Lie invariants in two and three variables

TL;DR

This paper uses computer algebra techniques to compute Lie invariants in free Lie algebras with two and three generators, focusing on degrees up to 12 and 9 respectively, for representations of sl(2,C) and sl(3,C).

Contribution

It provides explicit computations of Lie invariants for specific degrees and representations, advancing understanding of invariants in free Lie algebras through algorithmic methods.

Findings

Computed Lie invariants of degree <= 12 for two generators in sl(2,C)

Determined Lie invariants of degree <= 7 for three generators in the adjoint representation

Found Lie invariants of degree <= 9 for three generators in the natural representation of sl(3,C)

Abstract

We use computer algebra to determine the Lie invariants of degree <= 12 in the free Lie algebra on two generators corresponding to the natural representation of the simple 3-dimensional Lie algebra sl(2,C). We then consider the free Lie algebra on three generators, and compute the Lie invariants of degree <= 7 corresponding to the adjoint representation of sl(2,C), and the Lie invariants of degree <= 9 corresponding to the natural representation of sl(3,C). We represent the action of sl(2,C) and sl(3,C) on Lie polynomials by computing the coefficient matrix with respect to the basis of Hall words. We then use algorithms for linear algebra (row canonical form, Hermite normal form, lattice basis reduction) to compute a basis of the nullspace.