Abstract
Evolution algebras are currently widely studied due to their importance not only “per se” but also for their many applications to different scientific disciplines, such as Physics or Engineering, for instance. This paper deals with these types of algebras and their applications. A criterion for classifying those satisfying certain conditions is given and an algorithm to obtain degenerate evolution algebras starting from those of smaller dimensions is also analyzed and constructed.
Highlights
Among the most recent references referring to the applications of the evolution operator, the two following ones can be cited: that by Absalamov and Rozikov [22], who investigate in the field of biology the dynamical system generated by a gonosomal evolution operator of sex linked inheritance depending on parameters and the one by Padmanabhan, related to physics, who studies the Planck scale by the modification of the time evolution operator due to the quantum structure of spacetime [23]
The characterization theorem given in this paper allows for obtaining the complete classification in the case of derivations, in any dimension
When the evolution operator is a derivation, it is easy to check that the equality
Summary
Evolution algebras were firstly introduced by Tian in his Ph.D. Thesis [1] in 2004, later published in a book in 2008 [2], in literature, researchers only usually cite [3]. Thesis [1] in 2004, later published in a book in 2008 [2], in literature, researchers only usually cite [3] These types of algebras belongs to the family of genetic algebras and have direct applications in non-Mendelian genetics [3]. The space of derivations of an evolution algebra is a frequent subject of study in the literature [14,15,16,17], obtaining a complete characterization of this space is still an open question. Our main goal is to obtain the complete classification of evolution algebras whose evolution operator is a derivation.
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