Introduction

While I was studying stochastic processes and genetics, it occurred to me that there exists an intrinsic and general mathematical structure behind the neutral Wright-Fisher models in population genetics, the reproduction of bacteria involved by bacteriophages, asexual reproduction or generally non-Mendelian inheritance, and Markov chains. Therefore, we defined it as a type of new algebra — the evolution algebra. Evolution algebras are nonassociative and non-power-associative Banach algebras. Indeed, they are natural examples of nonassociative complete normed algebras arising from science. It turns out that these algebras have many unique properties, and also have connections with other fields of mathematics, including graph theory (particularly, random graphs and networks), group theory, Markov processes, dynamical systems, knot theory, 3-manifolds, and the study of the Riemann-zeta function (or a version of it called the Ihara-Selberg zeta function). One of the unusual features of evolution algebras is that they possess an evolution operator. This evolution operator reveals the dynamical information of evolution algebras. However, what makes the theory of evolution algebras different from the classical theory of algebras is that in evolution algebras, we can have two different types of generators: algebraically persistent generators and algebraically transient generators.