψ

~

(

z

)

=

ψ

|

𝖦

z

=

ψ

|

𝖦

(

x

,

y

)

. Commencing from the

vector space

γ

=

{

ψ

~

:

ψ

∈

C

c

⁢

(

𝖦

)

}

, the set

{

ψ

~

⁢

(

z

)

:

ψ

~

∈

γ

}

is dense in

𝒜

z

. For
each

ψ

~

∈

γ

, the function

∥

ψ

~

⁢

(

z

)

∥

z

is continuous on

X

, and each

ψ

~

is a
continuous section of

p

:

𝒜

⟶

X

×

X

. These facts
follow from Seda (1982,

Theorem

1). Furthermore, under the

convolution
product

f

*

g

,

the space

C

c

⁢

(

G

)

forms an associative algebra
over

C

(cf. Seda, 1982, Theorem 3).

1.2

Groupoids

Recall that a groupoid

𝖦

is, loosely speaking, a small
category with

inverses

over its set of objects

X

=

O

⁢

b

⁢

(

𝖦

)

. One
often 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

)

=

Ob

⁢

(

𝖦

)

⊂

𝖦

,
called

the space of objects

of

𝖦

, together with maps

r

,

s

:

(1.3)

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