Lie groups and Lie algebras play an essential part in differential geometry and its applications. For this reason the DifferentialGeometry package provides Maple users with the DifferentialGeometry:-LieAlgebra package.

The LieAlgebra package contains a large number of commands for defining Lie algebras from a variety of sources and for creating new Lie algebras from existing Lie algebras.  These include DirectSum, Extension, LieAlgebraData, MatrixAlgebras, QuotientAlgebra, SimpleLieAlgebraData, SemiDirectSum. Especially noteworthy is the use of the LieAlgebraData command to convert a Lie algebra of vector fields on a manifold to an abstract Lie algebra.

The general structure of the Lie algebra can be investigated with the Decompose, Query, Series, Nilradical, and Radical commands.

The structure of a semi-simple Lie algebra can be explored with the commands CartanDecomposition, CartanSubalgebra, CartanMatrix, CompactRoots, PositiveRoots, RootSpaceDecomposition, RestrictedRootSpaceDecomposition, SimpleLieAlgebraProperties, SimpleRoots.

Homomorphisms between Lie algebras can be constructed using the DifferentialGeometry command Transformation.  

The matrix exponential of any derivation of a Lie algebra will define an automorphism of that Lie algebra.

Structure equations for general more general algebras such as the quaternions, octonions, Jordan algebras and Clifford algebras are available.  See AlgebraData and AlgebraLibraryData.

Cohomology of Lie algebras can be computed with the commands Cohomology, CohomologyDecomposition, KostantCodifferential, KostantLaplacian.

Deformations of Lie algebras can be studied with the commands Deformation and MasseyProduct.  

Lie algebras can be prolonged with TanakaProlongation.

Properties of Lie subalgebras can also be investigated with the Query command.

The LieAlgebra Lessons provide a systematic introduction to the commands in the LieAlgebra package.

The LieAlgebra package is a subpackage of the DifferentialGeometry package. Each command in the LieAlgebras package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

Complexify: find the complexification of a Lie algebra.

DirectSum: create the direct sum of a list of Lie algebras.

Extension: calculate a right or a central extension of a Lie algebra.

LieAlgebraData: convert different realizations of a Lie algebra to a Lie algebra data structure.

LieAlgebraWithCoefficientsData: calculate the structure equations for a Lie algebra with coefficients in a representation.

QuotientAlgebra: create the Lie algebra data structure for a quotient algebra of a Lie algebra by an ideal.

SemiDirectSum: create the semi-direct sum of two Lie algebras.

SimpleLieAlgebraData: obtain the structure equations for a classical matrix Lie algebra.

SymbolAlgebra: find the symbol algebra for a distribution.

Center: find the center of a Lie algebra.

Centralizer: find the centralizer of a list of vectors.

DerivedAlgebra: find the derived algebra of a Lie algebra.

GeneralizedCenter: calculate the generalized center of an ideal in a Lie algebra.

HomomorphismSubalgebras: find the kernel or image of a Lie algebra homomorphism.

MinimalIdeal: find the smallest ideal containing a given set.

MinimalSubalgebra: find the minimal subalgebra of a Lie algebra.

Nilradical: find the nilradical of a Lie algebra.

ParabolicSubalgebra: find the parabolic subalgebra defined by a set of simple roots or a set of restricted simple roots.

ParabolicSubalgebraRoots: find the simple roots which generate a parabolic subalgebra

Radical: find the radical of a Lie algebra.

Series: find the derived series, lower central series, upper central series of a Lie algebra or a Lie subalgebra.

SubalgebraNormalizer: find the normalizer of a subalgebra.

Adjoint: find the Adjoint Matrix for a vector in a Lie algebra.

AdjointExp: find the Exponential of the Adjoint Matrix for a vector in a Lie algebra.

ApplyHomomorphism: apply a Lie algebra homomorphism to a vector, form or tensor.

Derivations: find the inner and/or outer derivations of a Lie algebra.

BracketOfSubspaces: find the subspace generated by the bracketing of two subspaces.

ChangeLieAlgebraTo: change the current frame to the frame for a Lie algebra.

InfinitesimalCoadjointAction: find the vector fields defining the infinitesimal co-adjoint action of a Lie group on its Lie algebra.

Killing: find the Killing form of a Lie algebra.

KillingForm: find the Killing form, defined as a tensor, of a Lie algebra.

MultiplicationTable: display the multiplication table of a Lie algebra.

Query: check various properties of a Lie algebra, subalgebra, or transformation.

Decompose: decompose a Lie algebra into a direct sum of indecomposable Lie algebras.

LeviDecomposition: compute the Levi decomposition of a Lie algebra.

CartanDecomposition: find the Cartan decomposition defined by a Cartan involution, find the Cartan decomposition of a semi-simple matrix algebra.

CartanInvolution: find the Cartan involution defined by a Cartan decomposition of a non-compact, semi-simple, real Lie algebra.

CartanMatrix: find the Cartan matrix for a simple Lie algebra from a root space decomposition, display the Cartan matrix for a given root type.

