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

We find and discuss an unexpected (to us) order n cyclic group of automorphisms of the Lie algebra \(I{\mathfrak u}_n{:}{=}{\mathfrak u}_n < imes {\mathfrak u}_n^*\), where \({\mathfrak u}_n\) is the Lie algebra of upper triangular \(n\times n\) matrices. Our results also extend to \(\mathfrak {gl}_{n+}^\epsilon \), a “solvable approximation” of \(\mathfrak {gl}_n\), as defined within.