Images of multilinear graded polynomials on upper triangular matrix algebras

Abstract

In this paper we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices U Tn. For positive integers q ≤ n, we classify these images on U Tn endowed with a particular elementary Zq-grading. As a consequence, we obtain the images of multilinear graded polynomials on U Tn with the natural Zn-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 U T 2 and U T 3 , for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra U J 2 , and also for U J 3 endowed with the natural elementary Z 3 -grading.

WITHDRAWN: Images of arbitrary polynomials on upper triangular matrix algebras

“…We have that p(T n (K)) = T n (K) (t) for some integer −1 ≤ t ≤ m 2 . In 2022 Fagundes and Koshlukov [23] investigated the image of multilinear graded polynomials on upper triangular matrix algebras. Recently Luo and Chen [30] gave a complete description of the image of linear polynomials on upper triangular matrix algebras, which improves Theorem 1.1 and some results in [23].…”

Prediction of Pediatric Sepsis Using a Deep Encoding Network with Cross Features

“…The following technical result is a generalization of [7,Lemma 2.11]. We give its proof for completeness.…”

“…We organize the paper as follows: In Section 2 we shall give some prelimiaries. We shall slightly modify some results in [5,7,12], which will be used in the proof of Theorem 1.2. In Section 3 we shall give the proof of Theorem 1.2 by using some new arguments, for example, the compatibility of variables and recursive polynomials.…”