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 over a field K endowed with its canonical -grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on matrices, where n is a prime number. We confirm such conjecture for polynomials of degree 2 over 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 .
Images of graded polynomials on matrix algebras
Centrone, Lucio;
2022-01-01
Abstract
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 over a field K endowed with its canonical -grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on matrices, where n is a prime number. We confirm such conjecture for polynomials of degree 2 over 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 .I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
