Finite dimensional algebras supported in Magma are organised as follows. Any algebra may be defined as a structure constant algebra; that is, defined by specifying a multiplication table for the basis elements. This is how most algebras are represented in Magma; the main exceptions, which have their own special representations, are group algebras, matrix algebras, and commutative algebras (which are best represented using polynomial rings and modules).
Apart from the special types just mentioned, and apart from Lie algebras, most algebras supported fall under the banner of associative algebras. At this level of generality there is considerable functionality: support for orders and ideals, computation of radical, and idempotent decomposition. Depending on the base field, more nontrivial routines are provided. For an algebra over ℚ, an algorithm to compute a maximal order in the algebra is implemented (de Graaf).
An extensive package of algorithms is provided for the special case of quaternion algebras. In the future, it is likely that a package will be developed for central simple algebras.
In the rest of this page, tools for finite-dimensional associative algebras defined by structure constants (ASC-algebras) will be reviewed. This is a very general way of presenting a finite-dimensional algebra, though as the dimension increases the memory needed for the structure constant matrices can become substantial and so it is limited to dimensions of at most a few thousand. Certain other types of associative algebras such as quaternion and group algebras have different representations that make use of special features of the particular type of algebra.
A general finite-dimensional associative algebra can also be represented either as a matrix algebra or as a finitely-presented algebra. One can move between these representations. Thus, given a finite-dimensional finitely-presented algebra (fp-algebra) one can often construct its regular representation using a non-commutative Gröbner Basis algorithm.
Specifying a structure constant algebra A is straightforward. A sparse representation is available if it is believed that the structure constant vectors are sparse. There are a number of functions that create ASC-algebras directly from other algebraic structures. If F is a number field or finite field with subfield E, a function is provided which constructs the ASC-algebra of dimension [F : E] over E which is isomorphic to F. Given a simple algebra A with centre K, another function returns a K-algebra B which is K-isomorphic to A. Finally, a group algebra may be converted to an ASC-algebra.
Constructors are available to create subalgebras and quotient algebras of an ASC-algebra A. However, quotient algebras are only available for the case in which A is defined over a field. If A is an R-algebra and S is a ring such that there exists a monomorphism f : R→S then a function allows the user to construct the S-algebra B obtained by applying f to map the coefficients of elements of A into S.
Every algebra over a field comes with a basis and the usual operations are available: extract basis, extend basis, test independence of specified algebra elements, etc. Functions are provided to create various subalgebras: centre, centraliser, idealiser, left and right annihilators, commutator of two subalgebras (as a module or ideal), etc.
The functions described in this paragraph assume that A is defined over a finite field. An algebra A can be regarded as a (left- or right-) module for itself so that if A is defined over a finite field, the machinery for A-modules can be used to decompose modules arising from the regular module for A and so compute structural information for A. The information that can be computed includes: a composition series, the composition factors, the Jacobson radical (the largest nilpotent ideal), and the decomposution of A into a direct sum of indecomposable ideals. Tests for an algebra being simple or semisimple are available. In addition the minimal two-sided (left, right) ideals and maximal two-sided (left, right) ideals can be computed. The machinery is very efficient so that composition series (and factors) have been computed for algebras having dimension up to 10,000.
The functions described in this paragraph assume that A is defined over a number field. The Jacobson radical J(A) (the largest nilpotent ideal) of A can be computed using an algorithm due to A. Cohen, G. Ivanyos, and D. Wales. A function is provided which computes the decomposition of A into a direct sum of ideals of A such that each is indecomposable. A closely related function returns a sequence of primitive orthogonal idempotents in the centre Z of A such that their images in A/J(Z) span J(Z). This machinery will soon be replaced by a more powerful approach based on the use of a Meataxe that works for modules defined over number fields.
Let K be a number field and let R be the ring of integers of K. For an associative algebra A over a field K, an order O is a subring of A such that O contains a K-basis for A and is also a finitely generated R-module. Orders in an algebra are represented by a pseudobasis, similar to the way in which modules over Dedekind domains are represented.
Arbitrary orders of an algebra can be created in Magma. A function is also provided to construct a maximal order in an algebra. The standard operations for orders and their elements are provided. Left, right and two-sided ideals of orders are supported. Apart from the obvious arithmetic operations, sums, products, and intersections of ideals are supported as is the computation of colon ideals.