LieAlgebras[RootToCartanSubalgebraElementH] - associate to each positive root of a simple Lie algebra a vector in the Cartan subalgebra
Calling Sequences
RootToCartanSubalgebraElementH(α , RSD)
Parameters
α - a vector, defining a positive (or negative) root of a simple Lie algebra
RSD - a table, defining the root space decomposition of a simple Lie algebra
Description
Examples
Let g be a simple Lie algebra, h a Cartan subalgebra, and 𝔤 = 𝔥 ⊕⨁α ∈ ΔRα the root space decomposition of g with respect to h. For each root α ∈Δ, there are vectors Xα ∈Rα , X−α ∈R−α and Hα∈ 𝔥 such that
[Hα , Xα] = 2 Xα, [Hα , X−α] = −2 X−α and Xα , X−α = Hα .
These conditions uniquely determine Hα. Note that the vectors Xα , X−α , Hα define the 3-dimensional Lie algebra sl2. The assignment α → Hα is used to calculate the Cartan matrix for the Lie algebra 𝔤.
The procedure RootToCartanSubalgebraElementH(α , RSD) returns the vector Hα.
withDifferentialGeometry:withLieAlgebras:
Example 1.
We consider the Lie algebra su3,3. This is the 24-dimensional real Lie algebra of 6×6 complex matrices A which are trace-free and skew-Hermitian with respect to the quadratic form Q=0I3I30 . We use the command SimpleLieAlgebraData to initialize this Lie algebra.
LD1≔SimpleLieAlgebraDatasu(3,3),su33,labelformat=gl,labels=E,ω:
DGsetupLD1
Lie algebra: su33
We use the command SimpleLieAlgebraProperties to obtain the Cartan subalgebra, the root space decomposition, and the simple roots.
P≔SimpleLieAlgebraPropertiessu33:
The result P is a table. Here is the Cartan subalgebra for su3, 3.
CSA≔PCartanSubalgebra
CSA:=E11,E22,E33,Ei11,Ei22
Here is the root space decomposition for su3,3.
RSD≔evalPRootSpaceDecomposition
RSD:=table1,0,1,−2I,−I=E16+IEi16,1,1,0,I,−I=E15−IEi15,−1,0,1,−2I,−I=E31+IEi31,1,−1,0,−I,I=E12−IEi12,0,0,−2,0,0=Ei63,0,−1,−1,I,2I=E53−IEi53,0,−2,0,0,0=Ei52,−1,−1,0,I,−I=E42−IEi42,1,0,−1,2I,I=E13+IEi13,0,1,−1,−I,−2I=E23−IEi23,0,−1,−1,−I,−2I=E53+IEi53,2,0,0,0,0=Ei14,0,−1,1,I,2I=E32−IEi32,−1,0,1,2I,I=E31−IEi31,−1,−1,0,−I,I=E42+IEi42,0,1,−1,I,2I=E23+IEi23,0,1,1,−I,−2I=E26+IEi26,1,0,−1,−2I,−I=E13−IEi13,1,−1,0,I,−I=E12+IEi12,−1,0,−1,2I,I=E43−IEi43,−1,0,−1,−2I,−I=E43+IEi43,0,2,0,0,0=Ei25,1,0,1,2I,I=E16−IEi16,−2,0,0,0,0=Ei41,−1,1,0,I,−I=E21−IEi21,0,−1,1,−I,−2I=E32+IEi32,1,1,0,−I,I=E15+IEi15,0,0,2,0,0=Ei36,−1,1,0,−I,I=E21+IEi21,0,1,1,I,2I=E26−IEi26
Here are the positive roots.
PR≔PPositiveRoots
Let us find Hα,where α is the first root
α≔PR1
H≔RootToCartanSubalgebraElementHα,RSD
H:=−I2Ei11+I2Ei22+12E11−12E22
We check that H is in the Cartan subalgebra.
GetComponentsH,CSA
12,−12,0,−12I,12I
Here are the root spaces for α and −α .
X≔RootSpaceα,RSD
X:=E12+IEi12
Y≔RootSpace−α,RSD
Y:=E21+IEi21
We check that H , X, Y defines a Lie subalgebra.
LieAlgebraDataH,X,Y
e1,e2=2e2,e1,e3=−2e3,e2,e3=4e1
If we scale the vectors X and Y then the structure equations take the standard form for sl2.
LieAlgebraDataH,12X,12Y
e1,e2=2e2,e1,e3=−2e3,e2,e3=e1
Example 2.
We illustrate how to use RootToCartanSubalgebraElementH(α , RSD) to calculate the Cartan matrix for su3, 3. We first calculate the Hα for the simple roots α.
SR≔PSimpleRoots
Halpha≔mapRootToCartanSubalgebraElementH,SR,RSD
Halpha:=−I2Ei11+I2Ei22+12E11−12E22,−I2Ei22+12E22−12E33,E33,I2Ei22+12E22−12E33,I2Ei11−I2Ei22+12E11−12E22
Then we calculate the Killing form , restricted to subspace [H1, H2, H3, H4, H5].
B≔KillingHalpha
The Cartan matrix is given by normalizing the entries of B.
C≔Matrix5,5,i,j↦2⋅Bi,jBi,i
The Lie algebra su3,3 is a rank 5 simple Lie algebra of type "A". The matrix in is therefore correct.
CartanMatrixA,5
See Also
DifferentialGeometry
CartanMatrix
Killing
LieAlgebraData
RootSpace
SimpleLieAlgebraData
SimpleLieAlgebraProperties
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