In this book, we introduce a new type of algebra, which we call evolution
algebras. These are algebras in which the multiplication tables are of a spe-
cial type. They are motivated by evolution laws of genetics. We view alleles
(or organelles or cells, etc,) as generators of algebras. Therefore we define the
multiplication of two “alleles” Gi and Gj by Gi · Gj = 0 if i à= j. However,
Gi · Gi is viewed as “self-reproduction,” so that Gi · Gi =
pij Gj , where the j
summation is taken over all generators Gj. Thus, reproduction in genetics is represented by multiplication in algebra. It seems obvious that this type of algebra is nonassociative, but commutative. When the pijs form Markovian transition probabilities, the properties of algebras are associated with prop- erties of Markov chains. Markov chains allow us to develop an algebra the- ory at deeper hierarchical levels than standard algebras. After we introduce several new algebraic concepts, particularly algebraic persistency, algebraic transiency, algebraic periodicity, and their relative versions, we establish hier- archical structures for evolution algebras in Chapter 3. The analysis developed in this book, particularly in Chapter 4, enables us to take a new perspective on Markov process theory and to derive new algebraic properties for Markov chains at the same time. We see that any Markov chain has a dynamical hi- erarchy and a probabilistic flow that is moving with invariance through this hierarchy. We also see that Markov chains can be classified by the skeleton- shape classification of their evolution algebras. Remarkably, when applied to non-Mendelian genetics, particularly organelle heredity, evolution algebras can explain establishment of homoplasmy from heteroplasmic cell population and the coexistence of mitochondrial triplasmy, and can also predict all possible mechanisms to establish the homoplasmy of cell population. Actually, these mechanisms are hypothetical mechanisms in current mitochondrial disease research. By using evolution algebras, it is easy to identify different genetic patterns from the complexity of the progenies of Phytophthora infectans that cause the late blight of potatoes and tomatoes. Evolution algebras have many connections with other fields of mathematics, such as graph theory, group theory, knot theory, 3-manifolds, and Ihara-Selberg zeta functions.