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Graded polynomial identities for the Lie algebra of upper triangular matrices of order 3

Felipe Yukihide Yasumura

Abstract

We compute the graded polynomial identities and its graded codimension sequence for the elementary gradings of the Lie algebra of upper triangular matrices of order 3.

Graded polynomial identities for the Lie algebra of upper triangular matrices of order 3

Abstract

We compute the graded polynomial identities and its graded codimension sequence for the elementary gradings of the Lie algebra of upper triangular matrices of order 3.
Paper Structure (14 sections, 17 theorems, 40 equations)

This paper contains 14 sections, 17 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elementary gradings of $UT_3^{(-)}$
  4. Notation and preliminaries
  5. Universal grading
  6. Almost Universal grading
  7. Almost Canonical grading
  8. Remaining grading
  9. Canonical grading
  10. Conclusions
  11. Type 2 Gradings on $UT_3^{(-)}$
  12. Canonical T2 grading
  13. Almost Canonical T2 grading
  14. Trivial T2 grading

Key Result

Theorem 1

The $G$-graded identities for an elementary $G$-grading $\epsilon$ on $UT_n$ follow from all $f_\mu$ where $\mu$ runs over the $\epsilon$-bad sequences of length at most $n$.

Theorems & Definitions (35)

  • Definition 1: Definition 2.1 of VinKoVa2004
  • Theorem 1: Theorem 2.8 of VinKoVa2004
  • Definition 2
  • Definition 3
  • Conjecture 1
  • Example
  • Conjecture 2
  • Lemma 2
  • proof
  • Theorem 3
  • ...and 25 more