Degree-inverting involutions on matrix algebras

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 $\ast$ on $A$ satisfying $(A_g{)}^{\ast }\subseteq A_{g^{-1}}$, 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.

Title changed. Final section about $U{T}_n$ added