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Details for Cartan matrices and Dynkin diagrams

 

Description

 

• 

This document contains a list of all Cartan matrices and Dynkin diagrams for all classical root types of rank 6 and for all exceptional root types.

• 

 Let Δ0 = α1, α2, ... , αm be a set of simple roots and let (⋅,⋅) be the inner product on the roots induced by the Killing form. The Cartan matrix is given by Cij = 2 αi ,αj αj , αj.

• 

 From the Cartan matrix one can calculate the the number of lines connecting αi to αj as Eij = Cij Cji  (*) . The relative lengths of the root vectors can be found as the ratios αi ,αi αj , αj = Cji Cij  (**). Set  Li = αi ,αi αi+1 , αi+1.

• 

 The edge matrix Eij and the root length vector Li  clearly determine the Dynkin diagram.  Conversely, the equations (*) and (**), together with the facts that Cii = 2 and 3 Cij  0 for ij uniquely determine the Cartan matrix from the edge matrix Eij and the root length vector Li. For additional details see, for example, W. A. de Graaf, Lie Algebras: Theory and Algorithms, pages 167-168.

Code

 

with(DifferentialGeometry): with(LieAlgebras):

 

We give 3 simple programs. The first calculates the edge matrix Eij and the second the relative lengths Li of the roots. The third program re-constructs the Cartan matrix from the edge matrix and the relative length vector.

 

EdgeMatrix := proc(C) local n;

description `a procedure to find the adjacency matrix for the Dynkin diagram from the Cartan matrix`;

n := LinearAlgebra:-ColumnDimension(C);

Matrix(n, n, (i, j) -> C[i, j]*C[j, i]);

end:

RootLengths := proc(C) local n, Eq, soln;

description `a procedure to find the ratio of the root lengths for the Dynkin diagram from the Cartan matrix`;

n := LinearAlgebra:-ColumnDimension(C);

Eq := {seq(seq(C[j, i]*x||i/x||j = C[i, j], i = 1 .. n) ,j = 1 .. n)}:

soln := solve(Eq, {seq(x||i , i = 1 .. n)});

eval(Vector([seq(x||i/x||(i+1), i = 1 .. n-1)]), soln)

end:

DynkinDiagramDataToCartanMatrix := proc(Edges, L) local n, C, vars, Eq1, Eq2, Eq3, soln;

description `a procedure to find the Cartan matrix from the Dynkin diagram (edge matrix and root length rations)`;

n := LinearAlgebra:-ColumnDimension(Edges);

C := Matrix(n, n, proc(i, j) if i=j then 2 else c||i||j fi end); vars := indets(C);

Eq1:= {seq(seq( C[i,j]*C[j,i] = Edges[i,j], j = i+1..n), i = 1..n)};

Eq2 := {seq(seq(C[j,i]*mul(L[k], k = i.. j-1) = C[i,j], j = i+1..n) , i = 1..n-1)};

Eq3 := {seq(v<=0 ,v =vars), seq(v >= -3, v=vars)};

soln := solve(Eq1 union Eq2 union Eq3, vars);

eval(C, [soln][1]);

end:

Root Type A

Root Type

Cartan Matrix

CijCji 

Ci&comma;i&plus;1Ci&plus;1&comma;i

 Dynkin Diagram

   A1

2

--

--

    A2

2112

 

4114

1

   A3

 

210121012

410141014

11

  

 

A4

 

2100121001210012

4100141001410014

 111

 A5

2100012100012100012100012

4100014100014100014100014

   1111

  A6

210000121000012100001210000121000012

410000141000014100001410000141000014

11111

 

Here is the Cartan matrix for A4  and the corresponding edge matric and root length vector

C := CartanMatrix("A", 4);

C:=2100121001210012

(3.1)

E := EdgeMatrix(C); L := RootLengths(C);

E:=4100141001410014

L:=111

(3.2)

 

Re -construct the A4 Cartan matrix.

DynkinDiagramDataToCartanMatrix(E, L);

2100121001210012

(3.3)

Root Type B

Root

Type

Cartan Matrix C

CijCji   

Ci&comma;i&plus;1Ci&plus;1&comma;i

 Dynkin Diagram

B2

2212

0200

2

B3

210122012

010002000&comma;

12

B4

2100121001220012

0100001000020000

112

B5

2100012100012100012200012

0100000100000100000200000

1112

B6

210000121000012100001210000122000012

010000001000000100000010000002000000&comma;

11112

 

Here is the Cartan matrix for B4  and the corresponding edge matrix and root length vector.

C := CartanMatrix("B", 4);

C:=2100121001220012

(4.1)

E := EdgeMatrix(C); L := RootLengths(C);

E:=4100141001420024

L:=112

(4.2)

 

Re -construct the B4 Cartan matrix.

DynkinDiagramDataToCartanMatrix(E, L);

2100121001220012

(4.3)

Root Type C

Root Type

Cartan Matrix

CijCji   

Ci&comma;i&plus;1Ci&plus;1&comma;i

 Dynkin Diagram

C3

210121022

410142024

112

C4

2100121001210022

4100141001420024

1112

C5

2100012100012100012100022

4100014100014100014200024&comma;

11112

C6

210000121000012100001210000121000022

410000141000014100001410000142000024

111112

Here is the  Cartan matrix for C4  and the corresponding edge matrix and root length vector.

C := CartanMatrix("C", 4);

C:=2100121001210022

(5.1)

E := EdgeMatrix(C); L := RootLengths(C);

E:=4100141001420024

L:=1112

(5.2)

 

Re -construct the C4 Cartan matrix. 

DynkinDiagramDataToCartanMatrix(E, L);

2100121001210022

(5.3)

Root Type D

 

 

 

Root

Type

Cartan Matrix

CijCji   

Ci&comma;i&plus;1Ci&plus;1&comma;i

 Dynkin Diagram

D3

211120102

411140104

11

D4

2100121101200102

4100141101400104&comma;

 111

D5

2100012100012110012000102

4100014100014110014000104

 1111

D6

210000121000012100001211000120000102

4100014100014110014000104

11111

 

 

Here is the Cartan matrix for D4  and the corresponding edge matrix and root length vector.

C := CartanMatrix("D", 4);

C:=2100121101200102

(6.1)

E := EdgeMatrix(C); L := RootLengths(C);

E:=4100141101400104

L:=111

(6.2)

 

Re -construct the D4 Cartan matrix.

DynkinDiagramDataToCartanMatrix(E, L);

2100121101200102

(6.3)

Exceptional Root Types

 

 

Root Type

Cartan Matrix

CijCji   

Ci&comma;i&plus;1Ci&plus;1&comma;i

 Dynkin Diagram

E6

201000020100102100011210000121000012

401000040100104100011410000141000014

--

E7

2010000020100010210000112100000121000001210000012

4010000040100010410000114100000141000001410000014

--

 

 

E8

2010000002010000102100000112100000012100000012100000012100000012

4010000004010000104100000114100000014100000014100000014100000014

 

F4

 

2100122001210012

4100142002410014

121

G2

 

2132

4334

13

 

Here is the Cartan matrix for E6  and the corresponding edge matrix and root length vector.

C := CartanMatrix("E", 6);

C:=201000020100102100011210000121000012

(7.1)

E := EdgeMatrix(C); L := RootLengths(C);

E:=401000040100104100011410000141000014

L:=11111

(7.2)

 

Re -construct the E6 Cartan matrix.

DynkinDiagramDataToCartanMatrix(E, L);

201000020100102100011210000121000012

(7.3)

 

See Also

DifferentialGeometry, CartanMatrix, CartanSubalgebra, DynkinDiagram, RootSpaceDecomposition, SimpleRoots


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