New central elements in free alternative algebras

Abstract

A new series of central elements is found in the free alternative algebra. More exactly, let Alt[X] and SMalc[X] ⊂ Alt[X] be the free alternative algebra and the free special Malcev algebra over a field of characteristic 0 on a set of free generators X, and let f (x, y, x₁,…, x_n) ∈ SMalc[X] be a multilinear element which is trivial in the free associative algebra. Then the element u_n = u_n (x, x₁,…,x_n) = f (x², x, x₁,…,x_n) − f (x, x²,x₁,…,x_n) lies in the center of the algebra Alt[X]. The elements u_n(x, x₁,…, x_n) are uniquely defined up to a scalar for a given n (that is, they do not depend on f but only on deg f), and they are skew-symmetric on the variables x₁,…,x_n. Moreover, u_n = 0 for n = 4m + 2, 4m + 3 and u_n ≠ 0 for n = 4m, 4m + 1. The ideals generated by the elements u_4m, u_4m+1 lie in the associative center of the algebra Alt[X] and have trivial multiplication.