# 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.

## Author

• Dave Swinarski

## Certification

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 0.5 of LieTypes.

## Source code

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

## Exports

• Types
• Functions and commands
• casimirScalar -- computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module
• dualCoxeterNumber -- returns the dual Coxeter number 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
• isIsomorphic -- tests whether two Lie algebra modules are isomorphic
• KillingForm -- computes the scaled Killing form applied to two weights
• positiveRoots -- returns the positive roots of a simple Lie algebra
• simpleLieAlgebra -- construct a simple Lie algebra
• starInvolution -- computes w* for a weight w
• tensorCoefficient -- computes the multiplicity of W in U tensor V
• 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
• "casimirScalar(LieAlgebraModule)" -- see casimirScalar -- computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module
• "casimirScalar(String,ZZ,List)" -- see casimirScalar -- computes the scalar by which the Casimir operator acts on an irreducible 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
• "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
• "highestRoot(String,ZZ)" -- see highestRoot -- returns the highest root of a simple Lie algebra
• "irreducibleLieAlgebraModule(List,LieAlgebra)" -- see irreducibleLieAlgebraModule -- construct the irreducible Lie algebra module with given highest weight
• "isIsomorphic(LieAlgebraModule,LieAlgebraModule)" -- see isIsomorphic -- tests whether two Lie algebra modules are isomorphic
• "KillingForm(LieAlgebra,List,List)" -- see KillingForm -- computes the scaled Killing form applied to two weights
• "KillingForm(String,ZZ,List,List)" -- see KillingForm -- computes the scaled Killing form applied to two weights
• LieAlgebra == LieAlgebra -- tests equality of LieAlgebra
• LieAlgebraModule ** LieAlgebraModule -- tensor product of LieAlgebraModules
• LieAlgebraModule ++ LieAlgebraModule -- direct sum of LieAlgebraModules
• multiplicity(List,LieAlgebraModule) -- compute the multiplicity of a weight in a Lie algebra module
• "positiveRoots(LieAlgebra)" -- see positiveRoots -- returns the positive roots of a simple Lie algebra
• "positiveRoots(String,ZZ)" -- see positiveRoots -- returns the positive roots of a simple Lie algebra
• "simpleLieAlgebra(String,ZZ)" -- see simpleLieAlgebra -- construct a simple Lie algebra
• "starInvolution(List,LieAlgebra)" -- see starInvolution -- computes w* for a weight w
• "starInvolution(String,ZZ,List)" -- see starInvolution -- computes w* for a weight w
• "tensorCoefficient(LieAlgebraModule,LieAlgebraModule,LieAlgebraModule)" -- see tensorCoefficient -- computes the multiplicity of W in U tensor V
• "weightDiagram(LieAlgebraModule)" -- see weightDiagram -- computes the weights in a Lie algebra module and their multiplicities
• "weightDiagram(String,ZZ,List)" -- see weightDiagram -- computes the weights in a Lie algebra module and their multiplicities
• "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
• Symbols
• MaxWordLength -- Optional argument to specify the allowable length of words in the affine Weyl group when computing fusion products.

## For the programmer

The object LieTypes is .