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LieAlgebras[CartanMatrixToStandardForm] - transform a Cartan matrix to standard form

Calling Sequences

     CartanMatrixToStandardForm(C,SR)

Parameters

     C   - a square matrix

   SR  - (optional) a list of vectors, the simple roots used to determine the Cartan matrix for a simple Lie algebra

 

Description

Examples

Description

• 

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= 2αi, αj  αij, αj  =  2 Hαi, Hαj  Hαi, Hαi  .

(See CartanMatrix for the definition of the vectors Hαi )

• 

A permutation of the roots leads to a different but equivalent Cartan matrix.

• 

The command CartanMatrixToStandardForm transforms a Cartan matrix to the standard form for each root type.

• 

The command returns the Cartan matrix in standard form, a permutation matrix, and a string denoting the root type. The permutation matrix will transform the given Cartan matrix to its standard form by a similarity transformation.

• 

If the second calling is invoked, then the second element of the output is the permuted set of simple roots which will generate the standard form of the Cartan matrix.

 

Examples

> 

with⁡DifferentialGeometry:with⁡LieAlgebras:

 

Example 1.

We define 4 different Cartan matrices and calculate their standard forms and root type.

> 

CM1≔Matrix⁡2,−1,0,−1,−1,0,−1,2,0,0,0,0,0,0,2,0,0,−1,−1,0,0,2,0,−1,−1,0,0,0,2,0,0,0,−1,−1,0,2

> 

CM2≔Matrix⁡2,0,0,0,−1,0,0,2,−1,0,0,0,0,−1,2,0,−1,−1,0,0,0,2,0,−1,−1,0,−1,0,2,0,0,0,−1,−1,0,2

> 

CM3≔Matrix⁡2,0,0,−1,−1,0,0,2,0,−1,0,−1,0,0,2,0,−2,0,−1,−1,0,2,0,0,−1,0,−1,0,2,0,0,−1,0,0,0,2

> 

CM4≔Matrix⁡2,−2,0,0,−1,0,−1,2,0,0,0,0,0,0,2,−1,0,0,0,0,−1,2,0,−1,−1,0,0,0,2,−1,0,0,0,−1,−1,2

 

Here are the standard forms, permutation matrices and root types.

> 

C1,P1,T1≔CartanMatrixToStandardForm⁡CM1

> 

C2,P2,T2≔CartanMatrixToStandardForm⁡CM2

> 

C3,P3,T3≔CartanMatrixToStandardForm⁡CM3

> 

C4,P4,T4≔CartanMatrixToStandardForm⁡CM4

alg > 

C1,P1,T1≔CartanMatrixToStandardForm⁡CM1

 

For each example the second output is a permutation matrix which transforms the given input Cartan matrix to its standard form.

> 

LinearAlgebra:-Equal⁡P1−1·CM1·P1,C1

true

(2.1)
> 

LinearAlgebra:-Equal⁡P2−1·CM2·P2,C2

true

(2.2)
> 

LinearAlgebra:-Equal⁡P3−1·CM3·P3,C3

true

(2.3)
> 

LinearAlgebra:-Equal⁡P4−1·CM4·P4,C4

true

(2.4)

 

Example 2.

We define a 21-dimensional simple Lie algebra and calculate its root type.

 

