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. The hecke package allows dealing with Specht modules.

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.

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.

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.

Four new packages for Lie algebras appeared in GAP 4.7 distribution:

• The package CoReLG for calculations in real semisimple Lie algebras.
• The package LieRing for constructing finitely-presented Lie rings and calculating the Lazard correspondence. The package also provides a database of small $n$-Engel Lie rings.
• The package LiePRing, introducing a new datastructure for nilpotent Lie rings of prime-power order. This allows to define such Lie rings for specific primes as well as for symbolic primes and other symbolic parameters. The package also includes a database of nilpotent Lie rings of order at most $p^7$ for all primes $p> 3$.
• The package SLA for computations with simple Lie algebras. The main topics of the package are nilpotent orbits, theta-groups and semisimple subalgebras.

The package LAGUNA allows to investigate unit groups of the modular group algebra of a $p$-group and Lie algebras associated with associative algebras. It is extended by the UnitLib package providing a library of unit groups of modular group algebras of $p$-groups of small order.

The package Wedderga computes the simple components of the Wedderburn decomposition of semisimple group algebras of finite groups over finite fields and over subfields of finite cyclotomic extensions of rationals. It also contains functions that produce the primitive central idempotents of semisimple group algebras. It also provides the functionality to construct crossed products.

The package ModIsom contains various methods for computing with nilpotent associative algebras. In particular, it contains a method to determine the automorphism group and to test isomorphisms of such algebras over finite fields and of modular group algebras of finite $p$-groups. Further, it contains a nilpotent quotient algorithm for finitely presented associative algebras and a method to determine Kurosh algebras.