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 supercomplex 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 interrelations 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 wellestablished mathematical examples.
Definition 1.1.
Mathematical representations are defined as associations $\mathrm{\Re}\mathrm{:}{S}^{\mathrm{*}}\mathrm{\to}C$ between abstract structures ${S}^{\mathrm{*}}$ 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^{} “$\cdot $” 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 ${C}_{3v}$ that admits a numerical representation by the sextet of numbers $(1,1,1,1,1,1)$ (or line matrix) for the group symmetry elements $(E,{C}_{3},{\overline{C}}_{3},{\sigma}_{1},{\sigma}_{2},{\sigma}_{3})$, 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 ${C}_{3v}$ 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 threedimensional (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 ${\overline{C}}_{3}$), and of course, multiplication by $E$ leaves all symmetry operations unchanged. (This is also true for nonAbelian^{}, 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 $\mathcal{G}$ whose representations are, however, defined as morphisms (or functors^{}), to either families or fiber bundles^{} of spaces such as Hilbert spaces^{} $\mathscr{H}$. 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 $U(1)\times SU(2)\times SU(3)$ 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 ultrafast 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, manyvalued logic algebras, higher dimensional algebra^{} (HDA) and organismic supercategories^{}. Without such mathematical approaches and the use of ultrafast 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$\mathrm{\xe8}$bre commutative^{} in $\mathrm{\xc9}$l$\mathrm{\xe9}$ments de Math$\mathrm{\xe9}$matique, Chs. 16, Hermann: Paris.
Title  ARB symmetry and groupoid representations 
Canonical name  ARBSymmetryAndGroupoidRepresentations 
Date of creation  20130322 18:11:51 
Last modified on  20130322 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 