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LieTypes -- Common types for Lie groups and Lie algebras

Description

This package defines types used by the ConformalBlocks package and may someday be used by other packages as well. If you would like to see a type or function added to this package (or better yet, if you would like to write types or functions for this package), please contact Dan Grayson, Mike Stillman, or Dave Swinarski.

Authors

Certification a gold star

Version 0.5 of this package was accepted for publication in volume 8 of The Journal of Software for Algebra and Geometry on 2 August 2018, in the article Software for computing conformal block divisors on bar M_0,n. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 0.81 of LieTypes.

Source code

The source code from which this documentation is derived is in the file LieTypes.m2.

Exports

  • Types
  • Functions and commands
    • adams -- Computes the action of the nth Adams operator on a Lie algebra module
    • adjointModule -- The adjoint module of a Lie algebra
    • branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
    • cartanMatrix -- Provide the Cartan matrix of a simple Lie algebra
    • casimirScalar -- computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module
    • character -- Computes the character of a Lie algebra module
    • dualCoxeterNumber -- returns the dual Coxeter number of a simple Lie algebra
    • dynkinDiagram -- Provide the Dynkin diagram of a simple Lie algebra
    • fusionCoefficient -- computes the multiplicity of W in the fusion product of U and V
    • fusionProduct -- computes the multiplicities of irreducibles in the decomposition of the fusion product of U and V
    • highestRoot -- returns the highest root of a simple Lie algebra
    • irreducibleLieAlgebraModule -- construct the irreducible Lie algebra module with given highest weight
    • isIrreducible -- Whether a Lie algebra module is irreducible or not
    • killingForm -- computes the scaled Killing form applied to two weights
    • LieAlgebraModuleFromWeights -- finds a Lie algebra module based on its weights
    • positiveCoroots -- see positiveRoots -- returns the positive (co)roots of a simple Lie algebra
    • positiveRoots -- returns the positive (co)roots of a simple Lie algebra
    • qdim -- Compute principal specialization of character or quantum dimension
    • simpleLieAlgebra -- construct a simple Lie algebra
    • simpleRoots -- returns the simple roots of a simple Lie algebra
    • starInvolution -- computes w* for a weight w
    • subLieAlgebra -- Define a sub-Lie algebra of an existing one
    • tensorCoefficient -- computes the multiplicity of W in U tensor V
    • trivialModule -- The trivial module of a Lie algebra
    • weightDiagram -- computes the weights in a Lie algebra module and their multiplicities
    • weylAlcove -- the dominant integral weights of level less than or equal to l
  • Methods
    • adams(ZZ,LieAlgebraModule) -- see adams -- Computes the action of the nth Adams operator on a Lie algebra module
    • adjointModule(LieAlgebra) -- see adjointModule -- The adjoint module of a Lie algebra
    • branchingRule(LieAlgebraModule,LieAlgebra) -- see branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
    • branchingRule(LieAlgebraModule,List) -- see branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
    • branchingRule(LieAlgebraModule,Matrix) -- see branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
    • cartanMatrix(LieAlgebra) -- see cartanMatrix -- Provide the Cartan matrix of a simple Lie algebra
    • casimirScalar(LieAlgebraModule) -- see casimirScalar -- computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module
    • character(LieAlgebra,List) -- see character -- Computes the character of a Lie algebra module
    • character(LieAlgebra,Vector) -- see character -- Computes the character of a Lie algebra module
    • character(LieAlgebraModule) -- see character -- Computes the character of a Lie algebra module
    • dim(LieAlgebraModule) -- computes the dimension of a Lie algebra module as a vector space over the ground field
    • dualCoxeterNumber(LieAlgebra) -- see dualCoxeterNumber -- returns the dual Coxeter number of a simple Lie algebra
    • dualCoxeterNumber(String,ZZ) -- see dualCoxeterNumber -- returns the dual Coxeter number of a simple Lie algebra
    • dynkinDiagram(LieAlgebra) -- see dynkinDiagram -- Provide the Dynkin diagram of a simple Lie algebra
    • fusionCoefficient(LieAlgebraModule,LieAlgebraModule,LieAlgebraModule,ZZ) -- see fusionCoefficient -- computes the multiplicity of W in the fusion product of U and V
    • fusionProduct(LieAlgebraModule,LieAlgebraModule,ZZ) -- see fusionProduct -- computes the multiplicities of irreducibles in the decomposition of the fusion product of U and V
    • highestRoot(LieAlgebra) -- see highestRoot -- returns the highest root of a simple Lie algebra
