This chapter introduces the data structures and functions for algebras in GAP. The word algebra in this manual means always associative algebra.
At the moment GAP supports only finitely presented algebras and matrix algebras. For details about implementation and special functions for the different types of algebras, see More about Algebras and the chapters Finitely Presented Algebras and Matrix Algebras.
The treatment of algebras is very similar to that of groups. For example, algebras in GAP are always finitely generated, since for many questions the generators play an important role. If you are not familiar with the concepts that are used to handle groups in GAP it might be useful to read the introduction and the overview sections in chapter Groups.
Algebras are created using
Algebra (see Algebra) or
(see UnitalAlgebra), subalgebras of a given algebra using
(see Subalgebra) or
UnitalSubalgebra (see UnitalSubalgebra).
See Parent Algebras and Subalgebras, and the corresponding section
More about Groups and Subgroups in the chapter about groups for details
about the distinction between parent algebras and subalgebras.
Parent Algebras and Subalgebras).
The next sections describe the functions for the construction of algebras, and the tests for algebras (see Algebra, UnitalAlgebra, IsAlgebra, IsUnitalAlgebra, Subalgebra, UnitalSubalgebra, IsSubalgebra, AsAlgebra, AsUnitalAlgebra, AsSubalgebra, AsUnitalSubalgebra).
The next sections describe the different types of functions for algebras Vector Space Functions for Algebras, Algebra Functions for Algebras, TrivialSubalgebra).
Operation for Algebras, OperationHomomorphism for Algebras).
Algebra Homomorphisms, Mapping Functions for Algebra Homomorphisms).
The next sections describe algebra elements (see Algebra Elements, IsAlgebraElement).
The last section describes the implementation of the data structures (see Algebra Records).
At the moment there is no implementation for ideals, cosets, and factors of algebras in GAP, and the only available algebra homomorphisms are operation homomorphisms.
Also there is no implementation of bases for general algebras, this will be available as soon as it is for general vector spaces.