LieAlgebra -- class for Lie algebras

Description

This class represents Lie algebras. Currently only semi-simple Lie algebras over the complex numbers are supported. An object of type LieAlgebra is a hash table whose keys record the rank of the Lie algebra and the type of the root system.

i1 : g=simpleLieAlgebra("A",1)

o1 = g

o1 : simple LieAlgebra

i2 : h=simpleLieAlgebra("E",6)

o2 = h

o2 : simple LieAlgebra

i3 : g++h

o3 = 𝔞  ++ 𝔢
      1     6

o3 : LieAlgebra

If you have access to unicode fraktur, you can use the shorthand

i4 : 𝔣_4

o4 = 𝔣
      4

o4 : simple LieAlgebra

Functions and methods returning an object of class LieAlgebra:

simpleLieAlgebra -- construct a simple Lie algebra
subLieAlgebra -- Define a sub-Lie algebra of an existing one

Methods that use an object of class LieAlgebra:

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
cartanMatrix(LieAlgebra) -- see cartanMatrix -- Provide the Cartan matrix of a simple Lie algebra
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
dualCoxeterNumber(LieAlgebra) -- 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
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
isIsomorphic(LieAlgebra,LieAlgebra) -- see isIsomorphic -- tests whether two Lie algebra are isomorphic
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
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
simpleRoots(LieAlgebra) -- see simpleRoots -- returns the simple roots of a simple Lie algebra
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
toExternalString(LieAlgebra) (missing documentation)
toString(LieAlgebra) (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
weylAlcove(LieAlgebra,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

For the programmer

The object LieAlgebra is a type, with ancestor classes HashTable < Thing.

The source of this document is in LieTypes.m2:1519:0.