Sur une classe d'algèbres à puissances associatives

Power-associative algebras that are train algebras

“…One of the present authors explored in [20] the cases in which A 0 = 0 or A 1 = 0. Here we consider the general situation.…”

“…(i) Assume that A is a nth-order Bernstein algebra. Then by [20,Théorème 3.7], the identity x 2 n +1 − ω(x)x 2 n = 0 holds in A. Hence A is a train algebra whose train polynomial P (X) divides Q (X) = X 2 n +1 − X 2 n = X 2 n (X − 1).…”

“…By parts (a) and (b) of Lemma 3.7, we have S x λ S y λ + S y λ S x λ = S x λ y λ . Thus, one easily obtains as in [20,Lemme 1.4] the following Lemma 3.8. For all k 2, λ ∈ {0, 1} and x λ ∈ A λ , we have:…”

“…Some results in this direction were given in [6,14,33] for ranks 4. The main goal of the present paper is to develop a structure theory for power-associative algebras that are train algebras. Our point of departure is the previous paper [20], where two special cases have been examined. This paper is organized as follows.…”

Derivations in power-associative algebras

“…It is known ( [5], [16]) that any power-associative n th -order Bernstein algebra is a train algebra satisfying x 2 n +1 − ω(x)x 2 n = 0, hence the train rank is ≤ 2 n + 1. We have recently shown ([3], Theorem 5.5) that, if A is a power-associative train algebra of rank m + 1, with train equation x m+1 − ω(x)x m = 0, i.e., of presentation (m, 1), then A is a n th -order Bernstein, where n ≥ 0 is the unique integer satisfying 2 n−1 < m ≤ 2 n .…”