The current elements of the machinery for Lie theory comprise:
Coxeter systems
Root systems
Root data
Coxeter groups
Reflection groups
Finite dimensional Lie algebras
Quantized enveloping algebras (aka quantum groups)
Groups of Lie type
The standard descriptions for Coxeter systems and reflection groups are all supported: Coxeter matrices, Coxeter graphs, Cartan matrices, Dynkin digraphs, and Cartan names.
Conversion between the different descriptions.
Testing isomorphism and Cartan equivalence of Coxeter systems.
Testing for properties such as finite, affine, hyperbolic, and compact hyperbolic Coxeter systems.
Construction of any finite, affine and hyperbolic system.
Determining the size and number of roots of (finite) Coxeter systems.
Dynkin diagrams for finite systems.
Predicates: IsIrreducible, IsCrystallographic, IsSimplyLaced.
A root system describes the reflections in a reflection group and plays an essential role in the theory of finite Coxeter groups and Lie algebras.
Any finite root system can be constructed by giving its simple roots and coroots, including nonsemisimple systems (where the dimension of the vector space is larger than the rank).
Semisimple root systems may be constructed from a Coxeter matrix, Coxeter graph, Cartan matrix, Dynkin digraph, or Cartan name.
Standard root systems may be constructed – these are systems whose pairing is the Coxeter form, and is the way in which root systems are frequently given in the literature.
Direct sums and duals of root systems are supported.
Test equality, isomorphism and Cartan equivalence.
Determine any of the following descriptions for a root system: Cartan name, Coxeter diagram, Dynkin diagram, Coxeter matrix, Coxeter graph, Cartan matrix, or Dynkin digraph.
Invariants such as base field, rank, dimension, Coxeter group order.
Properties such being irreducible, semisimple or simply laced.
The (co)roots are stored in an indexed set, with positive roots first. The can be described and manipulated via their index, or as vectors with respect to either the standard basis or the basis of simple (co)roots.
Root space and coroot space.
Construction of the complete set of roots or coroots.
Conversion between indices and vectors.
Highest long or short root.
Reflection actions of the (co)roots: given as matrices, permutations, or words in the simple reflections.
Basic arithmetic with (co)root indices: sum, negation, positivity, heights, norms.
Coxeter form and dual Coxeter form.
Root data are fundamental to the study of Lie algebras and groups of Lie type whereas the closely related concept of a root system discussed above is normally used when working with Coxeter groups or reflection groups.
A split (untwisted) root datum can be constructed by giving its simple roots and coroots.
A semisimple system may be constructed from a Cartan matrix, Dynkin digraph, or Cartan name. By default the adjoint datum is returned, but the isogeny type can be specified.
Standard root systems may be constructed – these are systems whose pairing is the Coxeter form, and is the way in which root systems are frequently given in the literature.
Constructions are provided for direct sums, duals and subdata of root data.
Equality, isomorphism, Cartan equivalence, and isogeny.
Determination any of the following descriptions for a root datum: Cartan name, Coxeter diagram, Dynkin diagram, Coxeter matrix, Coxeter graph, Cartan matrix, Dynkin digraph
Elementary invariants: Base field, rank, dimension, Coxeter group order, group of Lie type order.
Determination of the fundamental and (co)isogeny groups.
Determination of properties such as being irreducible, semisimple, crystallographic, simply laced, adjoint and simply connected.
The standard constants used to define Lie algebras and groups of Lie type can be computed: p, q, N, ε, M, C, and eta.
The (co)roots are stored in an indexed set, with positive roots first. The can be described and manipulated via their index, or as vectors with respect to the standard basis or the basis of simple (co)roots or the basis of fundamental weights.
Root space and coroot space.
Construction of the complete set of roots or coroots.
Conversion between indices and vectors.
Highest long or short root.
Reflection actions of the (co)roots: given as matrices, permutations, or words in the simple reflections.
Basic arithmetic with (co)root indices: sum, negation, positivity, heights, norms.
Left and right strings through one root in the direction of another.
Coxeter form and dual Coxeter form.
(Co)weight lattice and fundamental (co)weights
General Coxeter groups are implemented as a subclass of finitely presented groups so that they inherit all the operations for finitely presented groups as well as having many specialized functions. The main difference is that every word is automatically converted into normal form using an algorithm designed and implemented by Bob Howlett. This module was implemented by Bob Howlett, Scott Murray, and Don Taylor.
