## 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.

##### Degree

Master of Science (M.Sc.)##### Department

Mathematics and Statistics##### Program

Mathematics and Statistics##### Supervisor

Murray, Bremner##### Committee

Chris, Soteros; Martin, John##### Copyright Date

2009##### Subject

Free Lie algebra

Invariant theory

Representation theory