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Graded polynomial identities of the infinite-dimensional upper triangular matrices over an arbitrary field

Micael Said Garcia, Felipe Yukihide Yasumura

Abstract

We compute the graded polynomial identities of the infinite dimensional upper triangular matrix algebra over an arbitrary field. If the grading group is finite, we prove that the set of graded polynomial identities admits a finite basis. We find conditions under which a grading on such an algebra satisfies a nontrivial graded polynomial identity. Finally, we provide examples showing that two nonisomorphic gradings can have the same set of graded polynomial identities.

Graded polynomial identities of the infinite-dimensional upper triangular matrices over an arbitrary field

Abstract

We compute the graded polynomial identities of the infinite dimensional upper triangular matrix algebra over an arbitrary field. If the grading group is finite, we prove that the set of graded polynomial identities admits a finite basis. We find conditions under which a grading on such an algebra satisfies a nontrivial graded polynomial identity. Finally, we provide examples showing that two nonisomorphic gradings can have the same set of graded polynomial identities.
Paper Structure (15 sections, 32 theorems, 39 equations)

This paper contains 15 sections, 32 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group gradings
  4. Graded polynomial identities
  5. Upper triangular matrices
  6. Infinite-dimensional upper triangular matrices
  7. Graded polynomial identities
  8. General results
  9. Graded polynomial identities of $\mathrm{UT}$
  10. Conditions for the existence of graded identities
  11. Finite basis property
  12. Nonisomorphic gradings satisfying the same identities
  13. Example
  14. Example
  15. Example

Key Result

Theorem 2.1

Let $\varepsilon$ be a $G$-grading on $\mathrm{UT}_n$, over an arbitrary field $\mathbb{F}$. Then $\mathrm{Id}_G(\mathrm{UT}_n,\varepsilon)$ follows from all $f_1\cdots f_r$, where each $f_i\in\mathcal{S}$ (defined above), $(\deg_G f_1,\ldots,\deg_G f_r)$ is $\varepsilon$-bad, and $r\le n$.

Theorems & Definitions (74)

  • Theorem 2.1: VinKoVa2004GR2020
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Definition 3.5
  • Proposition 3.6
  • ...and 64 more