On homogeneous involutions on matrix algebras

TL;DR

This paper classifies homogeneous involutions on matrix algebras with various gradings, focusing on Pauli grading and graded-division algebra entries, expanding understanding of algebraic involutions in graded settings.

Contribution

It provides a classification of homogeneous involutions on matrix algebras with specific gradings, including Pauli and general group gradings, in algebraically closed fields.

Findings

Classification of isomorphism and equivalence classes for Pauli grading.

Analysis of homogeneous involutions on matrices over graded-division algebras.

Extension of involution classification to arbitrary group gradings.

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.