Images of multilinear polynomials of degree up to four on upper triangular matrices

Abstract

We describe the images of multilinear polynomials of degree up to four on the upper triangular matrix algebra.A famous open problem known as Lvov-Kaplansky's conjecture asserts: the image of a multilinear polynomial in noncommutative variables over a field K on the matrix algebra M n (K) is always a vector space [4].Recently, Kanel-Belov, Malev and Rowen [7] made a major breakthrough and solved the problem for n = 2.A special case on polynomials of degree two has been known for long time ([9] and [1]). Recently, Mesyan [8] and Buzinski and Winstanley [3] extended this result for nonzero multilinear polynomials of degree three and four, respectively.We will study the following variation of the Lvov-Kaplansky's conjecture:Conjecture 1 The image of a multilinear polynomial on the upper triangular matrix algebra is a vector space.In this paper, we will answer Conjecture 1 for polynomials of degree up to four. We point out that whereas in [3] and [8] the results describe conditions under which the image of a multilinear polynomial p, Im(p), contains a certain subset of M n (K), our results give the explicit forms of Im(p) on the upper triangular matrix algebra in each case.Throughout the paper UT n will denote the set of upper triangular matrices. The set of all strictly upper triangular matrices will be denoted by UT (0) n . More generally, if k ≥ 0, 1

The image of multilinear polynomials evaluated on 3 × 3 upper triangular matrices

“…A general result in the direction was given in [3], where the author proves the conjecture for polynomials of arbitrary degree evaluated on strictly upper triangular matrices.…”

“…We remark that in the papers [3], [4] J k was denoted by U T (k−1) n . This is the set of matrices whose entries (i, j) are zero if i ≥ j + k, for k ≥ 1.…”

“…In [4], the authors prove the conjecture for polynomials of degree up to 4 over fields of zero characteristic.…”

“…For instance, papers on images of Lie, and Jordan polynomials, and also for other algebras have been published since then. The particular case of n × n upper triangular matrices over a field K, U T n (K) (or simply U T n ), was studied by the author and Fagundes in [4], where the Lvov-Kaplansky's conjecture for U T n (K) was proved to be equivalent to the following Conjecture 1. The image of a multilinear polynomial on U T n (K) is either {0}, U T n or J k for some integer k ≥ 1, where J = J(U T n ) is the Jacobson radical of U T n , i.e., the set of strictly upper triangular matrices.…”

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

“…In particular, Im p is a vector space. In [40] Fagundes and de Mello describe the images of multilinear polynomials of degree ≤ 4 on the upper triangular matrices.…”