Degree-inverting involutions on matrix algebras

TL;DR

This paper classifies degree-inverting involutions on matrix algebras over algebraically closed fields, focusing on full and upper triangular matrices, enriching the understanding of involution symmetries in graded algebras.

Contribution

It provides a complete description of degree-inverting involutions on matrix and upper triangular matrix algebras over algebraically closed fields.

Findings

Classified degree-inverting involutions on full matrix algebras.

Described degree-inverting involutions on upper triangular matrices.

Extended understanding of involution symmetries in graded algebra structures.

Abstract

Let $F$ be an algebraically closed field of characteristic zero, and $G$ be a finite abelian group. If $A=\oplus_{g\in G} A_g$ is a $G$-graded algebra, we study degree-inverting involutions on $A$, i.e., involutions $*$ on $A$ satisfying $(A_g)^*\subseteq A_{g^{-1}}$, for all $g\in G$. We describe such involutions for the full $n\times n$ matrix algebra over $F$ and for the algebra of $n\times n$ upper triangular matrices.