In this section we are mainly concerned with associative algebras – the facilities for Lie algebras and quantum algebras may be found in the section on Lie theory. The three chief ways of defining algebras in Magma are in terms of a finite presentation, in terms of structure constants, or as a matrix (linear) algebra.
Finitely presented associative algebras
General finite dimensional algebras (defined by structure constants)
Finite dimensional associative algebras (defined by structure constants)
Group algebras
Matrix algebras
Quaternion algebras
Basic algebras
Finitely-presented (FP) associative algebras (or noncommutative polynomial rings) are defined by taking R-linear combinations of elements of a semigroup, where R is some ring. Since V2.11, these are handled by an extension of the commutative algebra machinery to noncommutative data structures and algorithms, where applicable. These include a noncommutative analogue for Gröbner bases.
Construction of free algebras over arbitrary fields
Arithmetic
Mappings into other associative algebras
Definition of left, right, two-sided ideals
Noncommutative Gröbner bases of ideals, with specialized algorithms for different coefficient fields (fraction-free methods for the rational field and rational function fields)
Gröbner bases of ideals over finite fields and rationals, using noncommutative extension of the Faugère F4 algorithm
Construction of degree-d (truncated) Gröbner bases
Normal form of a polynomial with respect to an ideal
Construction of FP-algebras as quotient rings
Enumeration of the basis of finite-dimensional FP algebras
Matrix and structure-constant representations of finite-dimensional FP algebras
Construction of a matrix representation (Linton's vector enumerator)
There are two major tools for computing with these algebras. The main approach is to apply a noncommutative version of Buchberger's algorithm to construct a Gröbner basis for an ideal. This technique has been developed chiefly by Teo Mora in Genova and Ed Green in Virginia. An extension of Faugère's F4 algorithm, due to Allan Steel, works by sparse linear algebra and is often much quicker.
Linton's vector enumerator uses the Todd-Coxeter algorithm in an attempt to construct a matrix representation. If the user has some idea as to how to select ideals that might give rise to matrix representations of reasonable degree, this approach is very successful.
A special type is provided for working with an exterior algebra. Such an algebra is a special kind of FP algebra which is skew-commutative and is represented as a quotient of the free algebra K < x1,…, xn > by the relations xi2 = 0 and xi xj = -xj xi for 1 ≤ i,j ≤ n, i ≠ j.
Because of the above relations, elements of an exterior algebra can be written in terms of commutative monomials in the variables (via a collection algorithm), and the associated algorithms are much more efficient than for the general noncommutative case. Gröbner bases of ideals can be computed very efficiently (the Faugere F4 algorithm has been specially adapted for this). Furthermore, the extensive module theory also works over exterior algebras.
These algebras are presented in terms of a basis for a free module M together with a set of structure constants defining the multiplication of these basis elements. It is assumed that we have an echelonization algorithm for M so that standard bases may be constructed for submodules.
Creation of algebras in terms of structure constants
Direct sum
Arithmetic including Lie bracket operation
Identities: associative, commutative, Lie, etc
Properties of elements: idempotent, unit, zero-divisor, nilpotent
Trace and minimal polynomial
Creation of subalgebras, ideals and quotient algebras
Ideal arithmetic: Sum, product, powers, intersection
Ideal structure: Jacobson radical, maximal (minimal) left, right, two-sided ideals
Decomposition: Simplicity, semi-simplicity, composition series
These algebras are presented in terms of a basis for a free module M together with a set of structure constants defining the multiplication of these basis elements. It is assumed that we have an echelonization algorithm for M so that standard bases may be constructed for submodules. We shall refer to these algebras as ASC-algebras.
Creation of algebras in terms of structure constants
Direct sum
Arithmetic including Lie bracket operation
Properties of elements: idempotent, unit, zero-divisor, nilpotent
Trace and minimal polynomial
Creation of subalgebras, ideals and quotient algebras
Ideal arithmetic: Sum, product, powers, intersection
Centralizer, idealizer
Characteristic ideals: Centre, commutator ideal, Jacobson radical
Ideal structure: Maximal (minimal) left, right, two-sided ideals
Decomposition: Simplicity, semi-simplicity, composition series
Construction of the (left, right) regular matrix representation
Lie algebra defined by the Lie product
Functions relating to the ideal structure (Jacobson radical, composition series, maximal and minimal ideals, etc) are implemented by applying the module theory machinery to the regular representation of the algebra.
