associate to each positive root of a simple Lie algebra a vector in the Cartan subalgebra - Maple Programming Help

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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

Description

• 

 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 , Xα Rα and Hα 𝔥  such that

 [Hα , Xα] = 2 Xα,  [Hα , Xα]  = 2 Xα  and Xα , Xα = Hα .

These conditions uniquely determine Hα.  Note that the vectors Xα , Xα , 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.

Examples

with(DifferentialGeometry): with(LieAlgebras):

 

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 := SimpleLieAlgebraData("su(3,3)", su33, labelformat = "gl", labels = ['E', 'omega']):

DGsetup(LD1);

Lie algebra: su33

(2.1)

 

We use the command SimpleLieAlgebraProperties to obtain the Cartan subalgebra, the root space decomposition, and the simple roots.

su33 > 

P := SimpleLieAlgebraProperties(su33):

 

The result P is a table. Here is the Cartan subalgebra for su3, 3.

su33 > 

CSA := P["CartanSubalgebra"];

CSA:=E11,E22,E33,Ei11,Ei22

(2.2)

 

Here is the root space decomposition for su3,3.

su33 > 

RSD := eval(P["RootSpaceDecomposition"]);

RSD:=table1,0,1,2I,I=E16+IEi16,1,1,0,I,I=E15IEi15,1,0,1,2I,I=E31+IEi31,1,1,0,I,I=E12IEi12,0,0,2,0,0=Ei63,0,1,1,I,2I=E53IEi53,0,2,0,0,0=Ei52,1,1,0,I,I=E42IEi42,1,0,1,2I,I=E13+IEi13,0,1,1,I,2I=E23IEi23,0,1,1,I,2I=E53+IEi53,2,0,0,0,0=Ei14,0,1,1,I,2I=E32IEi32,1,0,1,2I,I=E31IEi31,1,1,0,I,I=E42+IEi42,0,1,1,I,2I=E23+IEi23,0,1,1,I,2I=E26+IEi26,1,0,1,2I,I=E13IEi13,1,1,0,I,I=E12+IEi12,1,0,1,2I,I=E43IEi43,1,0,1,2I,I=E43+IEi43,0,2,0,0,0=Ei25,1,0,1,2I,I=E16IEi16,2,0,0,0,0=Ei41,1,1,0,I,I=E21IEi21,0,1,1,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=E26IEi26

(2.3)

 

Here are the positive roots.

su33 > 

PR := P["PositiveRoots"];

PR:=110II,011I2I,00200,011I2I,110II,1012II,011I2I,011I2I,1012II,1012II,02000,1012II,110II,110II,20000

(2.4)

 

Let us find Hα,where α is the first root (2.4) 

su33 > 

alpha := PR[1];

α:=110II

(2.5)
su33 > 

H := RootToCartanSubalgebraElementH(alpha,RSD);

H:=I2Ei11+I2Ei22+12E1112E22

(2.6)

 

We check that H is in the Cartan subalgebra.

su33 > 

GetComponents(H, CSA);

12,12,0,12I,12I

(2.7)

 

Here are the root spaces for α and α .

su33 > 

X := RootSpace(alpha, RSD);

X:=E12+IEi12

(2.8)
su33 > 

Y := RootSpace(-alpha, RSD);

Y:=E21+IEi21

(2.9)

 

We check that H , X, Y defines a Lie subalgebra.

su33 > 

LieAlgebraData([H, X, Y]);

e1,e2=2e2,e1,e3=2e3,e2,e3=4e1

(2.10)

 

If we scale the vectors X and Y then the structure equations take the standard form for sl2. 

su33 > 

LieAlgebraData([H, 1/2*X, 1/2*Y]);

e1,e2=2e2,e1,e3=2e3,e2,e3=e1

(2.11)

 

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.

su33 > 

SR := P["SimpleRoots"];

SR:=110II,011I2I,00200,011I2I,110II

(2.12)
su33 > 

Halpha := map(RootToCartanSubalgebraElementH, SR, RSD);

Halpha:=I2Ei11+I2Ei22+12E1112E22,I2Ei22+12E2212E33,E33,I2Ei22+12E2212E33,I2Ei11I2Ei22+12E1112E22

(2.13)

 

Then we calculate the Killing form , restricted to subspace [H1, H2, H3, H4, H5].

su33 > 

B := Killing(Halpha);

B:=24120001224120001224120001224120001224

(2.14)

 

The Cartan matrix is given by normalizing the entries of B.

su33 > 

C := Matrix(5, 5, (i, j) -> 2*B[i ,j]/B[i, i]);

C:=2100012100012100012100012

(2.15)

 

The Lie algebra su3,3 is a rank 5 simple Lie algebra of type "A". The matrix in (2.15) is therefore correct.

su33 > 

CartanMatrix("A", 5);

2100012100012100012100012

(2.16)

 

See Also

DifferentialGeometry

CartanMatrix

Killing

LieAlgebraData

RootSpace

SimpleLieAlgebraData

SimpleLieAlgebraProperties