ARB symmetry and groupoid representations

ARB symmetry and groupoid representations

1

Symmetry and groupoid representations in functional and abstract relational biology (ARB)

Let us consider first the modelling of

functional

biodynamics in

concrete categories

in

connection

with
mathematical representions of biological, or physiological

functions

of living organisms.
This will provide a

foundation

for the

introduction

of

groupoid

symmetry

and

groupoid
representations

in functional and

abstract relational biology

(ARB).

1.1

Categorical dynamics and mathematical representations in functional biology

Functional biology

is mathematically represented through models of integrated biological functions and activities that are expressed in terms of mathematical

relations

between the metabolic and repair

components

(Rashevsky, 1962

[

2

]

). Such

representations

of complex biosystems,

mappings

/functions, as well as their super-complex dynamics are important for understanding physiological dynamics and functional biology in terms of

algebraic topology

concepts

, concrete categories, and/or graphs; thus, they are describing or modeling theost important
inter-relations of biological functions in living organisms. This approach to biodynamics in terms of

category theory

representations of biological functions is part of the broader field of

categorical dynamics

.

In order to establish mathematical relations, or laws, in biology one needs to define the key concept of

mathematical representations

. A general

definition

of such representations as utilized by mathematical or theoretical biologists, as well as mathematical physicists, is specified next together with well-established mathematical examples.

Definition 1.1

.

Mathematical representations

are defined as

associations

ℜ

:

S

*

→

C

between abstract structures

S

*

and classes

C

, or sets (

S

) of concrete structures

S

c

, often satisfying
several additional conditions, or axioms imposed by the mathematical context (or

category

) to whom
the abstract structures

S

*

belong. Thus, in

representation theory

one is concerned with various

collections

of quantities which are

similar

to the abstract structure in regard to one or several mathematical

operations

.

Notes.

Abstract structures

are employed above in the sense defined by Bourbaki (1964)

[

4

]

.
Unlike

abstract categories

that may have only

morphisms

(or arrows) and ‘no objects’
(or vertices), other

abstract structures

are simply defined as ‘pure’

algebraic

objects with
no numerical

content

or direct physical

interpretation

, whereas the

concrete structures

do have
either a numerical content or a direct physical interpretation.

Examples

1.

An abstract

symmetry group

,

G

with

multiplication

“

⋅

” has mathematical representations by matrices, or numbers, that have the

same multiplication table

as the group (McWeeny, 2002

[

1

]

). In this example, such

similarity

in

structure

is called a

homomorphism

. As a specific illustration consider the symmetry group

C

3

⁢

v

that admits a numerical representation by the sextet of numbers

(

1

,

1

,

1

,

-

1

,

-

1

,

-

1

)

(or line matrix)
for the group symmetry elements

(

E

,

C

3

,

C

¯

3

,

σ

1

,

σ

2

,

σ

3

)

, where the latter
five are

rotations

(or the

generators

of this symmetry group) and

E

is the

unit element

of the
group. Note that the symmetry group

C

3

⁢

v

has the

obvious

geometric

interpretation as the
collection of

symmetry operations

of an

equilateral triangle

. Such symmetry operations are defined by
the

abstract group

elements

, with the group unit element playing the role of the ‘

identity

symmetry operation’
that leaves any physical object (or space on which it acts) unchanged, such as a

360

degree rotation in three-dimensional (real) space. Note that each such symmetry operation of the symmetry group has an

inverse

which ‘cancels out’ exactly the action of its opposite symmetry operation (e.g.,

C

3

and

C

¯

3

),
and of course, multiplication by

E

leaves all symmetry operations unchanged. (This is also true for

non-Abelian

, or noncommutative

groups with

E

acting either on the

left

or on the

right

of all the other

group operations

).

2.

The previous example extends to abstract groupoids

𝒢

whose representations are, however, defined as

morphisms (or

functors

)

, to either

families

or

fiber bundles

of spaces- such as

Hilbert spaces

ℋ

. Moreover, one notes that groupoids exhibit both

internal

and

external

symmetries
(

viz.

