LieAlgebras[CoRoot] - find the coroot of a root vector for a semi-simple Lie algebra
Calling Sequences
CoRoot(α, CSA, option)
Parameters
α - a vector, defining a root vector for a semi-simple Lie algebra
CSA - a list of r vectors in a Lie algebra, defining a Cartan subalgebra
option - an r ×r non-singular matrix, defining the restriction of the Killing form to the Cartan subalgebra
Description
Examples
Let 𝔤 be a semi-simple Lie algebra, 𝔥 a Cartan subalgebra, and Δ the associated set of roots. Let B be the Killing form. If α ∈ Δ, then the coroot of α is the unique vector Tα∈ 𝔥 such that αx = BTα,x. Let h1, h2 , ... , hr be a basis for 𝔥, αhi = ai,and bij = Bhi, hj with inverse bij. Then Tα= tihi, where ti = bijaj.
The calling sequence CoRoot(α, CSA) returns the vector Tα.
In a situation involving repeated calls to CoRoot, efficiency can be dramatically improved by using the optional 3rd argument to specify the restriction of the Killing form.
withDifferentialGeometry:withLieAlgebras:
Example 1.
We use the command SimpleLieAlgebraData to retrieve the structure equations for the rank 3 Lie algebra sl4, we initialize this algebra, and we calculate the coroots of several root vectors.
LD≔SimpleLieAlgebraDatasl(4),sl4
LD:=e1,e4=e4,e1,e5=e5,e1,e6=2e6,e1,e7=−e7,e1,e9=e9,e1,e10=−e10,e1,e12=e12,e1,e13=−2e13,e1,e14=−e14,e1,e15=−e15,e2,e4=−e4,e2,e6=e6,e2,e7=e7,e2,e8=e8,e2,e9=2e9,e2,e11=−e11,e2,e12=e12,e2,e13=−e13,e2,e14=−2e14,e2,e15=−e15,e3,e5=−e5,e3,e6=e6,e3,e8=−e8,e3,e9=e9,e3,e10=e10,e3,e11=e11,e3,e12=2e12,e3,e13=−e13,e3,e14=−e14,e3,e15=−2e15,e4,e7=e1−e2,e4,e8=e5,e4,e9=e6,e4,e10=−e11,e4,e13=−e14,e5,e7=−e8,e5,e10=e1−e3,e5,e11=e4,e5,e12=e6,e5,e13=−e15,e6,e7=−e9,e6,e10=−e12,e6,e13=e1,e6,e14=e4,e6,e15=e5,e7,e11=−e10,e7,e14=−e13,e8,e10=e7,e8,e11=e2−e3,e8,e12=e9,e8,e14=−e15,e9,e11=−e12,e9,e13=e7,e9,e14=e2,e9,e15=e8,e10,e15=−e13,e11,e15=−e14,e12,e13=e10,e12,e14=e11,e12,e15=e3
DGsetupLD
Lie algebra: sl4
We obtain the Cartan subalgebra and the positive roots using SimpleLieAlgebraProperties
P≔SimpleLieAlgebraPropertiessl4:
CSA≔PCartanSubalgebra
CSA:=e1,e2,e3
Δ≔PPositiveRoots
Calculate the coroot for the first root Δ1.
α≔Δ1
CoRootα,CSA
18e1−18e2
Calculate the coroot for the last root Δ6.
β≔Δ−1
CoRootβ,CSA
18e1
Example 2.
We repeat the calculation the first coroot from Example 1 using the optional calling sequence. The restriction of the Killing form to the Cartan subalgebra is needed.
B≔Killinge1,e2,e3
CoRootα,CSA,B
Here is the same computation in components.
T≔B−1·α
See Also
DifferentialGeometry
LieAlgebras
Killing
SimpleLieAlgebraData
SimpleLieAlgebraProperties
RootSpaceDecomposition
PositiveRoots
Download Help Document
