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Homogeneous involution on graded division algebras and their polynomial identities

Felipe Yukihide Yasumura

Abstract

In this short note, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by L. Fonseca and T. de Mello, a homogeneous involution naturally appears when dealing with graded polynomial identities and a compatible involution.

Homogeneous involution on graded division algebras and their polynomial identities

Abstract

In this short note, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by L. Fonseca and T. de Mello, a homogeneous involution naturally appears when dealing with graded polynomial identities and a compatible involution.
Paper Structure (9 sections, 8 theorems, 33 equations)

This paper contains 9 sections, 8 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graded algebra
  4. Factor sets
  5. Graded division algebra
  6. Free graded algebra with homogeneous involution
  7. Codimension sequence
  8. Homogeneous involution on graded division algebra
  9. Polynomial identities of graded division algebra

Key Result

Lemma 1

For each $\theta\in\overline{\mathrm{Aut}}(T)$ and $\sigma'\in Z^2(T,\mathbb{F}^\times)$, $\theta\sigma'\in Z^2(T,\mathbb{F}^\times)$.

Theorems & Definitions (15)

  • Definition
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • Definition
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 5 more