2022/01/21 by I. P. Shestakov, Sergey Sverchkov, Shestakov, Ivan +1
Mathematics · #17A70 #17D05 #17D10 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2201.08915
openalex publication_date 2022/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,…,xn)∈ SMalc[X] be a multilinear element which is trivial in the free associative algebra. Then the element un=un(x,x1,…,xn)=f(x2,x,x1,…,xn)-f(x,x2,x1,…,xn) lies in the center of the algebra Alt[X]. The elements un(x,x1,…,xn) are uniquely defined up to a scalar for a given n, and they are skew-symmetric on the variables x1,…,xn. Moreover, un=0 for n=4m+2, 4m+3. and un≠ 0 for n=4m,4m+1. The ideals generated by the elements u4m, u4m+1 lie in the associative center of the algebra Alt[X] and have trivial multiplication.