Algorithms for Lie algebras of algebraic groups

Abstract

In this thesis we present several new algorithms for dealing with simple algebraic groups and their Lie algebras. These groups and algebras have been studied for a long time, first in a theoretical sense and later with regards to effective calculations on the computer, including implementations in the GAP and Magma computer algebra systems. We build in particular on work by Arjeh Cohen, Willem de Graaf, Sergei Haller, Scott Murray, and Don Taylor. The work is partly stimulated by the matrix group recognition project: an international project which is aimed at the algorithmic analysis of problems with matrix groups over finite fields. Many algorithms that have been previously developed in this branch of research, however, apply only to groups and algebras over fields of characteristic 0 or at least 5. For instance, Cohen and Murray, and, independently, Ryba recently gave an algorithm for computing a split maximal toral subalgebra of a Lie algebra in all characteristics except 2 and (to a certain extent) 3. Unfortunately, not only their proofs but also their algorithms do not work in the excluded cases. Similarly, the algorithm for computing a Chevalley basis of a Lie algebra, when given a split toral subalgebra, is straightforward in almost all characteristics, and has consequently been implemented in major computer algebra systems such as GAP and Magma. In characteristics 2 and 3, however, the algorithm is much more involved. This thesis starts with an extensive introduction to the mathematical objects occurring in this thesis, such as root data, algebraic groups, and Lie algebras. The new results in this thesis are a heuristic algorithm for computing split maximal toral subalgebras of Lie algebras of split simple algebraic groups over fields of characteristic 2, and an algorithm for computing Chevalley bases of Lie algebras of split simple algebraic groups over any field. The latter algorithm is proved to be polynomial in the case where the field is finite. These algorithms are applied to the problem of recognizing these Lie algebras among all Lie algebras, and they help in the analysis of the associated algebraic groups. We also apply these algorithms in the computer aided proof that there is no graph on which a certain group acts distance transitively. All of the algorithms presented in this thesis have been implemented in the Magma computer algebra system.