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Vector Spaces, Modules and AlgebrasVector 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 package Specht for dealing with Specht modules has been ported to GAP 4 in the hecke package. 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 LAGUNA allows to investigate unit groups of the modular group algebra of a pgroup 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 pgroups 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 pgroups. Further, it contains a nilpotent quotient algorithm for finitely presented associative algebras and a method to determine Kurosh algebras. 

The GAP Group Last updated: Mon Oct 21 14:45:21 2019 