Images of multilinear graded polynomials on upper triangular matrix algebras

TL;DR

This paper classifies the images of multilinear graded polynomials on upper triangular matrix algebras with various gradings, providing new insights into their structure and applications to traceless matrices and graded Jordan algebras.

Contribution

It offers a comprehensive classification of polynomial images on graded upper triangular matrices and extends results to graded Jordan algebras, advancing understanding of polynomial identities.

Findings

Classified images of multilinear graded polynomials on UT_n with elementary Z_q-gradings

Derived conditions for images to include traceless matrices in full matrix algebra

Described polynomial images on UT_2, UT_3, and graded Jordan algebras UJ_2, UJ_3

Abstract

In this paper we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices UT_n. For positive integers q \leq n, we classify these images on UT_n endowed with a particular elementary Z_q-grading. As a consequence, we obtain the images of multilinear graded polynomials on UT_n with the natural Z_n-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras UT_2 and UT_3, for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra UJ_2, and also for UJ_3 endowed with the natural elementary Z_3-grading.