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 withmathematical representions of biological, or physiological functions of living organisms.This will provide a foundation for the introduction of groupoid symmetry and groupoidrepresentations 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 importantinter-relations of biological functions in living organisms. This approach to biodynamics in terms ofcategory 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 ofmathematical 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 $\mathrm{\Re }\mathrm{:}{S}^{\mathrm{*}}\mathrm{\to }C$between abstract structures $S^{\mathrm{*}}$ and classes $C$, or sets ($S$) of concrete structures $S_c$, often satisfyingseveral additional conditions, or axioms imposed by the mathematical context (or category) to whomthe abstract structures $S^{*}$ belong. Thus, in representation theory one is concerned with various collectionsof 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 withno numerical content or direct physical interpretation, whereas the concrete structures do haveeither 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 latterfive are rotations (or the generators of this symmetry group) and $E$ is the unit element of thegroup. Note that the symmetry group $C_{3v}$ has the obvious geometric interpretation as thecollection of symmetry operations of an equilateral triangle. Such symmetry operations are defined bythe 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 ${\overline{C}}_3$),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 asmorphisms (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 invertiblemorphisms, 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 structureas 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 ofgraded ‘Lie’ algebras, or Lie superalgebras (Weinberg, 2004 [3]).

5. Simplified mathematical models of networks of interacting living cells were recently formulatedin terms of symmetry groupoid representations, and several interesting theorems were proven for suchtopological 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 humansare now being developed very rapidly because of the huge impact of mathematical representationsand ultra-fast numerical computations in medicine, biotechnology and all life sciences.Thus, biomathematical and bioinformatics approaches to functional biologyutilize 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 firstHuman genome projects would not have been possible, because it would have taken much longer andwould have been far more costly.