Cartan Matrices and Dynkin Diagram Details - Maple Help

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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 rankand for all exceptional root types.
 • Let 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
 • From the Cartan matrix one can calculate the number of lines connecting to ${\mathrm{α}}_{j}$ as . The relative lengths of the root vectors can be found as the ratios (**). Set
 • The edge matrix and the root length vector clearly determine the Dynkin diagram.  Conversely, the equations (*) and (**), together with the facts that ${C}_{\mathrm{ii}}$ = 2 and  for uniquely determine the Cartan matrix from the edge matrix and the root length vector 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 and the second the relative lengths ${L}_{i}$ 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 $\frac{{C}_{i,i+1}}{{C}_{i+1,i}}$ Dynkin Diagram  $\left[\begin{array}{r}2\end{array}\right]$ -- -- $\left[\begin{array}{rr}2& -1\\ -1& 2\end{array}\right]$ $\left[\begin{array}{rr}4& 1\\ 1& 4\end{array}\right]$ $\left[\begin{array}{r}1\end{array}\right]$ $\left[\begin{array}{rrr}2& -1& 0\\ -1& 2& -1\\ 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrr}4& 1& 0\\ 1& 4& 1\\ 0& 1& 4\end{array}\right]$ $\left[\begin{array}{r}1\\ 1\end{array}\right]$ ${A}_{4}$ $\left[\begin{array}{rrrr}2& -1& 0& 0\\ -1& 2& -1& 0\\ 0& -1& 2& -1\\ 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrr}4& 1& 0& 0\\ 1& 4& 1& 0\\ 0& 1& 4& 1\\ 0& 0& 1& 4\end{array}\right]$ $\left[\begin{array}{rrrrr}2& -1& 0& 0& 0\\ -1& 2& -1& 0& 0\\ 0& -1& 2& -1& 0\\ 0& 0& -1& 2& -1\\ 0& 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrrr}4& 1& 0& 0& 0\\ 1& 4& 1& 0& 0\\ 0& 1& 4& 1& 0\\ 0& 0& 1& 4& 1\\ 0& 0& 0& 1& 4\end{array}\right]$ $\left[\begin{array}{r}1\\ 1\\ 1\\ 1\end{array}\right]$ $\left[\begin{array}{rrrrrr}2& -1& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0\\ 0& -1& 2& -1& 0& 0\\ 0& 0& -1& 2& -1& 0\\ 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrrrr}4& 1& 0& 0& 0& 0\\ 1& 4& 1& 0& 0& 0\\ 0& 1& 4& 1& 0& 0\\ 0& 0& 1& 4& 1& 0\\ 0& 0& 0& 1& 4& 1\\ 0& 0& 0& 0& 1& 4\end{array}\right]$ $\left[\begin{array}{r}1\\ 1\\ 1\\ 1\\ 1\end{array}\right]$

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

 > C := CartanMatrix("A", 4);
 ${C}{:=}\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}\\ {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {-}{1}& {2}\end{array}\right]$ (3.1)
 > E := EdgeMatrix(C); L := RootLengths(C);
 ${E}{:=}\left[\begin{array}{rrrr}{4}& {1}& {0}& {0}\\ {1}& {4}& {1}& {0}\\ {0}& {1}& {4}& {1}\\ {0}& {0}& {1}& {4}\end{array}\right]$
 ${L}{:=}\left[\begin{array}{r}{1}\\ {1}\\ {1}\end{array}\right]$ (3.2)

Re -construct the Cartan matrix.

 > DynkinDiagramDataToCartanMatrix(E, L);
 $\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}\\ {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {-}{1}& {2}\end{array}\right]$ (3.3)

