On real chains of evolution algebras

Abstract

In this paper, we define a chain of -dimensional evolution algebras corresponding to a permutation of numbers. We show that a chain of evolution algebras (CEA) corresponding to a permutation is non-trivial if and only if the permutation has a fixed point. We show that a CEA is a chain of nilpotent algebras (independently on time) if it is trivial. We construct a wide class of chains of three-dimensional EAs and a class of symmetric -dimensional CEAs. A construction of arbitrary dimensional CEAs is given. Moreover, for a chain of three-dimensional EAs, we study the behaviour of the baric property, the behaviour of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

Keywords:

• evolution algebra
• time
• Chapman–Kolmogorov equation
• nilpotentcy

AMS Subject Classifications:

• 17D92
• 17D99
• 60J27

Notes

1 We do not know another solution of the equation.

The work is supported by the [grant number 0251/GF3] of Education and Science Ministry of Republic of Kazakhstan. The authors are grateful to referee for helpful suggestions.