Images of graded polynomials on matrix algebras

• Published in: Journal of algebra Vol. 614; pp. 650 - 669
• Main Authors: Centrone, Lucio, Castilho de Mello, Thiago
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15-01-2023
• Subjects: Graded polynomial identities; Graded structures; Images of polynomials on algebras; 16R10; 16W50; 16R99; Graded polynomial identities; 16R50; Images of polynomials on algebras; Graded structures
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2022.09.031

The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra Mn(K) over a field K endowed with its canonical Zn-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field Q of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on n×n matrices, where n is a prime number. We confirm such conjecture for polynomials of degree 2 over Mn(K) when K is a quadratically closed field of characteristic zero or greater than n and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on M2(K).