Computation of Invariants of Lie Algebras by Means of Moving Frames

TL;DR

This paper introduces a new algebraic algorithm leveraging moving frames and automorphism groups to compute invariants of Lie algebras, demonstrated on low-dimensional real Lie algebras with new invariant bases calculated.

Contribution

It presents a novel algebraic method using moving frames for calculating Lie algebra invariants, including new invariant bases for specific classes of real Lie algebras.

Findings

Successfully computed invariants for low-dimensional real Lie algebras.

Compared effectiveness with existing methods through multiple examples.

Provided new tables of invariants for certain classes of Lie algebras.

Abstract

A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras. It uses the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. The algorithm is applied, in particular, to computation of invariants of real low-dimensional Lie algebras. A number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature. Bases of invariants of the real solvable Lie algebras up to dimension five, the real six-dimensional nilpotent Lie algebras and the real six-dimensional solvable Lie algebras with four-dimensional nilradicals are newly calculated and listed in tables.