A Coxeter group can be constructed from a Cartan matrix, Dynkin digraph, Cartan name, root system, or root datum.
Test isomorphism of Coxeter groups.
Elementary operaions include determining the Cartan name, Coxeter diagram, Coxeter matrix, Coxeter graph, rank.
Determination of basic properties such as being finite, affine, hyperbolic, compact hyperbolic, irreducible or simply laced.
Arithmetic of words: identity, multiplications, inversion, powers.
Degrees of the basic invariant polynomials.
Coxeter element and Coxeter number.
Braid group and pure braid group
Conversion to and from permutation and reflection representations.
Construction of the standard parabolic subgroups
The growth function of a Coxeter group may be computed using a very fast algorithm due to R. Howlett.
A permutation Coxeter group can be constructed from a Cartan matrix, Dynkin digraph, Cartan name, root system, or root datum.
Finite Coxeter groups are implemented as a subclass of permutation groups so that they inherit all the operations for permutation groups.
In addition to the standard functions for groups, almost all of the functions for root systems and root data also apply to permutation Coxeter groups.
A reflection subgroup can be represented two ways: As a permutation group on the roots of the larger groups, or as a permutation group on its own roots.
Transversals of reflection subgroups may be computed using an efficient algorithm due to Don Taylor.
The "standard" permutation action of a Coxeter group (usually the smallest degree permutation action) may be computed. For example, the standard action of the group of type An, gives the symmetric group on n + 1 points.
Construction and identification of a reflection group over an arbitrary ring, given the simple roots, coroots and orders
Construction of real reflection groups from a Cartan matrix, Dynkin digraph, Cartan name, root system, or root datum
Construction of all finite complex reflection groups
The degrees of the fundamental invariants may be computed for any complex reflection group. Basic codegrees can also be computed.
Most of the functions available for Coxeter groups are also available for real reflection groups.
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
Given a Lie algebra L defined over a field of characteristic p > 0, construction of the Lie subalgebra M of L generated by any set of elements of L. Thus, M is closed under the restriction map.
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
Construction of a faithful module over a Lie algebra of characteristic zero
Construction of highest-weight modules over split semisimple Lie algebras
Construction of tensor products, symmetric powers, antisymmetric powers of Lie algebra modules
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
Machinery is provided which allows computation in split (untwisted) groups of Lie type with the Steinberg presentation. These groups can be defined over any Magma field. Elements can be normalised using the Bruhat decomposition.
A group of Lie type can be created from a field and a Cartan name, Weyl group, root datum, Cartan matrix or Dynkin digraph.
Most of the operations and properties for root data also apply to groups of Lie type.
Equality, algebraic isomorphism, isogeny.
Algebraic group generators; abstract group generators for certain fields
Element arithmetic and normalisation
Bruhat decomposition and multiplicative Jordan decomposition.
The order of a twisted finite group of Lie type can be computed.
The inner, diagram, diagonal and field automorphisms can be constructed. These include all algebraic group automorphisms, and in many cases all abstract group automorphisms.
Standard, regular and highest weight representations can be constructed.
The inverse image of a module with respect to a given representation can be computed using a generalised row reduction function.
The following operations exploit the bijection between modules of connected Lie groups and their highest weights. Their Magma implementation closely follows that in the Lie package of Cohen et al.
Convert between highest weight, dominant weights, or all weights of such a module,
Determine the dimension of a module.
Apply the Adams operator or the Demazure operator.
Compute plethysms,
Compute symmetric, alternating, Littlewood-Richardson, or regular tensor products,
Branch to a subgroup, or collect to a supergroup, and
Compute Kazhdan-Lusztig polynomials and R-polynomials.
The functions in this section use the theory of FGLT to determine the required information and consequently are applicable to groups far larger than those that can be handled by the generic matrix group machinery.
Functions allow the construction of generators for any FGLT in its natural representation.
The order of any ordinary or twisted FGLT can be computed.
The Sylow subgroups of any classical FGLT are contructed using an algorithm of Holt and Stather.
The conjugacy classes of elements of most of the classical FGLTs can be determined.
Likewise the conjugacy of any two elements can be determined.