An Unexpected Cyclic Symmetry of $I\mathfrak{u}_n$

TL;DR

This paper uncovers an unexpected cyclic symmetry of order n in the automorphism group of a specific Lie algebra constructed from upper triangular matrices, revealing new structural insights.

Contribution

It identifies and analyzes a novel order n cyclic automorphism group of the Lie algebra $Irak{u}_n$, extending the understanding of its symmetry properties.

Findings

Discovery of an order n cyclic automorphism group

Extension of results to a solvable approximation of $gl_n$

Enhanced understanding of Lie algebra symmetries

Abstract

We find and discuss an unexpected (to us) order $n$ cyclic group of automorphisms of the Lie algebra $I\mathfrak{u}_n := \mathfrak{u}_n\ltimes\mathfrak{u}_n^\ast$, where $\mathfrak{u}_n$ is the Lie algebra of upper triangular $n\times n$ matrices. Our results also extend to $gl_{n+}^\epsilon$, a ``solvable approximation'' of $gl_n$, as defined within.