Evolution Algebras

As a system of abstract algebra, evolution algebras are nonassociative algebras. There is no deep structure theorem for general nonassociative algebra. However, there are deep structure theorem and classification theorem for evolution algebras because we introduce concepts of dynamical systems to evolution algebras. In this chapter, we shall introduce the foundation of the evolution algebras. Section 1 contains basic definitions and properties. Section 2 introduces evolution operators and examines related algebras, including multiplication algebras and derived Lie algebras. Section 3 introduces a norm to an evolution algebra. In Section 4, we introduce the concepts of periodicity, algebraic persistency, and algebraic transiency. In the last section, we obtain the hierarchy of an evolution algebra. For illustration, there are examples in each section.