MAGMA Computational Algebra System

Lie Algebras and Groups of Lie Type

Lie algebras can be constructed in three different ways in magma: as structure constant algebras, as Lie matrix algebras, or as finitely presented algebras. Most of our functionality is for algebras of finite dimension over a field. Algorithms designed and implemented by de Graaf [dG00] are available for determining the structure of a Lie algebra. In particular, if the algebra is reductive, its root system and its highest-weight representations can be determined.

We provide functionality for computing with groups of Lie type (i.e. reductive algebraic groups and their split (untwisted) groups, given by the Steinberg presentation. A canonical form for words in this group, and algorithms for computing with these words are given in [CMT04], [CHM08]. Twisted groups are given by a modified version of this presentation using Galois cohomology [Hal05]. Efficient algorithms have been implemented for arithmetic with the Steinberg presentation and for converting between this presentation and matrix representations over the base field. Note that these presentations are not in the categBrinkHowlett93ory GrpFP since the generators are parametrised by field elements and so the groups involved are not necessarily finitely generated.

This is described in Chapter LIE ALGEBRAS.