Invariant Lie polynomials in two and three variables.

Abstract

In 1949, Wever observed that the degree d of an invariant Lie polynomial must be a multiple of the number q of generators of the free Lie algebra. He also found that there are no invariant Lie polynomials in the following cases: q = 2, d = 4; q = 3, d = 6; d = q ≥ 3. Wever gave a formula for the number of invariants for q = 2 in the natural representation of sl(2). In 1958, Burrow extended Wever’s formula to q > 1 and d = mq where m > 1. In the present thesis, we concentrate on finding invariant Lie polynomials (simply called Lie invariants) in the natural representations of sl(2) and sl(3), and in the adjoint representation of sl(2). We first review the method to construct the Hall basis of the free Lie algebra and the way to transform arbitrary Lie words into linear combinations of Hall words. To find the Lie invariants, we need to find the nullspace of an integer matrix, and for this we use the Hermite normal form. After that, we review the generalized Witt dimension formula which can be used to compute the number of primitive Lie invariants of a given degree. Secondly, we recall the result of Bremner on Lie invariants of degree ≤ 10 in the natural representation of sl(2). We extend these results to compute the Lie invariants of degree 12 and 14. This is the first original contribution in the present thesis. Thirdly, we compute the Lie invariants in the adjoint representation of sl(2) up to degree 8. This is the second original contribution in the present thesis. Fourthly, we consider the natural representation of sl(3). This is a 3-dimensional natural representation of an 8-dimensional Lie algebra. Due to the huge number of Hall words in each degree and the limitation of computer hardware, we compute the Lie invariants only up to degree 12. Finally, we discuss possible directions for extending the results. Because there are infinitely many different simple finite dimensional Lie algebras and each of them has infinitely many distinct irreducible representations, it is an open-ended problem.