# Invariants of Triangular Lie Algebras

## TL;DR

This paper computes invariants for classes of triangular Lie algebras using an algebraic algorithm, confirming a conjecture about the number and form of these invariants.

## Contribution

It extends an existing algebraic algorithm to find invariants for various classes of triangular Lie algebras and confirms a prior conjecture about their invariants.

## Key findings

- Invariants for strictly and non-strictly triangular Lie algebras are explicitly determined.
- The algebraic algorithm successfully computes generalized Casimir operators.
- A conjecture on the number and form of invariants is corroborated.

## Abstract

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.