Root Type B

 Root Type Cartan Matrix $C$ $\frac{{C}_{i,i+1}}{{C}_{i+1,i}}$ Dynkin Diagram ${B}_{2}$ $\left[\begin{array}{rr}2& -2\\ -1& 2\end{array}\right]$ $\left[\begin{array}{rr}0& 2\\ 0& 0\end{array}\right]$ $\left[\begin{array}{r}2\end{array}\right]$ ${B}_{3}$ $\left[\begin{array}{rrr}2& -1& 0\\ -1& 2& -2\\ 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrr}0& 1& 0\\ 0& 0& 2\\ 0& 0& 0\end{array}\right],$ $\left[\begin{array}{r}1\\ 2\end{array}\right]$ ${B}_{4}$ $\left[\begin{array}{rrrr}2& -1& 0& 0\\ -1& 2& -1& 0\\ 0& -1& 2& -2\\ 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrr}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 2\\ 0& 0& 0& 0\end{array}\right]$ $\left[\begin{array}{r}1\\ 1\\ 2\end{array}\right]$ ${B}_{5}$ $\left[\begin{array}{rrrrr}2& -1& 0& 0& 0\\ -1& 2& -1& 0& 0\\ 0& -1& 2& -1& 0\\ 0& 0& -1& 2& -2\\ 0& 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrrr}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 2\\ 0& 0& 0& 0& 0\end{array}\right]$ $\left[\begin{array}{r}1\\ 1\\ 1\\ 2\end{array}\right]$ ${B}_{6}$ $\left[\begin{array}{rrrrrr}2& -1& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0\\ 0& -1& 2& -1& 0& 0\\ 0& 0& -1& 2& -1& 0\\ 0& 0& 0& -1& 2& -2\\ 0& 0& 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrrrr}0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 2\\ 0& 0& 0& 0& 0& 0\end{array}\right],$ $\left[\begin{array}{r}1\\ 1\\ 1\\ 1\\ 2\end{array}\right]$

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

 > C := CartanMatrix("B", 4);
 ${C}{:=}\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}\\ {0}& {-}{1}& {2}& {-}{2}\\ {0}& {0}& {-}{1}& {2}\end{array}\right]$ (4.1)
 > E := EdgeMatrix(C); L := RootLengths(C);
 ${E}{:=}\left[\begin{array}{rrrr}{4}& {1}& {0}& {0}\\ {1}& {4}& {1}& {0}\\ {0}& {1}& {4}& {2}\\ {0}& {0}& {2}& {4}\end{array}\right]$
 ${L}{:=}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]$ (4.2)

Re -construct the Cartan matrix.

 > DynkinDiagramDataToCartanMatrix(E, L);
 $\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}\\ {0}& {-}{1}& {2}& {-}{2}\\ {0}& {0}& {-}{1}& {2}\end{array}\right]$ (4.3)

Root Type C



 Root Type Cartan Matrix $\frac{{C}_{i,i+1}}{{C}_{i+1,i}}$ Dynkin Diagram ${C}_{3}$ $\left[\begin{array}{rrr}2& -1& 0\\ -1& 2& -1\\ 0& -2& 2\end{array}\right]$ $\left[\begin{array}{rrr}4& 1& 0\\ 1& 4& 2\\ 0& 2& 4\end{array}\right]$ $\left[\begin{array}{c}1\\ \frac{1}{2}\end{array}\right]$ ${C}_{4}$ $\left[\begin{array}{rrrr}2& -1& 0& 0\\ -1& 2& -1& 0\\ 0& -1& 2& -1\\ 0& 0& -2& 2\end{array}\right]$ $\left[\begin{array}{rrrr}4& 1& 0& 0\\ 1& 4& 1& 0\\ 0& 1& 4& 2\\ 0& 0& 2& 4\end{array}\right]$ $\left[\begin{array}{c}1\\ 1\\ \frac{1}{2}\end{array}\right]$ ${C}_{5}$ $\left[\begin{array}{rrrrr}2& -1& 0& 0& 0\\ -1& 2& -1& 0& 0\\ 0& -1& 2& -1& 0\\ 0& 0& -1& 2& -1\\ 0& 0& 0& -2& 2\end{array}\right]$ $\left[\begin{array}{rrrrr}4& 1& 0& 0& 0\\ 1& 4& 1& 0& 0\\ 0& 1& 4& 1& 0\\ 0& 0& 1& 4& 2\\ 0& 0& 0& 2& 4\end{array}\right],$ $\left[\begin{array}{c}1\\ 1\\ 1\\ \frac{1}{2}\end{array}\right]$ ${C}_{6}$ $\left[\begin{array}{rrrrrr}2& -1& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0\\ 0& -1& 2& -1& 0& 0\\ 0& 0& -1& 2& -1& 0\\ 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& -2& 2\end{array}\right]$ $\left[\begin{array}{rrrrrr}4& 1& 0& 0& 0& 0\\ 1& 4& 1& 0& 0& 0\\ 0& 1& 4& 1& 0& 0\\ 0& 0& 1& 4& 1& 0\\ 0& 0& 0& 1& 4& 2\\ 0& 0& 0& 0& 2& 4\end{array}\right]$ $\left[\begin{array}{c}1\\ 1\\ 1\\ 1\\ \frac{1}{2}\end{array}\right]$





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

 > C := CartanMatrix("C", 4);
 ${C}{:=}\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}\\ {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {-}{2}& {2}\end{array}\right]$ (5.1)
 > E := EdgeMatrix(C); L := RootLengths(C);
 ${E}{:=}\left[\begin{array}{rrrr}{4}& {1}& {0}& {0}\\ {1}& {4}& {1}& {0}\\ {0}& {1}& {4}& {2}\\ {0}& {0}& {2}& {4}\end{array}\right]$
 ${L}{:=}\left[\begin{array}{c}{1}\\ {1}\\ \frac{{1}}{{2}}\end{array}\right]$ (5.2)

Re -construct the Cartan matrix.

