LieAlgebras[RootString] - find the sequence of roots through a given root of a semi-simple Lie algebra
Calling Sequences
RootString(α, β, Δ, option)
Parameters
α - a vector, defining a root vector for a semi-simple Lie algebra
β - a vector, defining a root vector for a semi-simple Lie algebra
Δ - a list of vectors, defining a list of root vectors for a semi-simple Lie algebra and containing α and β
option - the keyword argument output = "stringlengths"
Description
Examples
Let 𝔤 be a semi-simple Lie algebra, 𝔥 a Cartan subalgebra, and Δ the associated set of roots. If α, β ∈ Δ, then the α-string through β is the maximal sequence of roots of the form
β − p α, ... , β − 2 α, β − α, β, β+α, β+2 α, ..., β + q α where p,q are non-negative integers.
The calling sequence RootString(α, β, Δ) returns the α-string of roots through β. The calling sequence RootString(α, β, Δ, output = "stringlengths") returns the list of non-negative integers p, q.
withDifferentialGeometry:withLieAlgebras:
Example 1.
We initialize the split real form of the exceptional Lie algebra g2 and retrieve the root space decomposition and the list of all roots. We then calculate some root strings. The structure equations for g2 are obtained using SimpleLieAlgebraData.
LD≔SimpleLieAlgebraDatag(2,Split),g2
LD:=e1,e3=2e3,e1,e4=−3e4,e1,e5=−e5,e1,e6=e6,e1,e7=3e7,e1,e9=−2e9,e1,e10=3e10,e1,e11=e11,e1,e12=−e12,e1,e13=−3e13,e2,e3=−e3,e2,e4=2e4,e2,e5=e5,e2,e7=−e7,e2,e8=e8,e2,e9=e9,e2,e10=−2e10,e2,e11=−e11,e2,e13=e13,e2,e14=−e14,e3,e4=e5,e3,e5=2e6,e3,e6=−3e7,e3,e9=−e1,e3,e11=−3e10,e3,e12=−2e11,e3,e13=e12,e4,e7=−e8,e4,e10=−e2,e4,e11=e9,e4,e14=e13,e5,e6=−3e8,e5,e9=3e4,e5,e10=−e3,e5,e11=−e1−3e2,e5,e12=2e9,e5,e14=e12,e6,e9=2e5,e6,e11=−2e3,e6,e12=−2e1−3e2,e6,e13=−e9,e6,e14=−e11,e7,e9=−e6,e7,e12=e3,e7,e13=−e1−e2,e7,e14=−e10,e8,e10=−e7,e8,e11=−e6,e8,e12=e5,e8,e13=e4,e8,e14=−e1−2e2,e9,e10=e11,e9,e11=2e12,e9,e12=−3e13,e10,e13=−e14,e11,e12=−3e14
Initialize the Lie algebra with DGsetup.
DGsetupLD
Lie algebra: g2
The root space decomposition is retrieved, without calculation, using SimpleLieAlgebraProperties
P≔SimpleLieAlgebraPropertiesg2:
Here is the root space decomposition and the list of all positive roots.
RSD≔evalPRootSpaceDecomposition
RSD:=table−1,1=e5,−3,2=e4,0,1=e8,3,−1=e7,2,−1=e3,1,0=e6,0,−1=e14,−2,1=e9,−3,1=e13,3,−2=e10,−1,0=e12,1,−1=e11
Δ≔LieAlgebraRootsRSD
Define two roots α and β.
α≔2,−1
β≔−3,2
Calculate the α-string through β.
RootStringα,β,Δ
With the optional keyword argument output = "stringlengths", we obtain the lengths of the α-string through β in the negative and positive directions.
RootStringα,β,Δ,output=stringlengths
0,3
Thus the α-string through β in is given explicitly by.
β,β+α,β+2α,β+3α
Example 2.
Here is another example of a root string for the exceptional Lie algebra g2.
β≔1,0
The root α-string through β is now
and the string lengths are
2,1
Thus, the root string is explicitly given by
β−2α,β−α,β,β+α
Example 3.
Root strings can also be calculated for abstract roots systems, that is, a set of vectors satisfying the standard axioms of a root system and not explicitly defined from the root space decomposition of a semi-simple Lie algebra. The positive roots of an abstract root systems can be calculated with the PositiveRoots command.
AbstractRoots≔PositiveRootsB,3
Δ≔seq−v,v=AbstractRoots,seqv,v=AbstractRoots
Here are are 2 roots and their root string.
α≔Δ12
β≔Δ11
See Also
DifferentialGeometry
LieAlgebras
CartanSubalgebra
PositiveRoots
SimpleLieAlgebraData
SimpleLieAlgebraProperties
SimpleRoots
RootSpaceDecomposition
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