A class of nilpotent evolution algebras

Abstract

Recently, by A. Elduque and A. Labra a new technique and a type of an evolution algebra are introduced. Several nilpotent evolution algebras defined in terms of bilinear forms and symmetric endomorphisms are constructed. The technique then used for the classification of the nilpotent evolution algebras up to dimension five. In this article, we develop this technique for high dimensional evolution algebras. We construct nilpotent evolution algebras of any type. Moreover, we show that, except the cases considered by Elduque and Labra, this construction of nilpotent evolution algebras does not give all possible nilpotent evolution algebras.

Keywords:

• Evolution algebra
• nilpotent
• non-associative algebra

2010 Mathematics Subject Classifications:

• 17A60
• 17D92

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The concept of isotopism of algebras was introduced in [1] as a generalization of isomorphism. Two n-dimensional algebras A and B defined over a field K are isotopic if there exist three non-singular linear transformations f, g and h from A to B such that $f(u)g(v)=h(uv)$, for all $u,v\in A$.