Introduction

Algebras are viewed as free modules over a ring \(R\) with an additional multiplication. There are no a priori conditions imposed on the ring except that it must be unital, but some functions may require that an echelonization algorithm is available for modules over \(R\) and sometimes it is also required that \(R\) is a field. For example, quotients of algebras can only be constructed over fields, since otherwise the quotient module is not necessarily a free module over \(R\).

The most general way to define an algebra is by structure constants, but for special types of algebras Magma uses more efficient representations.

The Categories of Algebras

At present, Magma contains seven main categories of algebras:

(1)

General algebras represented by structure constants: category AlgGen;

(2)

Associative algebras represented by structure constants: category AlgAss;

(3)

Quaternion algebras as special types of associative algebras; category AlgQuat;

(4)

Lie algebras represented by structure constants: category AlgLie;

(5)

Group algebras: category AlgGrp with a special type AlgGrpSub for subalgebras of group algebras;

(6)

Matrix algebras: category AlgMat;

(7)

Finitely presented algebras: category AlgFP.

The hierarchy of these categories is such that AlgGen is on the top level and AlgAss and AlgLie are on the next level inheriting the functions available for AlgGen. The categories AlgQuat, AlgGrp and AlgMat are on a third level inheriting the functions available for AlgAss. Finitely presented algebras are independent of the other categories.