LieAlgebras[CartanMatrix] - find the Cartan matrix for a simple Lie algebra from a root space decomposition, display the Cartan matrix for a given root type
Calling Sequences
CartanMatrix(SimRts, RSD)
CartanMatrix(RT, m)
Parameters
SimRts - a list of column vectors, defining the simple roots of a simple Lie algebra
RSD - a table, defining the root space decomposition of an initialized Lie algebra
RT - a string, the root type of a simple Lie algebra "A", "B", "C", "D", "E", "F", "G"
m - a positive integer, the dimension of the Cartan matrix
Description
Examples
Let g be a simple Lie algebra, h a Cartan subalgebra, and 𝔤 = 𝔥 ⊕α ∈ Δ Rα the root space decomposition of g with respect to h. Let <⋅,⋅> be the Killing form of g. 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 H. The vector Hα can be computed using the command RootToCartanSubalgebraElementH.
Let Δ0 = α1 , α2, .... αm⊆ Δ be a set of simple roots for g. Then the associated Cartan matrix is the m×m matrix with entries Cij= 2<Hαi , Hαj>/ <Hαi, Hαi >. The entries of the Cartan matrix are 0, 1, -1 or 2. The Cartan matrix is independent of the choice of Cartan subalgebra h but is dependent upon the ordering of the simple roots in Δ0 .
The Cartan matrix is the fundamental invariant for semi-simple Lie algebras over C -- two complex semi-simple Lie algebras are isomorphic if and only if their Cartan matrices are the same, modulo a permutation of the vectors in the Cartan subalgebra. The command CartanMatrixToStandardForm will transform a given Cartan matrix to a standard form.
The Cartan matrix encodes the re-construction of the root system of the Lie algebra from its simple roots. See PositiveRoots .
The information contained in the Cartan matrix is also encoded in the Dynkin diagram of the Lie algebra.
The first calling sequence calculates the Cartan matrix of a Lie algebra from a set of simple roots and a root space decomposition.
The second calling sequence displays the standard form of the Cartan matrix for each possible root type of a simple Lie algebra.
withDifferentialGeometry:withLieAlgebras:
Example 1.
We use the command SimpleLieAlgebraData to obtain the Lie algebra data for the Lie algebra su4. This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices
We suppress the output of this command which is a lengthy list of structure equations.
LD≔SimpleLieAlgebraDatasu(4),su:
Initialize this Lie algebra -- the basis elements are given the default labels e1, e2, ... ,e15 .
DGsetupLD
Lie algebra: su
We remark that the command StandardRepresentation can be used to explicitly display the matrices defining su4.
StandardRepresentationsu
−I00000000000000I,00000−I000000000I,0000000000−I0000I,0−100100000000000,000000−1001000000,00000000000−10010,00−10000010000000,0000000−100000100,000−1000000001000,0I00I00000000000,000000I00I000000,00000000000I00I0,00I00000I0000000,0000000I00000I00,000I00000000I000
I0000−I0000000000,00000I0000−I00000,0000000000I0000−I,0−100100000000000,000000−1001000000,00000000000−10010,00−10000010000000,0000000−100000100,000−1000000001000,0I00I00000000000,000000I00I000000,00000000000I00I0,00I00000I0000000,0000000I00000I00,000I00000000I000
The first 3 matrices define a Cartan subalgebra. We can use the Query command to check this
CSA≔e1,e2,e3
CSA:=e1,e2,e3
QueryCSA,CartanSubalgebra
true
We use the command RootSpaceDecomposition to find the root space decomposition for su4 with respect to this Cartan subalgebra.
RSD≔RootSpaceDecompositionCSA
RSD:=tableI,2I,I=e8−Ie14,2I,I,I=e9−Ie15,I,I,2I=e6−Ie12,−I,−2I,−I=e8+Ie14,−2I,−I,−I=e9+Ie15,0,I,−I=e5−Ie11,−I,I,0=e4+Ie10,I,0,−I=e7−Ie13,−I,0,I=e7+Ie13,−I,−I,−2I=e6+Ie12,0,−I,I=e5+Ie11,I,−I,0=e4−Ie10
A choice of simple roots for this root space decomposition is:
Δ0≔I,I,2I,0,I,−I,I,−I,0
Δ0:=II2I,0I−I,I−I0
This set of simple roots can be determined by the command SimpleRoots. The Cartan matrix for this root space decomposition and choice of simple roots is :
CM≔CartanMatrixΔ0,RSD
CM:=2−10−12−10−12
We easily identify this as the standard Cartan matrix for A3 .
CartanMatrixA,3
2−10−12−10−12
Notice that a permutation of the simple roots gives a permuted Cartan matrix.
Δ1≔Δ03,Δ01,Δ02
Δ1:=I−I0,II2I,0I−I
CartanMatrixΔ1,RSD
20−102−1−1−12
Example 2.
For the exceptional Lie algebras E6, E7 and E8 there are two different conventions for the Cartan matrix. For E6 these are:
CartanMatrixE,6,version=I,CartanMatrixE,6,version=II
20−1000020−100−102−1000−1−12−10000−12−10000−12,2−10000−12−10000−12−10−100−12−10000−12000−1002
See Also
DifferentialGeometry
DynkinDiagram
CartanSubalgebra
LieAlgebras
RootSpaceDecomposition
SimpleRoots
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