IMAGES OF MULTILINEAR POLYNOMIALS ON n x n UPPER TRIANGULAR MATRICES OVER INFINITE FIELDS

Autor(es)

Gargate, Ivan Gonzales ; de Mello, Thiago Castilho

• Tipo de documento: Artigo Científico
• Fonte: Israel Journal of Mathematics; v. 252, n. 1, p. 18-pg., 2022-09-09.

Resumo

In this paper we prove that the image of multilinear polynomials evaluated on the algebra UTn(K) of n x n upper triangular matrices over an infinite field K equals J(r), a power of its Jacobson ideal J = J(UTn(K)). In particular, this shows that the analogue of the Lvov-Kaplansky conjecture for UTn(K) is true, solving a conjecture of Fagundes and de Mello. To prove that fact, we introduce the notion of commutator-degree of a polynomial and characterize the multilinear polynomials of commutator-degree r in terms of its coefficients. It turns out that the image of a multilinear polynomial f on UTn(K) is J(r) if and only if f has commutator-degree r. (AU)

• Processo FAPESP: 18/15627-2 - Graduações, automorfismos e identidades em álgebras
• Beneficiário: Thiago Castilho de Mello
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 18/23690-6 - Estruturas, representações e aplicações de sistemas algébricos
• Beneficiário: Ivan Chestakov
• Modalidade de apoio: Auxílio à Pesquisa - Temático