Invariants of the nilpotent and solvable triangular Lie algebras

TL;DR

This paper computes invariants of coadjoint representations for nilpotent and solvable triangular Lie algebras, providing explicit polynomial invariants for the nilpotent case and analyzing invariants for specific solvable cases.

Contribution

It explicitly determines polynomial invariants for nilpotent upper triangular Lie algebras and explores invariants of solvable extensions with nilradicals, including detailed cases.

Findings

Nilpotent Lie algebras T(M) have [M/2] polynomial invariants.

Explicit polynomial invariants are constructed for T(M).

Invariants for solvable Lie algebras L(M,f) with specific parameters are characterized.

Abstract

Invariants of the coadjoint representation of two classes of Lie algebras are calculated. The first class consists of the nilpotent Lie algebras $T(M)$, isomorphic to the algebras of upper triangular $M\times M$ matrices. The Lie algebra $T(M)$ is shown to have $[M/2]$ functionally independent invariants. They can all be chosen to be polynomials and they are presented explicitly. The second class consists of the solvable Lie algebras $L(M,f)$ with $T(M)$ as their nilradical and $f$ additional linearly nilindependent elements. Some general results on the invariants of $L(M,f)$ are given and the cases M=4 for all $f$ and $f=1$, or $f=M-1$ for all $M$ are treated in detail.