Invariants of triangular Lie algebras with one nilindependent diagonal element

TL;DR

This paper exhaustively studies the invariants of solvable triangular Lie algebras with one nilindependent diagonal element, introducing an algebraic algorithm based on Cartan's method to construct invariant bases and proving a conjecture on their structure.

Contribution

It develops a novel algebraic algorithm for constructing invariant bases of these Lie algebras and proves a conjecture regarding their invariants' number and form.

Findings

Constructed invariant bases for all such algebras.

Proved the conjecture of Tremblay and Winternitz on invariants.

Extended understanding of invariants in triangular Lie algebras.

Abstract

The invariants of solvable triangular Lie algebras with one nilindependent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartan's method of moving frames and the special technique developed for triangular and related algebras in [J. Phys. A: Math. Theor. 40 (2007), 7557-7572]. The conjecture of Tremblay and Winternitz [J. Phys. A: Math. Gen. 34 (2001), 9085-9099] on the number and form of elements in the bases is completed and proved.