Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field

Thiago Castilho de Mello, Felipe Yukihide Yasumura

Abstract

We study the graded polynomial identities with a homogeneous involution on the algebra of upper triangular matrices endowed with a fine group grading. We compute their polynomial identities and a basis of the relatively free algebra, considering an arbitrary base field. We obtain the asymptotic behaviour of the codimension sequence when the characteristic of the base field is zero. As a consequence, we compute the exponent and the second exponent of the same algebra endowed with any group grading and any homogeneous involution.

On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field

Abstract

We study the graded polynomial identities with a homogeneous involution on the algebra of upper triangular matrices endowed with a fine group grading. We compute their polynomial identities and a basis of the relatively free algebra, considering an arbitrary base field. We obtain the asymptotic behaviour of the codimension sequence when the characteristic of the base field is zero. As a consequence, we compute the exponent and the second exponent of the same algebra endowed with any group grading and any homogeneous involution.
Paper Structure (12 sections, 17 theorems, 40 equations)

This paper contains 12 sections, 17 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graded algebra
  4. Free graded algebra with homogeneous involution
  5. Gradings on $\mathrm{UT}_n$
  6. Involutions
  7. Notations
  8. Polynomial identities
  9. On codimension sequence
  10. Basic definitions
  11. Upper triangular matrices
  12. Other gradings and involution

Key Result

Lemma 2

$(\mathrm{UT}_n,\eta,\ast)$ satisfies the following polynomial identities: In addition, if $\mathbb{F}$ is finite with $q$ elements, then it satisfies:

Theorems & Definitions (37)

  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Definition 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 27 more