Construction of orders of algebras over the rationals or a number field
Construction of a maximal order of a central simple algebra defined over the rational numbers or a number field
Basis of an order
Construction of elements of an order of an algebra
Arithmetic of elements of an order of an algebra
Norm, trace, conjugate, minimal polynomial and representation matrix of elements
Construction of left, right and two-sided ideals of orders
Addition and multiplication of ideals
Left and right order of an ideal, colon ideal
Basis and basis matrix of an ideal
Matrix algebras arise naturally as the endomorphism ring of a module and consequently are a fundamental structure in mathematics. Much effort has been invested into developing highly efficient code for working with these algebras and their elements in Magma. A description of generic matrix operations may be found in the section on Vector Spaces and Matrices and will not be repeated in this section.
While a matrix algebra may be defined over any ring R, most non-trivial operations require R to be an Euclidean Domain.
Arithmetic
Extension and restriction of coefficient ring
Direct sum, tensor product, exterior square, symmetric square
Determinant (including modular algorithm), trace, characteristic polynomial, minimum polynomial
Order of a unit (Leedham-Green algorithm)
Canonical forms over a field: echelon, Jordan, rational, primary rational (asymptotically-fast Strassen-based algorithms over finite fields and modular algorithms over Q)
Canonical forms over an ED: echelon, Hermite, Smith (asymptotically-fast modular algorithms over Z)
Characteristic polynomial, minimal polynomial
Properties of an element: unit, zero-divisor, nilpotent
Standard basis for subalgebras, left, right and two-sided ideals
Quotient algebras
Sum, intersection, product, power of ideal
Radical of an ideal
Centre, commutator algebra
Centralizer of a subalgebra in the complete matrix algebra
Jacobson radical (Brooksbank-O'Brien algorithm used when base field is Fq)
Unit group of an algebra over Fq (Brooksbank-O'Brien algorithm)
Maximal (minimal) left, right, two-sided ideals
Construction of the (left, right) regular matrix representation
Diagonalisation of a commutative algebra over a field
Construction of Z-basis of a maximal order of a central simple algebra over Z
Algorithms have been developed by Carlson and Matthews which construct a presentation for a matrix algebra in terms of generators and relations. At present this and related machinery is restricted to algebras over finite fields. The ability to compute presentations is needed as part of the development of module theory over matrix algebras.
Primitive idempotents
The condensed algebra eAe where e is a sum of primitive idempotents, one for each simple A-module
A presentation for the algebra in terms of generators and relations
The facility to write any element of the algebra as a word in the generators of the above presentation
Cartan matrix
A group algebra may be created for a finite group of moderate order over a Euclidean Domain.
Creation of group algebras: a vector and term representation are provided allowing the construction of algebras for groups of arbitrary size.
Arithmetic including Lie bracket operation
Properties of elements: idempotent, unit, zero-divisor, nilpotent
Trace and minimal polynomial
Creation of subalgebras, ideals and quotient algebras
Ideal arithmetic: Sum, product, powers, intersection
Centralizer, idealizer
Augmentation ideal, augmentation map
Characteristic ideals: Centre, commutator ideal, Jacobson radical
Ideal structure: Maximal (minimal) left, right, two-sided ideals
Decomposition: Simplicity, semi-simplicity, composition series
Construction of the (left, right) regular matrix representation
A quaternion algebra is a central, simple algebra of dimension four over a field. Basic functions are provided for quaternion algebras over an arbitrary field. Higher level routines are available for algebras over ℚ, number fields, and rational function fields k(x), where k is a finite field. This includes support for orders and ideals, in particular enumeration of left, right and two-sided ideal classes.
Arithmetic of elements
Norm, trace, and conjugation
Minimal polynomial of elements
Discriminant, ramified primes and ramified places
Construction of maximal orders
Recognition of quaternion algebras
Computation of an isomorphism to the matrix algebra, in particular over completions of the base field
Embedding subfields into quaternion algebras
Left and right orders of ideals
Isomorphism testing for ideals and orders
Construction and recognition of Eichler orders
Enumeration of left, right and two-sided ideal classes of Eichler orders
Enumeration of conjugacy classes of orders
Unit groups of orders in definite algebras