Lie Theory
Magma supports Lie algebras and their representations - root systems, Cartan matrices, Weyl groups, Chevalley bases, and highest-weight modules. Both classical and exceptional Lie algebra types are fully covered.
Contributors
The major structural machinery for Lie algebras has been implemented for Magma by Willem de Graaf (Utrecht) and is based on his ELIAS package written in GAP. He has also implemented a separate package for finitely presented Lie rings.
A database of soluble Lie algebras of dimensions 2, 3 and 4 over all fields has been implemented by Willem de Graaf (Trento). Willem has also provided a database of all nilpotent Lie algebras of dimension up to 6 over all base fields (except characteristic 2 when the dimension is 6).
More recent extensions to the Lie algebra package developed by Willem de Graaf (Trento) include quantum groups, universal enveloping algebras, the semisimple subalgebras of a simple Lie algebra and nilpotent orbits for simple Lie algebras.
A fast algorithm for multiplying the elements of Coxeter groups based on their automatic structure has been designed and implemented by Bob Howlett. Bob has also contributed Magma code for computing the growth function of a Coxeter group.
Machinery for computing the W-graphs for Lie types An, E6, E7 and E8 has been supplied by Bob Howlett. Subsequently, Bob supplied code for working with directed W-graphs.
The current version of Lie groups in Magma has been implemented by Scott Murray (Sydney) and Sergei Haller with some assistance from Don Taylor (Sydney).
An extensive package for computing the combinatorial properties of highest weight representations of a Lie algebra has been written by Dan Roozemond (Eindhoven). This code is based on the LiE package with permission of the authors.
The original version of the code for root systems and permutation Coxeter groups was modelled, in part, on the Chevie package of GAP and implemented by Don Taylor (Sydney) with the assistance of Frank Lübeck (Aachen).
Functions that construct any finite irreducible unitary reflection group in ℂn have been implemented by Don Taylor (Sydney). Extension to the infinite case was implemented by Scott Murray (Sydney).
Code has been contributed by Robert Zeier (TU Munich) for determining the irreducible simple subalgebras of the Lie algebra SUk.