LieAlgebras[LieAlgebraRoots] - find a root or the roots for a semi-simple Lie algebra from a root space and the Cartan subalgebra; or from a root space decomposition
Calling Sequences
LieAlgebraRoots(X, CSA)
LieAlgebraRoots(RSD )
Parameters
X - a vector in a Lie algebra, defining a root space
CSA - a list of vectors in a semi-simple Lie algebra, defining a Cartan subalgebra
RSD - a table, defining a root space decomposition of a semi-simple Lie algebra
Description
Examples
Let g be a Lie algebra and h a Cartan subalgebra. Let h1, h2, ... , hm be a basis for 𝔥. A root for g with respect to this basis is a non-zero m-tuple of complex numbers α = α1, α2, ... , αm such that adhix = αi x (*) for some x∈ 𝔤.
The set of x∈ 𝔤 which satisfy (*) is called the root space of g defined by α and denoted by Rα. A basic theorem in the structure theory of semi-simple Lie algebras asserts that the root spaces Rα are 1-dimensional.
The first calling sequence calculates the root α for the given root space X. If X is not a root space, then an empty vector is returned.
The second calling sequence simply returns the indices, as column vectors, for the table defining the root space decomposition.
withDifferentialGeometry:withLieAlgebras:
Example 1.
Use the command SimpleLieAlgebraData to initialize the simple Lie algebra su4. This is a 15-dimensional Lie algebra of skew-Hermitian matrices.
LD≔SimpleLieAlgebraDatasu(4),su4,labelformat=gl,labels=E,θ
LD:=e1,e4=−2e10,e1,e5=e11,e1,e7=−e13,e1,e8=e14,e1,e9=−e15,e1,e10=2e4,e1,e11=−e5,e1,e13=e7,e1,e14=−e8,e1,e15=e9,e2,e4=e10,e2,e5=−2e11,e2,e6=e12,e2,e7=−e13,e2,e8=−e14,e2,e10=−e4,e2,e11=2e5,e2,e12=−e6,e2,e13=e7,e2,e14=e8,e3,e5=e11,e3,e6=−2e12,e3,e7=e13,e3,e8=−e14,e3,e9=−e15,e3,e11=−e5,e3,e12=2e6,e3,e13=−e7,e3,e14=e8,e3,e15=e9,e4,e5=−e7,e4,e7=e5,e4,e8=−e9,e4,e9=e8,e4,e10=−2e1,e4,e11=−e13,e4,e13=e11,e4,e14=−e15,e4,e15=e14,e5,e6=−e8,e5,e7=−e4,e5,e8=e6,e5,e10=e13,e5,e11=−2e2,e5,e12=−e14,e5,e13=−e10,e5,e14=e12,e6,e7=e9,e6,e8=−e5,e6,e9=−e7,e6,e11=e14,e6,e12=−2e3,e6,e13=e15,e6,e14=−e11,e6,e15=−e13,e7,e9=e6,e7,e10=e11,e7,e11=−e10,e7,e12=−e15,e7,e13=−2e1−2e2,e7,e15=e12,e8,e9=−e4,e8,e10=e15,e8,e11=e12,e8,e12=−e11,e8,e14=−2e2−2e3,e8,e15=−e10,e9,e10=e14,e9,e12=−e13,e9,e13=e12,e9,e14=−e10,e9,e15=−2e1−2e2−2e3,e10,e11=e7,e10,e13=e5,e10,e14=e9,e10,e15=e8,e11,e12=e8,e11,e13=−e4,e11,e14=e6,e12,e13=−e9,e12,e14=−e5,e12,e15=−e7,e13,e15=e6,e14,e15=−e4,Ei11,Ei22,Ei33,E12,E23,E34,E13,E24,E14,Ei12,Ei23,Ei34,Ei13,Ei24,Ei14,thetai11,thetai22,thetai33,θ12,θ23,θ34,θ13,θ24,θ14,thetai12,thetai23,thetai34,thetai13,thetai24,thetai14
DGsetupLD
Lie algebra: su4
The explicit matrices defining su4are given by the StandardRepresentation command.
StandardRepresentationsu4
The diagonal matrices determine a Cartan subalgebra.
CSA≔Ei11,Ei22,Ei33
CSA:=Ei11,Ei22,Ei33
We use the Query command to check that (2.3) is a Cartan subalgebra.
QueryCSA,CartanSubalgebra
true
Find the root for the root space E14− I Ei14.
LieAlgebraRootsE14−IEi14,CSA
Note that the command RootSpace performs the inverse operation to LieAlgebraRoot - given a root, the command RootSpace returns the corresponding root space.
RootSpace−I,0,−I,CSA
E14−IEi14
Example 2.
If the complete root space decomposition is given as a table, then the command LieAlgebraRoots returns the indices of that table as column vectors.
RSD≔RootSpaceDecompositionCSA
RSD:=table−I,I,I=E24+IEi24,0,I,−2I=E34−IEi34,0,−I,2I=E34+IEi34,I,−2I,I=E23−IEi23,−I,2I,−I=E23+IEi23,I,0,I=E14+IEi14,−I,0,−I=E14−IEi14,I,−I,−I=E24−IEi24,−2I,I,0=E12−IEi12,I,I,−I=E13+IEi13,−I,−I,I=E13−IEi13,2I,−I,0=E12+IEi12
LieAlgebraRootsRSD
See Also
DifferentialGeometry
CartanSubalgebra
Query
RootSpace
RootSpaceDecomposition
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