 > DynkinDiagramDataToCartanMatrix(E, L);
 $\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}\\ {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {-}{2}& {2}\end{array}\right]$ (5.3)



Root Type D

 Root Type Cartan Matrix $\frac{{C}_{i,i+1}}{{C}_{i+1,i}}$ Dynkin Diagram ${\mathrm{D}}_{3}$ $\left[\begin{array}{rrr}2& -1& -1\\ -1& 2& 0\\ -1& 0& 2\end{array}\right]$ $\left[\begin{array}{rrr}4& 1& 1\\ 1& 4& 0\\ 1& 0& 4\end{array}\right]$ $\left[\begin{array}{r}1\\ 1\end{array}\right]$ ${\mathrm{D}}_{4}$ $\left[\begin{array}{rrrr}2& -1& 0& 0\\ -1& 2& -1& -1\\ 0& -1& 2& 0\\ 0& -1& 0& 2\end{array}\right]$ $\left[\begin{array}{rrrr}4& 1& 0& 0\\ 1& 4& 1& 1\\ 0& 1& 4& 0\\ 0& 1& 0& 4\end{array}\right],$ ${\mathrm{D}}_{5}$ $\left[\begin{array}{rrrrr}2& -1& 0& 0& 0\\ -1& 2& -1& 0& 0\\ 0& -1& 2& -1& -1\\ 0& 0& -1& 2& 0\\ 0& 0& -1& 0& 2\end{array}\right]$ $\left[\begin{array}{rrrrr}4& 1& 0& 0& 0\\ 1& 4& 1& 0& 0\\ 0& 1& 4& 1& 1\\ 0& 0& 1& 4& 0\\ 0& 0& 1& 0& 4\end{array}\right]$ $\left[\begin{array}{r}1\\ 1\\ 1\\ 1\end{array}\right]$ ${\mathrm{D}}_{6}$ $\left[\begin{array}{rrrrrr}2& -1& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0\\ 0& -1& 2& -1& 0& 0\\ 0& 0& -1& 2& -1& -1\\ 0& 0& 0& -1& 2& 0\\ 0& 0& 0& -1& 0& 2\end{array}\right]$ $\left[\begin{array}{rrrrr}4& 1& 0& 0& 0\\ 1& 4& 1& 0& 0\\ 0& 1& 4& 1& 1\\ 0& 0& 1& 4& 0\\ 0& 0& 1& 0& 4\end{array}\right]$ $\left[\begin{array}{r}1\\ 1\\ 1\\ 1\\ 1\end{array}\right]$

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

 > C := CartanMatrix("D", 4);
 ${C}{:=}\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {-}{1}\\ {0}& {-}{1}& {2}& {0}\\ {0}& {-}{1}& {0}& {2}\end{array}\right]$ (6.1)
 > E := EdgeMatrix(C); L := RootLengths(C);
 ${E}{:=}\left[\begin{array}{rrrr}{4}& {1}& {0}& {0}\\ {1}& {4}& {1}& {1}\\ {0}& {1}& {4}& {0}\\ {0}& {1}& {0}& {4}\end{array}\right]$
 ${L}{:=}\left[\begin{array}{r}{1}\\ {1}\\ {1}\end{array}\right]$ (6.2)

Re -construct the Cartan matrix.

 > DynkinDiagramDataToCartanMatrix(E, L);
 $\left[\begin{array}{rrrr}{2}& {-}{1}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {-}{1}\\ {0}& {-}{1}& {2}& {0}\\ {0}& {-}{1}& {0}& {2}\end{array}\right]$ (6.3)