Weinstein, 1998). Whereas a group can be considered as a

one object

category with all invertible
morphisms, a groupoid can be defined as a category with all invertible morphisms but with

many objects

instead of just one. Therefore, the groupoid structure has a substantial advantage over the group structure
as it allows for the simultaneous representation of extended symmetries beyond the simpler symmetries represented by groups.

3.

The favorite family of

group representations

in the

current

, Standard Model of Physics (called

SUSY

) is that of the

U

⁢

(

1

)

×

S

⁢

U

⁢

(

2

)

×

S

⁢

U

⁢

(

3

)

product

of symmetry groups; this choice might explain some of the limitations encountered in high energy physics using SUSY and the corresponding physical representations of the symmetry associated with this product of groups, rather than

quantum groupoid

-related symmetries. It is also interesting that

noncommutative geometry

models of

quantum gravity

seem also to be ‘

consistent

with SUSY’ (

viz.

A. Connes, 2004).

4.

The quantum treatment of gravitational fields leads to extended quantum symmetries
(called

‘

supersymmetry

’

) that require mathematical representations of superfields in terms of

graded ‘Lie’

algebras

, or

Lie superalgebras

(Weinberg, 2004

[

3

]

).

5.

Simplified mathematical models of networks of interacting living cells were recently formulated
in terms of

symmetry groupoid representations

, and several interesting

theorems

were proven for such
topological structures (Stewart, 2007) that are relevant to relational and functional biology.

Several areas of functional biology, such as:

functional genomics, interactomics,
and

computer

modeling

of the physiological functions in living organisms, including humans
are now being developed very rapidly because of the huge impact of mathematical representations
and ultra-fast numerical computations in medicine, biotechnology and all life sciences.
Thus, biomathematical and bioinformatics approaches to functional biology
utilize a wide range of mathematical concepts, theories and tools, from ODE’s to biostatistics,
probability theory, graph theory,

topology

,

abstract algebra

,

set theory

, algebraic topology, categories,

many-valued logic

algebras,

higher dimensional algebra

(HDA) and

organismic supercategories

.
Without such mathematical approaches and the use of ultra-fast computers, the recent completion of the first
Human genome

projects

would not have been possible, because it would have taken much longer and
would have been far more costly.

References

1

R. McWeeney. 2002.

Symmetry : An Introduction to

Group Theory

and Its Applications.

Dover Publications Inc.: Mineola, New York, NY.

2

N. Rashevsky.1962.

Mathematical Biology

. Chicago University Press: Chicago.

3

S. Weinberg. 2004.

Quantum Field Theory

, vol.3. Cambridge University Press: Cambridge, UK.

4

N. Bourbaki. 1964. Alg

è

bre

commutative

in

É

l

é

ments de Math

é

matique, Chs. 1-6, Hermann: Paris.

Title

ARB symmetry and groupoid representations

Canonical

name

ARBSymmetryAndGroupoidRepresentations

Date of creation

2013-03-22 18:11:51

Last modified on

2013-03-22 18:11:51

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

58

Author

bci1 (20947)

Entry type

Topic

Classification

msc 92C35

Classification

msc 92C30

Synonym

integrative systems biology

Synonym

relational biology

Synonym

abstract relational biology

Related topic

MolecularSetTheory

Related topic

GroupoidRepresentation4

Related topic

FrameGroupoid

Related topic

CategoricalDynamics

Related topic

QuantumGroupoids2

Related topic

Groupoids

Related topic

LieSuperalgebra3

Related topic

SupercategoriesOfComplexSystems

Related topic

ComplexSystemsBiology

Related topic

AbstractRelationalBiology

Related topic

GeneticNetsOrNetworks

Related topic

OrganismicSets2

Defines

mathematical representations

Defines

C

3

⁢

v

symmetry group

Defines

symmetry operations

Defines

abstract structure

Defines

abstract category

Defines

concrete category

Defines

group representations

Defines

abstract groupoid representations

Defines

SUSY symmetry group product

Defines

functional biology

Defines

ARB

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