Some Properties of Evolution Algebras

Abstract

Abstract. The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criterium of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent n-dimensional complex evolution algebras the possible maximal nilpotency index is 1 + 2 n−1 .

On Evolution Algebras

“…, n + 1}. Thus, a ipip+1 a ip+1ip+2 · · · a iq−1ip = 0, a contradiction with (1). So there is a row π n which consists of zeros (n zeros).…”

“…Any nilpotent evolution algebra is solvable, and in [1], it is proved that the notions of nilpotent and right nilpotent are equivalent.…”

“…Consider the A πn -minor of A, which is obtained from A deleting the π n -th row and π n -th column. A πn is an (n − 1) × (n − 1) matrix satisfying condition (1). To prove that A has a row with n − 1 zeros, it is sufficient to prove that A πn has a row with all zeros.…”

Power-Associative Evolution Algebras

“…In [4,Example 4.8], for k = n = 4, we find a power-associative algebra which is not associative. Indeed, it is isomorphic to N 4,6 (see Table 1).…”

“…In [3, Table 1], the fifth, sixth and seventh algebras, of dimension 3, are associative. They are respectively isomorphic to E 3,4 , N 3,3 (1) and N 3,2 (see Table 1). In [9, Theorem 6.1], the algebras E 4,1 , E 4,2 , E 4,3 , E 4,5 and E 4,10 are associative.…”