LieAlgebras[RootSpace] - find a root space for a semi-simple Lie algebra from a Cartan subalgebra or a root space decomposition
Calling Sequences
RootSpace(RV, CSA)
RootSpace(RV, RSD)
Parameters
RV - a column vector
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 theorem of semi-simple Lie algebras asserts that the root spaces Rα are 1-dimensional.
The first call sequence calculates the root space Rα for a given root. If α is not a root, then the zero vector (in 𝔤) is returned.
The second calling sequence simply returns the table entry in the table of root spaces corresponding to the root α.
withDifferentialGeometry:withLieAlgebras:
Example 1.
Use the command SimpleLieAlgebraData to obtain the Lie algebra data for the simple Lie algebra su4. This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices.
LD≔SimpleLieAlgebraDatasu(4),su4,labelformat=gl,labels=U,η
LD:=e1,e4=e10,e1,e6=e12,e1,e7=e13,e1,e8=e14,e1,e9=2e15,e1,e10=−e4,e1,e12=−e6,e1,e13=−e7,e1,e14=−e8,e1,e15=−2e9,e2,e4=−e10,e2,e5=e11,e2,e6=e12,e2,e8=2e14,e2,e9=e15,e2,e10=e4,e2,e11=−e5,e2,e12=−e6,e2,e14=−2e8,e2,e15=−e9,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=2e1−2e2,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=2e2−2e3,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=2e1−2e3,e7,e15=e12,e8,e9=−e4,e8,e10=e15,e8,e11=e12,e8,e12=−e11,e8,e14=2e2,e8,e15=−e10,e9,e10=e14,e9,e12=−e13,e9,e13=e12,e9,e14=−e10,e9,e15=2e1,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,Ui11,Ui22,Ui33,U12,U23,U34,U13,U24,U14,Ui12,Ui23,Ui34,Ui13,Ui24,Ui14,etai11,etai22,etai33,η12,η23,η34,η13,η24,η14,etai12,etai23,etai34,etai13,etai24,etai14
Initialize the Lie algebra su4.
DGsetupLD
Lie algebra: su4
The command StandardRepresentation will produce the actual matrices defining su4. (This command only applies to Lie algebras constructed by the SimpleLieAlgebraData procedure.)
StandardRepresentationsu4
The Lie algebra elements corresponding to the complex diagonal matrices define a Cartan subalgebra.
CSA≔Ui11,Ui22,Ui33
CSA:=Ui11,Ui22,Ui33
We check this is indeed a Cartan subalgebra using the Query command
QueryCSA,CartanSubalgebra
true
Here is the root space corresponding to the root <I, I, -I>.
X≔RootSpaceI,I,2I,CSA
X:=U34−IUi34
We check that the X is an eigenvector for the elements of the Cartan subalgebra.
B≔seqLieBracketh,X,h=CSA
B:=IU34+Ui34,IU34+Ui34,2IU34+2Ui34
GetComponentsB,X
I,I,2I
The column vector <I, I, I> is not a root
RootSpaceI,I,I,CSA
0Ui11
Example 2.
Here is the full root space decomposition for the Lie algebra su4from Example 1.
RSD≔RootSpaceDecompositionCSA
RSD:=table2I,I,I=U14−IUi14,−I,I,0=U12+IUi12,I,2I,I=U24−IUi24,I,0,−I=U13−IUi13,0,I,−I=U23−IUi23,−I,−I,−2I=U34+IUi34,0,−I,I=U23+IUi23,−I,−2I,−I=U24+IUi24,−2I,−I,−I=U14+IUi14,−I,0,I=U13+IUi13,I,I,2I=U34−IUi34,I,−I,0=U12−IUi12
The second calling sequence for RootSpace simply converts the given root vector to a list and extracts the corresponding root space from the root space decomposition table.
RootSpaceI,I,2I,RSD
U34−IUi34
See Also
DifferentialGeometry
CartanSubalgebra
GetComponents
Query
RootSpaceDecomposition
SimpleLieAlgebraData
SimpleLieAlgebraProperties
StandardRepresentation
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