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On homogeneous involutions on matrix algebras

Micael Said Garcia, Cassia Ferreira Sampaio

TL;DR

The paper classifies homogeneous involutions on full matrix algebras over an algebraically closed field with division gradings, focusing on commutative-support gradings and Pauli-type gradings. It develops a unified framework for tau-homogeneous anti-automorphisms, ties their existence to automorphisms of the support, and provides explicit normal forms and equivalence classifications, including detailed results for Pauli gradings. Extending to matrix algebras with entries in graded-division algebras, it characterizes tau-homogeneous involutions via a decomposition into a graded-dimension data triplet (g0, psi0, B) and constructs the involution psi by conjugation with a degree-aware matrix Phi; the orthogonal/symplectic nature is read off from a balanced bilinear form. These results yield concrete descriptions of involution types and isomorphism classes across the graded setting, with implications for graded identities and graded-division algebra structures.

Abstract

We study the homogeneous involutions on the full square matrices over an algebraically closed field endowed with a division grading with commutative support. We obtain the classification of the isomorphism and equivalence classes for the Pauli grading. We also investigate the homogeneous involutions on the full square matrices with entries in a finite-dimensional graded-division algebra over an algebraically closed field of characteristic not $2$ endowed with an arbitrary grading by an arbitrary group.

On homogeneous involutions on matrix algebras

TL;DR

The paper classifies homogeneous involutions on full matrix algebras over an algebraically closed field with division gradings, focusing on commutative-support gradings and Pauli-type gradings. It develops a unified framework for tau-homogeneous anti-automorphisms, ties their existence to automorphisms of the support, and provides explicit normal forms and equivalence classifications, including detailed results for Pauli gradings. Extending to matrix algebras with entries in graded-division algebras, it characterizes tau-homogeneous involutions via a decomposition into a graded-dimension data triplet (g0, psi0, B) and constructs the involution psi by conjugation with a degree-aware matrix Phi; the orthogonal/symplectic nature is read off from a balanced bilinear form. These results yield concrete descriptions of involution types and isomorphism classes across the graded setting, with implications for graded identities and graded-division algebra structures.

Abstract

We study the homogeneous involutions on the full square matrices over an algebraically closed field endowed with a division grading with commutative support. We obtain the classification of the isomorphism and equivalence classes for the Pauli grading. We also investigate the homogeneous involutions on the full square matrices with entries in a finite-dimensional graded-division algebra over an algebraically closed field of characteristic not endowed with an arbitrary grading by an arbitrary group.
Paper Structure (8 sections, 19 theorems, 74 equations)

This paper contains 8 sections, 19 theorems, 74 equations.

Table of Contents

  1. Preliminaries
  2. Graded algebras
  3. Homogeneous anti-automorphisms
  4. Factor sets
  5. Graded-division algebras
  6. Homogeneous involutions on graded-division algebras
  7. Pauli grading
  8. Homogeneous involutions on matrix algebras

Key Result

Lemma 1.1

Let $T$ be a group, $\tau\colon T\to T$ be a bijection, and $\mathcal{D}=\bigoplus _{t\in T}\mathcal{D}_{t}$ be a graded-division algebra with support $T$. If there exists a $\tau$-homogeneous anti-automorphism $\psi$ on $\mathcal{D}$, then for all $g,h \in T$ we have If $\psi$ is also an involution, then

Theorems & Definitions (36)

  • Lemma 1.1
  • Proposition 2.1
  • proof
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Corollary 2.6: FSY22 and EK2013
  • Proposition 2.7
  • Proposition 2.8
  • ...and 26 more