TY - JOUR

T1 - New central elements in free alternative algebras

AU - Shestakov, Ivan

AU - Sverchkov, Sergei

N1 - The first author was supported by the Mathematical Center in Akademgorodok, agreement with the Ministry of Science and Higher Education of the Russian Federation No. 075-15-2022-281. He was also partially supported by FAPESP, Proc. 2018/23690-6 and CNPq, Proc. 304313/2019-0 of Brazil. The second author was partially supported by FAPESP, Proc. 2018/03717-7 of Brazil.

PY - 2024/8/21

Y1 - 2024/8/21

N2 - 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 (that is, they do not depend on f but only on deg f), 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.

AB - 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 (that is, they do not depend on f but only on deg f), 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.

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85200965418&origin=inward&txGid=480d91cc34c17dca74d980f24e2a1c1e

UR - https://www.webofscience.com/wos/woscc/full-record/WOS:001291207800002

UR - https://www.mendeley.com/catalogue/d149de34-2517-3f32-adf1-0b178f04760c/

U2 - 10.1007/s11856-024-2650-9

DO - 10.1007/s11856-024-2650-9

M3 - Article

SP - 1

EP - 26

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -