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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

• 

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

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

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

CM1:=Matrix2,1,0,1,1,0,1,2,0,0,0,0,0,0,2,0,0,1,1,0,0,2,0,1,1,0,0,0,2,0,0,0,1,1,0,2

CM1:=210110120000002001100201100020001102

(2.1)

CM2:=Matrix2,0,0,0,1,0,0,2,1,0,0,0,0,1,2,0,1,1,0,0,0,2,0,1,1,0,1,0,2,0,0,0,1,1,0,2

CM2:=200010021000012011000201101020001102

(2.2)

CM3:=Matrix2,0,0,1,1,0,0,2,0,1,0,1,0,0,2,0,2,0,1,1,0,2,0,0,1,0,1,0,2,0,0,1,0,0,0,2

CM3:=200110020101002020110200101020010002

(2.3)

CM4:=Matrix2,2,0,0,1,0,1,2,0,0,0,0,0,0,2,1,0,0,0,0,1,2,0,1,1,0,0,0,2,1,0,0,0,1,1,2

CM4:=220010120000002100001201100021000112

(2.4)

 

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

C1,P1,T1:=CartanMatrixToStandardFormCM1

C1,P1,T1:=210000121000012100001211000120000102,000100000001100000001000000010010000,D

(2.5)

C2,P2,T2:=CartanMatrixToStandardFormCM2

C2,P2,T2:=210000121000012101001210000120001002,100000000001001000000010010000000100,E

(2.6)

C3,P3,T3:=CartanMatrixToStandardFormCM3

C3,P3,T3:=210000121000012100001210000121000022,000100010000000001001000000010100000,C

(2.7)

C4,P4,T4:=CartanMatrixToStandardFormCM4

C4,P4,T4:=210000121000012100001210000122000012,000010000001100000010000000100001000,B

(2.8)
alg > 

C1,P1,T1:=CartanMatrixToStandardFormCM1

C1,P1,T1:=210000121000012100001211000120000102,000100000001100000001000000010010000,D

(2.9)

 

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

LinearAlgebra:-EqualP11.CM1.P1,C1

true

(2.10)

LinearAlgebra:-EqualP21.CM2.P2,C2

true

(2.11)

LinearAlgebra:-EqualP31.CM3.P3,C3

true

(2.12)

LinearAlgebra:-EqualP41.CM4.P4,C4

true

(2.13)

 

Example 2.

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

 

LD:=_DGLieAlgebra,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.

DGsetupLD:

 

Find a Cartan subalgebra.

alg > 

CSA:=CartanSubalgebra

CSA:=e1,e16+e19,e21

(2.14)

 

Find the root space decomposition.

alg > 

RSD:=RootSpaceDecompositionCSA

RSD:=table0,0,2I=e9+Ie15,2I,2I,0=e4+Ie5e7Ie10+e11+Ie13,I,I,I=e6Ie8Ie12e14,I,I,I=e6+Ie8+Ie12e14,I,I,I=e2Ie3Ie18e20,I,I,I=e2+Ie3+Ie18e20,2I,2I,0=e4Ie5e7Ie10e11+Ie13,2I,2I,0=e4Ie5e7+Ie10+e11Ie13,2I,0,0=e16+Ie17e19,I,I,I=e2Ie3+Ie18+e20,0,0,2I=e9Ie15,I,I,I=e2+Ie3Ie18+e20,I,I,I=e6Ie8+Ie12+e14,2I,0,0=e16Ie17e19,0,2I,0=e4+e7+Ie10+Ie13,I,I,I=e6+Ie8Ie12+e14,0,2I,0=e4+e7Ie10Ie13,2I,2I,0=e4+Ie5e7+Ie10e11Ie13

(2.15)

 

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

alg > 

RT:=LieAlgebraRootsRSD

RT:=002I,2I2I0,III,III,III,III,2I2I0,2I2I0,2I00,III,002I,III,III,2I00,02I0,III,02I0,2I2I0

(2.16)
alg > 

PR:=PositiveRootsRT,7I,3I,I

PR:=002I,2I2I0,III,III,2I00,III,02I0,III,2I2I0

(2.17)
alg > 

SR:=SimpleRootsPR

 

Find the Cartan matrix.

alg > 

CM:=CartanMatrixSR,RSD

CM:=202021112

(2.18)

 

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:=CartanMatrixToStandardFormCM,SR

C1,S1,T1:=210121022,02I0,III,002I,C

(2.19)

 

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

alg > 

CartanMatrixS1,RSD

210121022

(2.20)

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

 

See Also

DifferentialGeometry, CartanMatrix, CartanSubalgebra, PositiveRoots, RootSpaceDecomposition, SimpleRoots 


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