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)
Quaternion algebras
Group algebras
Matrix algebras
Finite dimensional Lie algebras (defined by structure constants)
Quantized enveloping algebras (aka quantum groups)
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.
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
A quaternion algebra is a central, simple algebra of dimension four over a field. A special type for quaternion algebras is released in Magma V2.7. Support for orders over ℤ, k[x] and orders of number fields is provided for quaternions over the rational field ℚ, k(x) or a number field. Special functions for enumeration all ideals in definite quaternion algebras over ℚ, with connections to modular forms.
Arithmetic of elements
Norm, trace, and conjugation
Minimal polynomial of elements
Discriminant and ramified primes
Creation of prime ideals
Testing for principal ideals
Enumeration of left and right ideals of an definite order over ℤ
Left and right orders of an ideal in a definite order over ℤ
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
While a matrix algebra may be defined over any ring R, most non-trivial computations require R to be an Euclidean Domain.
Arithmetic
Extension and restriction of coefficient ring
Direct sum, tensor product
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
Canonical forms over an ED: echelon, Hermite, Smith
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, Jacobson radical
Centralizer of a subalgebra in the complete matrix algebra
Maximal (minimal) left, right, two-sided ideals
Construction of the (left, right) regular matrix representation
The order of a unit over a finite field is found using the very efficient algorithm of Leedham-Green.
A finite-dimensional Lie algebra L over a field K is presented in terms of a basis for a K-vector space V together with a set of structure constants defining the multiplication of these basis elements.
The major structural machinery for Lie algebras has been implemented for Magma by Willem de Graaf.
Creation of Lie algebras in terms of structure constants
Construction of a Lie algebra from an associative algebra via the Lie bracket product
Construction of a Lie algebra given by generators and relations
Construction of a Lie algebra from a p-group, by using its Jennings series.
Construction of a specified simple Lie algebra
Direct sum
Arithmetic
Trace and minimal polynomial
Test for abelian, nilpotent, solvable, restricted
Test for simple, semisimple
Killing form
Adjoint representation of an element; Associated adjoint algebra
Root system of a semisimple Lie algebra with a split Cartan subalgebra
Creation of subalgebras, ideals and quotient algebras
Ideal arithmetic: Sum, product, powers, intersection
Centre
Centralizer, normalizer
Jacobson radical, nil radical, solvable radical
Composition series
Derived series, lower central series, upper central series
Nilradical, solvable radical
Cartan subalgebra, Levi subalgebra
Maximal (minimal) left, right, two-sided ideals
Decomposition of a Lie algebra into a direct sum of ideals
Type of a simple or semisimple algebra
A quantized enveloping algebra (corresponding to a given root datum) is represented with respect to an integral basis, as defined by Lusztig.
Constructing of quantized enveloping algebras with respect to a given root datum
Arithmetic: sum and product
Representations: construction of highest-weight modules, and tensor products of them
Construction of the canonical basis of a highest-weight module
Construction of elements of the canonical basis of the negative part of a quantized enveloping algebra
Action of the Kashiwara operators
Littelmann's path model: action of the path operators, construction of the crystal graph