Classifications of evolution algebras over finite fields

Abstract

In this communication we deal with a class of emerging algebras called evolution algebras. These algebras were firstly introduced by J. P. Tian, and then jointly presented with Vojtechovsky in 2006 [6], and later appeared as a book by Tian in 2008 [5]. The motivation to consider these algebras, which lie between algebras and dynamical systems, is due to the fact that at present, the study of them is very extended (see [2, 3, 4], for instance), due to the numerous connections between these algebras and many other branches of Mathematics, such as Graph Theory, Group Theory, Markov Processes, Dynamic Systems and Theory of Knots, among others. An n-dimensional evolution algebra is an algebra (E, •) over a field K which admits a basis {e1, e2, . . . , en} such that ei • ej = 0, if i 6= j, and ei • ei = Pn k=1 aikek, for any 1 ≤ i ≤ n. Regarding these algebras, it is known the classification of evolution algebras into isomorphism classes. At this respect, the main goal of our study is to know the distribution of such algebras in the case of lower dimensions and finite fields, not only into isomorphism classes (the usual criterion), but also into isotopism classes, which constitutes a novel contribution on this subject (a first paper on this topic can be consulted in [1]). To do this we firstly introduce the concept of evolution isotopism as a triple (f, g, h), where f, g and h are non singular linear transformations between E and E 0 such that h(x•y) = f(x)• g(y) for all x, y ∈ E and that for a natural basis {ei}i of E, {f(ei)}i and {g(ei)}i span an evolution subalgebra of E 0 . Next, taking into consideration that the isotopic relation between evolution algebras implies the isomorphic relation, we begin our study starting from the classification of the 2-dimensional complex evolution algebra E. We know that E will be isomorphic to one of the following pairwise non isomorphic algebras E1 : e1 • e1 = e1, E2 : e1 • e1 = e1, e2 • e2 = e1, E3 : e1 • e1 = e1 +e2, e2 • e2 = −e1−e2, E4 : e1•e1 = e2, E5 : e1•e1 = e1+a2e2, e2•e2 = a3e1+e2, 1−a2a3 6= 0, where E5(a2, a3) ' E 0 5 (a3, a2), E6 : e1 • e1 = e2, e2 • e2 = e1 + a4e2, a4 6= 0, where E6(a4) ' E 0 6 (a 0 4 ) ⇔ (a4/a0 4 ) = cos (2πk/3) + i sin (2πk/3), for some k = 0, 1, 2. This classification allows us to study the isotopic relations between these classes. Indeed, we find that algebra E1 is isotopic to algebra E4, E2 to E3, E5 to E6, and we also obtain that algebras E1, E2 and E5 are not isotopic. Finally, we complete this work with the study of the classifications of low dimensional evolution algebras over Z/pZ, with p prime. We find that the number of isomorphic (or isotopic) classes depends on the dimension of the algebra and on the the value of the integer p.