New elements in the center of free alternative algebra

TL;DR

This paper introduces a new series of central elements in free alternative algebras, characterized by specific multilinear properties and symmetry, expanding understanding of their algebraic structure.

Contribution

It identifies and characterizes a new series of central elements in free alternative algebras, detailing their properties and conditions for non-vanishing.

Findings

The elements $u_n$ are central in the free alternative algebra.

$u_n$ are skew-symmetric in certain variables.

The elements vanish for specific values of n, namely $4m+2$ and $4m+3$..

Abstract

A new series of central elements is found in the free alternative algebra. More exactly, let $Alt[X]$ and $SMalc[X]\subset Alt[X]$ 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 $f(x,y,x_1,\ldots,x_n)\in SMalc[X]$ be a multilinear element which is trivial in the free associative algebra. Then the element $u_n=u_n(x,x_1,\ldots,x_n)=f(x^2,x,x_1,\ldots,x_n)-f(x,x^2,x_1,\ldots,x_n)$ lies in the center of the algebra $Alt[X]$. The elements $u_n(x,x_1,\ldots,x_n)$ are uniquely defined up to a scalar for a given $n$, and they are skew-symmetric on the variables $x_1,\ldots,x_n$. Moreover, $u_n=0$ for $n=4m+2,\,4m+3$. and $u_n\neq 0$ 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.