Commutative Power-Associative Nilalgebras of Nilindex 5

Abstract

In this paper we study the structure of commutative power-associative nilalgebras of dimension 8 and nilindex ≤ 5 over a field of characteristic different from 2, 3 and 5. We prove that every algebra in this class verifies the identities x4y = 0 and x(x(x(x(xy)))) = 0. In particular, we finish the proof of the Albert’s problem [0] in the following case: every commutative power-associative nilalgebra of dimension ≤ 8 over a field of characteristic ≠ 2, 3 and 5 is solvable. The solvability of these algebras for dimension 4, 5 and 6 were proved in [0], [0] and [0] respectively.