Vector Spaces, Modules and Algebras

Vector spaces over fields and modules over rings can be defined when the coefficient domain is available in GAP. Note, however, that the range of implemented methods will depend on the coefficient domain.

There are algorithms for the efficient calculation of Hermite and Smith normal forms over the integers (see also the package EDIM).

Computations concerning special modules arising in representation theory are possible. However the package Specht for dealing with Specht modules is still (2005) only available in GAP 3.

Lie algebras can be given by structure constants, by generating matrices or by a finite presentation. There are routines for computing the structure of finite dimensional Lie algebras, in particular there are functions for computing Cartan subalgebras, the direct sum decomposition, a Levi decomposition, the solvable radical and nil radicals.

Much of the support for Lie algebras is based on more general methods using an implementation of the arithmetic operations via structure constants, which works for any finite dimensional algebra. In particular, associative algebras (e.~g., group rings, cf the manual chapter Magma Rings) are also supported.

The package LAGUNA allows to investigate unit groups of the modular group ring of a p-group and Lie algebras associated with associative algebras.

The package Sophus deals with nilpotent Lie algebras over prime fields allowing to construct central extensions and to determine their automorphism groups.

The package QuaGroup allows to investigate quantum groups.

On the home page of Jan Draisma functions for working with the Weyl algebra and for the realisation of Lie algebras by means of derivations are found.

Investigation of algebras given by presentations are currently restricted to Lie algebras using the package FPLSA; associative algebras will have to wait for a GAP 4 implementation of the vector enumeration method.