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MR2362673 (2008i:17042) 17D92 17A05 17D05 Mallol, Cristi´an (RCH-FRN-EM); Varro, Richard (F-MONT3-MI) Train alg`ebresalternatives: relation entre nilindice et degr´e.(French. English summary) [Alternative train algebras: relation between nilindex and degree] Comm. Algebra 35 (2007), no. 11, 3603–3610.

In this paper train algebras are considered. These algebras appear in a biological context, and so they are baric algebras. A baric algebra A over a field K is a K-algebra with a weight homomorphism ω: A → K, that is, a nonzero algebra homomorphism. In general, algebras appearing in genetics are non-associative, but commutative. A train algebra of degree n + 1 is a baric algebra (A, ω) satisfying n n+1 X n−k+1 k x = αkω(x) x , k=1 for every x ∈ A. k k+1 k Here, x denotes the left principal power of x; that is, x = Lxx, where Lx: A → A denotes the usual left . In a previous paper, the first author studied associative train algebras, proving that the only train identities of degree 3 are x3 = ω(x)x2 and x3 = 2ω(x)x2 − ω(x)2x and the ones of degree 4 are x4 = ω(x)x3, x4 = 2ω(x)x3 − ω(x)2x2, x4 = 3ω(x)x3 − 3ω(x)2x2 + ω(x)3x. In the present paper the associativity assumption is removed. Instead, the authors consider power-associativity (an algebra is power associative if the subalgebra generated by one element is associative). So the main result of the paper proves that, assuming char K 6= 0, the only associated to power-associative train algebras are of the type xp(x − 1)n−p, with 0 ≤ p ≤ n − 1. The existence of a nonzero idempotent element is also proved. If the algebra considered is alternative (recall that an can be characterized as an algebra in which every two elements generate an ), then using the previously proved existence of a nonzero idempotent and the associated Peirce decomposition, A = A00 + A01 + A10 + A11, the following result can be proved: p n−p Theorem. If A is a train algebra of x (x − 1) , then U00 is nil of index ≤ p + 1 and U11 is nil of nilindex ≤ n − p. And, conversely, if at least one among the subspaces Uij is zero, U00 and U11 are nil of nilindex r, s respectively, then A is a train algebra of degree r + s. Consuelo Mart´ınez

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