    • irreducibleLieAlgebraModule(LieAlgebra,List) -- see irreducibleLieAlgebraModule -- construct the irreducible Lie algebra module with given highest weight
    • irreducibleLieAlgebraModule(LieAlgebra,Vector) -- see irreducibleLieAlgebraModule -- construct the irreducible Lie algebra module with given highest weight
    • isIrreducible(LieAlgebraModule) -- see isIrreducible -- Whether a Lie algebra module is irreducible or not
    • killingForm(LieAlgebra,List,List) -- see killingForm -- computes the scaled Killing form applied to two weights
    • killingForm(LieAlgebra,Vector,Vector) -- see killingForm -- computes the scaled Killing form applied to two weights
    • directSum(LieAlgebra) -- see LieAlgebra ++ LieAlgebra -- Take the direct sum of Lie algebras
    • LieAlgebra ++ LieAlgebra -- Take the direct sum of Lie algebras
    • LieAlgebra == LieAlgebra -- tests equality of LieAlgebra
    • LieAlgebraModule ** LieAlgebraModule -- tensor product of LieAlgebraModules
    • directSum(LieAlgebraModule) -- see LieAlgebraModule ++ LieAlgebraModule -- direct sum of LieAlgebraModules
    • LieAlgebraModule ++ LieAlgebraModule -- direct sum of LieAlgebraModules
    • LieAlgebraModule @ LieAlgebraModule -- Take the tensor product of modules over different Lie algebras
    • LieAlgebraModule ^** ZZ -- Computes the nth tensor power of a Lie algebra module
    • LieAlgebraModuleFromWeights(RingElement,LieAlgebra) -- see LieAlgebraModuleFromWeights -- finds a Lie algebra module based on its weights
    • LieAlgebraModuleFromWeights(VirtualTally,LieAlgebra) -- see LieAlgebraModuleFromWeights -- finds a Lie algebra module based on its weights
    • multiplicity(List,LieAlgebraModule) -- compute the multiplicity of a weight in a Lie algebra module
    • multiplicity(Vector,LieAlgebraModule) -- see multiplicity(List,LieAlgebraModule) -- compute the multiplicity of a weight in a Lie algebra module
    • new LieAlgebra from Matrix -- Define a Lie algebra from its Cartan matrix
    • positiveCoroots(LieAlgebra) -- see positiveRoots -- returns the positive (co)roots of a simple Lie algebra
    • positiveRoots(LieAlgebra) -- see positiveRoots -- returns the positive (co)roots of a simple Lie algebra
    • qdim(LieAlgebraModule) -- see qdim -- Compute principal specialization of character or quantum dimension
    • qdim(LieAlgebraModule,ZZ) -- see qdim -- Compute principal specialization of character or quantum dimension
    • simpleLieAlgebra(String,ZZ) -- see simpleLieAlgebra -- construct a simple Lie algebra
    • simpleRoots(LieAlgebra) -- see simpleRoots -- returns the simple roots of a simple Lie algebra
    • simpleRoots(String,ZZ) -- see simpleRoots -- returns the simple roots of a simple Lie algebra
    • dual(LieAlgebraModule) -- see starInvolution -- computes w* for a weight w
    • starInvolution(LieAlgebraModule) -- see starInvolution -- computes w* for a weight w
    • subLieAlgebra(LieAlgebra,List) -- see subLieAlgebra -- Define a sub-Lie algebra of an existing one
    • subLieAlgebra(LieAlgebra,Matrix) -- see subLieAlgebra -- Define a sub-Lie algebra of an existing one
    • exteriorPower(ZZ,LieAlgebraModule) -- see symmetricPower(ZZ,LieAlgebraModule) -- Computes the nth symmetric / exterior tensor power of a Lie algebra module
    • symmetricPower(ZZ,LieAlgebraModule) -- Computes the nth symmetric / exterior tensor power of a Lie algebra module
    • tensorCoefficient(LieAlgebraModule,LieAlgebraModule,LieAlgebraModule) -- see tensorCoefficient -- computes the multiplicity of W in U tensor V
    • trivialModule(LieAlgebra) -- see trivialModule -- The trivial module of a Lie algebra
    • weightDiagram(LieAlgebra,List) -- see weightDiagram -- computes the weights in a Lie algebra module and their multiplicities
    • weightDiagram(LieAlgebra,Vector) -- see weightDiagram -- computes the weights in a Lie algebra module and their multiplicities
    • weightDiagram(LieAlgebraModule) -- see weightDiagram -- computes the weights in a Lie algebra module and their multiplicities
    • weylAlcove(LieAlgebra,ZZ) -- see weylAlcove -- the dominant integral weights of level less than or equal to l
    • weylAlcove(String,ZZ,ZZ) -- see weylAlcove -- the dominant integral weights of level less than or equal to l
    • weylAlcove(ZZ,LieAlgebra) -- see weylAlcove -- the dominant integral weights of level less than or equal to l
  • Other things
    • ฯ‰ -- construct the irreducible Lie algebra module with given highest weight
    • ๐”ž (missing documentation)
    • ๐”Ÿ (missing documentation)
    • ๐”  (missing documentation)
    • ๐”ก (missing documentation)
    • ๐”ข (missing documentation)
    • ๐”ฃ (missing documentation)
    • ๐”ค (missing documentation)
    • LL -- construct the irreducible Lie algebra module with given highest weight

For the programmer

The object LieTypes is a package.