Lie Algebras and Groups of Lie Type

Lie algebras are given as structure constant algebras. We provide a number of algorithms designed and implemented by de Graaf [dG00] for determining the structure of a Lie algebra. In particular if the algebra is reductive, we can find its root system and also its highest-weight representations.

In 2000 a major project was commenced by Scott Murray, Don Taylor and Arjeh Cohen [CMT] to develop algorithms and software for computing with groups of Lie type (i.e. reductive algebraic groups and their group schemes). At this a stage we only support split (untwisted) groups, which are given by the Steinberg presentation. We have implemented efficient algorithms for arithmetic in the Steinberg presentation and for converting between this presentation and matrix representations over the base field. Note that this presentation is not in the category GrpFP since our generators are parametrised by field elements and the groups involved are not necessarily finitely generated.