
Description


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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.

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

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

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

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

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

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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.



Commands for creating Lie algebras


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Complexify: find the complexification of a Lie algebra.

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

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

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

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



Commands for finding subalgebras


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Center: find the center of a Lie algebra.

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Centralizer: find the centralizer of a list of vectors.

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MinimalIdeal: find the smallest ideal containing a given set.

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Nilradical: find the nilradical of a Lie algebra.

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

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Radical: find the radical of a Lie algebra.

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



Commands for working with mappings of Lie algebras


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

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

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



Utilities


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Killing: find the Killing form of a Lie algebra.

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

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



Commands for general structure theory of Lie algebras


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



Commands for studying semisimple Lie algebras


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CartanDecomposition: find the Cartan decomposition defined by a Cartan involution, find the Cartan decomposition of a semisimple matrix algebra.

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CartanInvolution: find the Cartan involution defined by a Cartan decomposition of a noncompact, semisimple, real Lie algebra.

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

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ChevalleyBasis: find the Chevalley basis for a real, split semisimple Lie algebra.

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CompactRoots: find the compact roots in a root system for a noncompact semisimple real Lie algebra.

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CoRoot: find the coroot of a root vector for a semisimple Lie algebra.

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

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

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

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RootSpace: find the root space for a semisimple Lie algebra from a Cartan subalgebra.

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RootSpaceDecomposition: find the root space decomposition for a semisimple Lie algebra from a Cartan subalgebra.

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RootString: find the sequence of roots through a given root of a semisimple Lie algebra.

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SatakeAssociate: find the noncompact simple root associated to a given noncompact root in the Satake diagram.

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SatakeDiagram: display the Satake diagram for a noncompact, real, simple matrix algebra.

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

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SplitAndCompactForms: find the real split and real compact forms of a real semisimple Lie algebra.



Commands for calculating Lie algebra cohomology


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

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Cohomology: find the cohomology of a Lie algebra.

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

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KostantCodifferential: calculate the Kostant codifferential of a pform or a list of pforms defined on a nilpotent Lie algebra with coefficients in a representation.

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

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



Commands for calculating deformations of Lie algebras


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

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



Commands for working with matrix algebras


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

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

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



Commands for working with general algebras


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

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

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AlgebraNorm: find the norm of a quaternion or octonion.

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



Commands for working with representations of Lie algebras


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

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

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

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

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



Commands for working with prolongations of Lie algebras


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

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



Alphabetical listing of all LieAlgebra commands