CartanMatrixToStandardForm: transform a Cartan matrix to standard form.

CartanSubalgebra: find a Cartan subalgebra of a Lie algebra.

ChevalleyBasis: find the Chevalley basis for a real, split semi-simple Lie algebra.

CompactRoots: find the compact roots in a root system for a non-compact semi-simple real Lie algebra.

CoRoot: find the coroot of a root vector for a semi-simple Lie algebra.

GradeSemiSimpleLieAlgebra: find the grading of a semi-simple Lie algebra defined by a set of simple roots or restricted simple roots.

LieAlgebraRoots: find a root or the roots for a semi-simple Lie algebra from a root space and a Cartan subalgebra; or from a root space decomposition.

PositiveRoots: find the positive roots from a set of roots or a root space decomposition, list the positive roots for a given root type.

RestrictedRootSpaceDecomposition: find the real root space decomposition of a non-compact semi-simple Lie algebra with respect to an Abelian subalgebra.

RootSpace: find the root space for a semi-simple Lie algebra from a Cartan subalgebra.

RootSpaceDecomposition: find the root space decomposition for a semi-simple Lie algebra from a Cartan subalgebra.

RootString: find the sequence of roots through a given root of a semi-simple Lie algebra.

RootToCartanSubalgebraElementH: associate to each positive root of a simple Lie algebra a vector in the Cartan subalgebra.

SatakeAssociate: find the non-compact simple root associated to a given non-compact root in the Satake diagram.    

SatakeDiagram: display the Satake diagram for a non-compact, real, simple matrix algebra.

SimpleLieAlgebraProperties: provide a table of properties for any real simple Lie algebra.

SimpleRoots: find the simple roots for a set of positive roots.

SplitAndCompactForms: find the real split and real compact forms of a real semi-simple Lie algebra.

Codifferential: calculate the codifferential of a multi-vector defined on a Lie algebra with coefficients in a representation.

Cohomology: find the cohomology of a Lie algebra.

CohomologyDecomposition: decompose a closed form into the sum of an exact form and a form defining a cohomology class.

KostantCodifferential: calculate the Kostant co-differential of a p-form or a list of p-forms defined on a nilpotent Lie algebra with coefficients in a representation.

KostantLaplacian: calculate the Kostant Laplacian of a form defined on a nilpotent Lie algebra with coefficients in a representation.

PositiveDefiniteMetricOnRepresentationSpace: find a positive-definite inner product on a representation space which is compatible with a Cartan involution.

RelativeChains: find the vector space of forms on a Lie algebra relative to a given subalgebra.

Deformation: find the deformation of a Lie algebra defined by a list of 2-forms.

MasseyProduct: calculate the Massey product of a pair of forms.

JacobsonRadical: find the Jacobson radical for a matrix Lie algebra.

MatrixAlgebras: create a Lie algebra data structure for a matrix Lie algebra.

MatrixCentralizer: find the matrix centralizer of a list of matrices.

MatrixNormalizer: find the matrix normalizer of a list of matrices.

MatrixSubalgebra: find the subalgebra of a Lie algebra which preserves a collection of tensors or subspaces of tensors.

MinimalIdeal: find the smallest ideal containing a given set of vectors.

MinimalSubalgebra: find the smallest subalgebra containing a given set of vectors.

AlgebraData: find the structure equations for a real algebra defined by a list of matrices and a multiplication procedure.

AlgebraLibraryData: retrieve the structure equations for various classical algebras (quaternions, octonions, Clifford algebras, and low dimensional Jordan algebras).

AlgebraNorm: find the norm of a quaternion or octonion.

AlgebraInverse: find the multiplicative inverse of a quaternion or octonion.    

JordanMatrices: find a basis for a Jordan algebra of matrices.

JordanProduct: find the Jordan product of two Jordan matrices.    

ApplyRepresentation: apply a Lie algebra representation to a vector in a Lie algebra.

AscendingIdealsBasis: find a basis for a solvable Lie algebra which defines an ascending chain of ideals.

ChangeRepresentationBasis: change the basis for a representation, either in the Lie algebra or in the representation space.

Invariants: calculate the invariant vectors and tensors for a representation of a Lie algebra

Representation: define a representation of a Lie algebra.

RepresentationEigenvector: find a simultaneous eigenvector for the representation of a solvable Lie algebra.

RestrictedRepresentation: find the restriction of a representation of a subalgebra.

SolvableRepresentation: given a representation of a solvable algebra, find a basis for the representation space in which the representation matrices are Upper triangular matrices.

StandardRepresentation: find the standard matrix representation or linear vector field representation of a classical matrix algebra.

SubRepresentation: find the induced representation on an invariant subspace.

TensorProductOfRepresentations: form a tensor product representation from a list of representations.

ChangeGradedComponent: change one or more components of a graded Lie algebra.

Rank1Elements: calculate the rank 1 matrices in the span of a given list of matrices.

TanakaProlongation: calculate the Tanaka prolongation, to a specified order, of a graded nilpotent Lie algebra.