Evolution algebra theory

Behind the phenomena of genetics and stochastic processes, we find there is an intrinsic algebraic structure. We call this algebraic structure---evolution algbera. Evolution algebras are non-associative (non-power-associative) Banach algebras and have many connections with other mathematical fields including graph theory, group theory, Markov chains, dynamic systems, knot theory, 3-manifold and the study of the Riemann-zeta function. In the present dissertation, we will develop the foundation of the theory of evolution algebras and establish a hierarchical structure theorem for evolution algebras.; One of the unusual features of an evolution algebra is that it possesses an evolution operator. This evolution operator reveals the dynamic information of an evolution algebra. However, what makes the theory of evolution algebras different from the classical theory of algebras is that in an evolution algebra, we can have two different kinds of generators: algebraically persistent generators and algebraically transient generators. The basic notions of algebraic persistency and algebraic transiency, and their relative versions, lead to a hierarchical structure on an evolution algebra. Dynamically, this hierarchical structure displays the direction of the flow induced by the evolution operator. Algebraically, this hierarchical structure is given in the form of a sequence of semi-direct-sum decompositions of a general evolution algebra. Thus, this hierarchical structure demonstrates that an evolution algebra is a mixed algebraic and dynamic object. The algebraic nature of this hierarchical structure allows us to have a rough skeleton-shape classification of evolution algebras. On the other hand, the dynamic nature of this hierarchical structure is what makes the notion of an evolution algebra applicable to the study of stochastic processes and many other objects in different fields. For example, when we apply our structure theorem to evolution algebras induced by Markov chains, we see that any general Markov chain has a dynamic hierarchy and the probabilistic flow is moving with invariance on this hierarchy, and that all Markov chains can be classified by the skeleton-shape classification of their evolution algebras.