Images of multilinear graded polynomials on upper triangular matrix algebras

Abstract

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 ${\mathbb {Z}}_{q}$ -grading. As a consequence, we obtain the images of multilinear graded polynomials on $UT_{n}$ with the natural ${\mathbb {Z}}_{n}$ -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 ${\mathbb {Z}}_{3}$ -grading.