Semilinear automorphisms of classical groups and quivers

TL;DR

This paper characterizes fixed point subgroups and eigenspaces of automorphisms of classical groups using cyclic quivers with involution, providing detailed classifications especially for loop Lie algebras over complex Laurent series.

Contribution

It introduces a novel description of automorphism fixed points and eigenspaces in classical groups via cyclic quivers with involution, including explicit classifications for loop Lie algebras.

Findings

Fixed point subgroups described via cyclic quivers with involution

Eigenspaces of automorphisms characterized in terms of quivers

Complete classification for loop Lie algebras over a6

Abstract

For a classical group $G$ over a field $F$ together with a finite-order automorphism $\theta$ that acts compatibly on $F$, we describe the fixed point subgroup of $\theta$ on $G$ and the eigenspaces of $\theta$ on the Lie algebra $\mathfrak{g}$ in terms of cyclic quivers with involution. More precise classification is given when $\mathfrak{g}$ is a loop Lie algebra, i.e., when $F=\mathbb{C}((t))$.