Exceptional Root Types

 Root Type Cartan Matrix $\frac{{C}_{i,i+1}}{{C}_{i+1,i}}$ Dynkin Diagram ${E}_{6}$ $\left[\begin{array}{rrrrrr}2& 0& -1& 0& 0& 0\\ 0& 2& 0& -1& 0& 0\\ -1& 0& 2& -1& 0& 0\\ 0& -1& -1& 2& -1& 0\\ 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrrrr}4& 0& 1& 0& 0& 0\\ 0& 4& 0& 1& 0& 0\\ 1& 0& 4& 1& 0& 0\\ 0& 1& 1& 4& 1& 0\\ 0& 0& 0& 1& 4& 1\\ 0& 0& 0& 0& 1& 4\end{array}\right]$ -- ${E}_{7}$ $\left[\begin{array}{rrrrrrr}2& 0& -1& 0& 0& 0& 0\\ 0& 2& 0& -1& 0& 0& 0\\ -1& 0& 2& -1& 0& 0& 0\\ 0& -1& -1& 2& -1& 0& 0\\ 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrrrrr}4& 0& 1& 0& 0& 0& 0\\ 0& 4& 0& 1& 0& 0& 0\\ 1& 0& 4& 1& 0& 0& 0\\ 0& 1& 1& 4& 1& 0& 0\\ 0& 0& 0& 1& 4& 1& 0\\ 0& 0& 0& 0& 1& 4& 1\\ 0& 0& 0& 0& 0& 1& 4\end{array}\right]$ -- ${E}_{8}$ $\left[\begin{array}{rrrrrrrr}2& 0& -1& 0& 0& 0& 0& 0\\ 0& 2& 0& -1& 0& 0& 0& 0\\ -1& 0& 2& -1& 0& 0& 0& 0\\ 0& -1& -1& 2& -1& 0& 0& 0\\ 0& 0& 0& -1& 2& -1& 0& 0\\ 0& 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrrrrrr}4& 0& 1& 0& 0& 0& 0& 0\\ 0& 4& 0& 1& 0& 0& 0& 0\\ 1& 0& 4& 1& 0& 0& 0& 0\\ 0& 1& 1& 4& 1& 0& 0& 0\\ 0& 0& 0& 1& 4& 1& 0& 0\\ 0& 0& 0& 0& 1& 4& 1& 0\\ 0& 0& 0& 0& 0& 1& 4& 1\\ 0& 0& 0& 0& 0& 0& 1& 4\end{array}\right]$ ${F}_{4}$ $\left[\begin{array}{rrrr}2& -1& 0& 0\\ -1& 2& -2& 0\\ 0& -1& 2& -1\\ 0& 0& -1& 2\end{array}\right]$ $\left[\begin{array}{rrrr}4& 1& 0& 0\\ 1& 4& 2& 0\\ 0& 2& 4& 1\\ 0& 0& 1& 4\end{array}\right]$ $\left[\begin{array}{r}1\\ 2\\ 1\end{array}\right]$ ${G}_{2}$ $\left[\begin{array}{rr}2& -1\\ -3& 2\end{array}\right]$ $\left[\begin{array}{rr}4& 3\\ 3& 4\end{array}\right]\begin{array}{r}\end{array}$ $\left[\begin{array}{c}\frac{1}{3}\end{array}\right]$

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

 > C := CartanMatrix("E", 6);
 ${C}{:=}\left[\begin{array}{rrrrrr}{2}& {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {2}& {0}& {-}{1}& {0}& {0}\\ {-}{1}& {0}& {2}& {-}{1}& {0}& {0}\\ {0}& {-}{1}& {-}{1}& {2}& {-}{1}& {0}\\ {0}& {0}& {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {0}& {0}& {-}{1}& {2}\end{array}\right]$ (7.1)
 > E := EdgeMatrix(C); L := RootLengths(C);
 ${E}{:=}\left[\begin{array}{rrrrrr}{4}& {0}& {1}& {0}& {0}& {0}\\ {0}& {4}& {0}& {1}& {0}& {0}\\ {1}& {0}& {4}& {1}& {0}& {0}\\ {0}& {1}& {1}& {4}& {1}& {0}\\ {0}& {0}& {0}& {1}& {4}& {1}\\ {0}& {0}& {0}& {0}& {1}& {4}\end{array}\right]$
 ${L}{:=}\left[\begin{array}{r}{1}\\ {1}\\ {1}\\ {1}\\ {1}\end{array}\right]$ (7.2)

Re -construct the Cartan matrix.

 > DynkinDiagramDataToCartanMatrix(E, L);
 $\left[\begin{array}{rrrrrr}{2}& {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {2}& {0}& {-}{1}& {0}& {0}\\ {-}{1}& {0}& {2}& {-}{1}& {0}& {0}\\ {0}& {-}{1}& {-}{1}& {2}& {-}{1}& {0}\\ {0}& {0}& {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {0}& {0}& {-}{1}& {2}\end{array}\right]$ (7.3)