Lie Invariants in Two and Three Variables

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 𝔰𝔩2(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 𝔰𝔩2(ℂ), and the Lie invariants of degree ≤ 9 corresponding to the natural representation of 𝔰𝔩3(ℂ). We represent the action of 𝔰𝔩2(ℂ) and 𝔰𝔩3(ℂ) on Lie polynomials by computing the coefficient matrix with respect to the basis of Hall words.