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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 (DOI: 10.2140/jsag.2018.8.81). That version can be obtained from the journal.

Version

This documentation describes version 0.9 of LieTypes.

Citation

If you have used this package in your research, please cite it as follows:

@misc{LieTypesSource,
  title = {{LieTypes: common types and methods for Lie groups and Lie algebras. Version~0.9}},
  author = {Dave Swinarski and Paul Zinn-Justin},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

@article{LieTypesArticle,
  title = {{Software for computing conformal block divisors on bar M_0,n}},
  author = {Dave Swinarski and Paul Zinn-Justin},
  journal = {The Journal of Software for Algebra and Geometry},
  volume = {8},
  year = {2018},
}

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
    • embedding -- gives the embedding of Cartan subalgebras of one Lie algebra into another
    • 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
    • zeroModule -- The zero module of a Lie algebra
  • 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
    • casimirScalar(LieAlgebra,List) (missing documentation)
    • 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
    • embedding(LieAlgebra,LieAlgebra) -- see embedding -- gives the embedding of Cartan subalgebras of one Lie algebra into another
    • 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
    • isIsomorphic(LieAlgebra,LieAlgebra) -- see isIsomorphic -- tests whether two Lie algebra are isomorphic
    • isIsomorphic(LieAlgebraModule,LieAlgebraModule) (missing documentation)
    • 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
    • LieAlgebra _ ZZ -- selects one summand of a semi-simple Lie Algebra
    • LieAlgebra _* -- gives the list of summands of a semi-simple Lie Algebra
    • 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 (missing documentation)
    • LieAlgebraModule ^** ZZ -- Computes the nth tensor power of a Lie algebra module
    • LieAlgebraModule _ LieAlgebraModule -- see LieAlgebraModule _ ZZ -- Pick out one irreducible submodule of a Lie algebra module
    • LieAlgebraModule _ List -- see LieAlgebraModule _ ZZ -- Pick out one irreducible submodule of a Lie algebra module
    • LieAlgebraModule _ Vector -- see LieAlgebraModule _ ZZ -- Pick out one irreducible submodule of a Lie algebra module
    • LieAlgebraModule _ ZZ -- Pick out one irreducible submodule of a Lie algebra module
    • LieAlgebraModule _* -- List irreducible submodules 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
    • new LieAlgebra from Matrix -- Define a Lie algebra from its Cartan matrix
    • new LieAlgebra from Sequence (missing documentation)
    • 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
    • subLieAlgebra(LieAlgebra,String) -- 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
    • toExternalString(LieAlgebra) (missing documentation)
    • toExternalString(LieAlgebraModule) (missing documentation)
    • toString(LieAlgebra) (missing documentation)
    • toString(LieAlgebraModule) (missing documentation)
    • 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
    • zeroModule(LieAlgebra) -- see zeroModule -- The zero module of a Lie algebra
  • 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, defined in LieTypes.m2.

The source of this document is in LieTypes.m2:1498:0.