Lie identities and images of Lie polynomials for the skew-symmetric elements of UT_m

ABSTRACT

Let UT_m be the algebra of all m × m upper triangular matrices over a field $\mathbb{F}$ whose characteristic is different from 2. Given an involution * of the first kind on UT_m, we will obtain the minimal integer t such that the Lie polynomial [[z₁, z₂], …, [z_2t−1, z_2t]] is an identity for $K(m,\ast )=\{U\in U{T}_m\mid U^{\ast }=-U\}$. Afterwards, under a mild technical restriction on $\mathbb{F}$, we will describe all multilinear Lie polynomials whose image evaluated on K(m, *) is the space formed by all strictly upper triangular matrices of K(m, *).

Keywords:

• Lie identities
• Lie polynomials
• Images of polynomials
• Upper triangular matrices
• Involutions
• Skew-symmetric elements
• Lvov–Kaplansky conjecture

2010 Mathematics Subject Classifications:

• 16W10
• 17B01
• 16S50
• 15A24

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The Ronald Ismael Quispe Urure was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.