Mathematical Concepts of Evolution Algebras in Non-mendelian Genetics

Evolution algebras are not necessarily associative algebras satisfying eiej = 0 whenever ei, ej are two distinct basis elements. They mimic the self-reproduction of alleles in non-Mendelian genetics. We present elementary mathematical properties of evolution algebras that are of importance from the biological point of view. Several models of Mendelian [2, 4, 12, 6, 8, 11] and non-Mendelian genetics [1, 5] exist. Based on the self-reproduction rule of non-Mendelian genetics [1, 7], the first author introduced a new type of algebra [10], called evolution algebra. In this paper we discuss some basic properties of evolution algebras. 1. Evolution algebras and subalgebras Let K be a field. A vector space E over K equipped with multiplication is an algebra (not necessarily associative) if u(v +w) = uv +uw, (u+ v)w = uw + vw, (αu)v = α(uv) = u(αv) for every u, v, w ∈ E and α ∈ K. Let {ei; i ∈ I} be a basis of an algebra E. Then eiej = ∑ k∈I aijkek for some aijk ∈ K, where only finitely many structure constants aijk are nonzero for a fixed i, j ∈ I. The multiplication in E is fully determined by the structure constants aijk, thanks to the distributive laws. Let E be an algebra. Then F ⊆ E is a subalgebra of E if F is a subspace of E closed under multiplication. It is not difficult to show that the intersection of subalgebras is a subalgebra. Thus, given a subset S of E, there is the smallest subalgebra of E containing S. We call it the subalgebra generated by S, and denote it by 〈S〉. As usual: Lemma 1.1. Let S be a subset of an algebra E. Then 〈S〉 consists of all elements of the form α1(s1,1 · · · s1,m1)+ · · ·+αk(sk,1 · · · sk,mk), where k ≥ 1, 1991 Mathematics Subject Classification. Primary: 17A20, 92C15; Secondary: 20N05, 37N25.