> 

LD≔_DG⁡LieAlgebra,alg,21,1,2,3,1,1,3,2,−1,1,4,5,1,1,5,4,−2,1,5,7,2,1,6,8,1,1,7,5,−1,1,8,6,−1,1,10,11,1,1,11,10,−2,1,11,13,2,1,12,14,1,1,13,11,−1,1,14,12,−1,1,16,17,1,1,17,16,−2,1,17,19,2,1,18,20,1,1,19,17,−1,1,20,18,−1,2,3,1,1,2,4,6,1,2,5,8,1,2,6,4,−2,2,6,9,2,2,8,5,−1,2,9,6,−1,2,10,12,1,2,11,14,1,2,12,10,−2,2,12,15,2,2,14,11,−1,2,15,12,−1,2,16,18,1,2,17,20,1,2,18,16,−2,2,18,21,2,2,20,17,−1,2,21,18,−1,3,5,6,1,3,6,5,−1,3,7,8,1,3,8,7,−2,3,8,9,2,3,9,8,−1,3,11,12,1,3,12,11,−1,3,13,14,1,3,14,13,−2,3,14,15,2,3,15,14,−1,3,17,18,1,3,18,17,−1,3,19,20,1,3,20,19,−2,3,20,21,2,3,21,20,−1,4,5,1,1,4,6,2,1,4,10,16,2,4,11,17,1,4,12,18,1,4,16,10,−2,4,17,11,−1,4,18,12,−1,5,6,3,1,5,7,1,1,5,8,2,1,5,10,17,1,5,11,16,2,5,11,19,2,5,12,20,1,5,13,17,1,5,14,18,1,5,16,11,−1,5,17,10,−2,5,17,13,−2,5,18,14,−1,5,19,11,−1,5,20,12,−1,6,8,1,1,6,9,2,1,6,10,18,1,6,11,20,1,6,12,16,2,6,12,21,2,6,14,17,1,6,15,18,1,6,16,12,−1,6,17,14,−1,6,18,10,−2,6,18,15,−2,6,20,11,−1,6,21,12,−1,7,8,3,1,7,11,17,1,7,13,19,2,7,14,20,1,7,17,11,−1,7,19,13,−2,7,20,14,−1,8,9,3,1,8,11,18,1,8,12,17,1,8,13,20,1,8,14,19,2,8,14,21,2,8,15,20,1,8,17,12,−1,8,18,11,−1,8,19,14,−1,8,20,13,−2,8,20,15,−2,8,21,14,−1,9,12,18,1,9,14,20,1,9,15,21,2,9,18,12,−1,9,20,14,−1,9,21,15,−2,10,11,1,1,10,12,2,1,10,16,4,2,10,17,5,1,10,18,6,1,11,12,3,1,11,13,1,1,11,14,2,1,11,16,5,1,11,17,4,2,11,17,7,2,11,18,8,1,11,19,5,1,11,20,6,1,12,14,1,1,12,15,2,1,12,16,6,1,12,17,8,1,12,18,4,2,12,18,9,2,12,20,5,1,12,21,6,1,13,14,3,1,13,17,5,1,13,19,7,2,13,20,8,1,14,15,3,1,14,17,6,1,14,18,5,1,14,19,8,1,14,20,7,2,14,20,9,2,14,21,8,1,15,18,6,1,15,20,8,1,15,21,9,2,16,17,1,1,16,18,2,1,17,18,3,1,17,19,1,1,17,20,2,1,18,20,1,1,18,21,2,1,19,20,3,1,20,21,3,1:

 

 

Initialize this Lie algebra.

> 

DGsetup⁡LD:

 

Find a Cartan subalgebra.

alg > 

CSA≔CartanSubalgebra⁡

CSA:=e1,e16+e19,e21

(2.5)

 

Find the root space decomposition.

alg > 

RSD≔RootSpaceDecomposition⁡CSA

RSD:=table0,0,−2⁢I=e9+I⁢e15,−2⁢I,2⁢I,0=e4+I⁢e5−e7−I⁢e10+e11+I⁢e13,I,I,I=e6−I⁢e8−I⁢e12−e14,−I,−I,−I=e6+I⁢e8+I⁢e12−e14,I,−I,I=e2−I⁢e3−I⁢e18−e20,−I,I,−I=e2+I⁢e3+I⁢e18−e20,2⁢I,2⁢I,0=e4−I⁢e5−e7−I⁢e10−e11+I⁢e13,2⁢I,−2⁢I,0=e4−I⁢e5−e7+I⁢e10+e11−I⁢e13,−2⁢I,0,0=e16+I⁢e17−e19,I,I,−I=e2−I⁢e3+I⁢e18+e20,0,0,2⁢I=e9−I⁢e15,−I,−I,I=e2+I⁢e3−I⁢e18+e20,I,−I,−I=e6−I⁢e8+I⁢e12+e14,2⁢I,0,0=e16−I⁢e17−e19,0,−2⁢I,0=e4+e7+I⁢e10+I⁢e13,−I,I,I=e6+I⁢e8−I⁢e12+e14,0,2⁢I,0=e4+e7−I⁢e10−I⁢e13,−2⁢I,−2⁢I,0=e4+I⁢e5−e7+I⁢e10−e11−I⁢e13

(2.6)

 

Find the roots, positive roots and a choice of simple roots.

alg > 

RT≔LieAlgebraRoots⁡RSD

alg > 

PR≔PositiveRoots⁡RT,7⁢I,3⁢I,I

alg > 

SR≔SimpleRoots⁡PR

 

Find the Cartan matrix.

alg > 

CM≔CartanMatrix⁡SR,RSD

 

Transform the Cartan matrix to standard form. Here we use the second calling sequence. The command CartanMatrixToStandardForm now returns a permuted set of simple roots for which the Cartan matrix will be in standard form.

alg > 

C1,S1,T1≔CartanMatrixToStandardForm⁡CM,SR

 

Check the result by re-calculating the Cartan matrix with respect to the permuted set of roots. We get the standard form immediately.

alg > 

CartanMatrix⁡S1,RSD

The root type of our 21-dimensional Lie algebra is C3 .

 

See Also

DifferentialGeometry

CartanMatrix

CartanSubalgebra

PositiveRoots

RootSpaceDecomposition

SimpleRoots