Cartan Matrices and Dynkin Diagram Details

All Products Maple MapleSim

Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : Cartan Matrices and Dynkin Diagram Details

Details for Cartan matrices and Dynkin diagrams

Exceptional Root Types

Description

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

•	Let $Δ_{0} = \{α_{1}, α_{2}, .. &period;, α_{m}\}$ be a set of simple roots and let $(\cdot, \cdot)$ be the inner product on the roots induced by the Killing form. The Cartan matrix is given by $C_{ij} = 2 \frac{(α_{i}, α_{j})}{(α_{j}, α_{j})} &period;$

•

From the Cartan matrix one can calculate the number of lines connecting $α_{i}$ to $α_{j}$ as $E_{ij} = C_{ij} C_{ji} (*)$ . The relative lengths of the root vectors can be found as the ratios $\frac{(α_{i}, α_{i})}{(α_{j}, α_{j})} = \frac{C_{ji}}{C_{ij}}$ (**). Set $L_{i} = \frac{(α_{i}, α_{i})}{(α_{i + 1}, α_{i + 1})} &period;$

•

The edge matrix $E_{ij}$ and the root length vector $L_{i}$ clearly determine the Dynkin diagram. Conversely, the equations (*) and (**), together with the facts that $C_{ii}$ = 2 and $- 3 \leq C_{ij} \leq 0$ for $i \neq j$ uniquely determine the Cartan matrix from the edge matrix $E_{ij}$ and the root length vector $L_{i} &period;$ 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 $E_{ij}$ 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	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$A_{1}$	$[\begin{array}{r} 2 \end{array}]$	--	--
$A_{2}$	$[\begin{array}{r} 2 & - 1 \\ - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 \\ 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \end{array}]$
$A_{3}$	$[\begin{array}{r} 2 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 \\ 1 & 4 & 1 \\ 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \end{array}]$
$A_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 \\ 1 & 4 & 1 & 0 \\ 0 & 1 & 4 & 1 \\ 0 & 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \end{array}]$
$A_{5}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \end{array}]$
$A_{6}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 & 0 \\ 0 & 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}]$

Here is the Cartan matrix for $A_{4}$ and the corresponding edge matric and root length vector

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

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

Re -construct the $A_{4}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

Root Type B

Root Type	Cartan Matrix $C$	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$B_{2}$	$[\begin{array}{r} 2 & - 2 \\ - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 2 \\ 0 & 0 \end{array}]$	$[\begin{array}{r} 2 \end{array}]$
$B_{3}$	$[\begin{array}{r} 2 & - 1 & 0 \\ - 1 & 2 & - 2 \\ 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{array}],$	$[\begin{array}{r} 1 \\ 2 \end{array}]$
$B_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 2 \\ 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]$
$B_{5}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & - 1 & 2 & - 2 \\ 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 2 \end{array}]$
$B_{6}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 2 \\ 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}],$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \\ 2 \end{array}]$

Here is the Cartan matrix for $B_{4}$ and the corresponding edge matrix and root length vector.

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

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

Re -construct the $B_{4}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

Root Type C

Root Type	Cartan Matrix	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$C_{3}$	$[\begin{array}{r} 2 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 2 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 \\ 1 & 4 & 2 \\ 0 & 2 & 4 \end{array}]$	$[\begin{array}{c} 1 \\ \frac{1}{2} \end{array}]$
$C_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 2 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 \\ 1 & 4 & 1 & 0 \\ 0 & 1 & 4 & 2 \\ 0 & 0 & 2 & 4 \end{array}]$	$[\begin{array}{c} 1 \\ 1 \\ \frac{1}{2} \end{array}]$
$C_{5}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & - 2 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 1 & 4 & 2 \\ 0 & 0 & 0 & 2 & 4 \end{array}],$	$[\begin{array}{c} 1 \\ 1 \\ 1 \\ \frac{1}{2} \end{array}]$
$C_{6}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & - 2 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 & 0 \\ 0 & 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 & 4 & 2 \\ 0 & 0 & 0 & 0 & 2 & 4 \end{array}]$	$[\begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \\ \frac{1}{2} \end{array}]$

Here is the Cartan matrix for $C_{4}$ and the corresponding edge matrix and root length vector.

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

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

Re -construct the $C_{4}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

Root Type D

Root Type	Cartan Matrix	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$D_{3}$	$[\begin{array}{r} 2 & - 1 & - 1 \\ - 1 & 2 & 0 \\ - 1 & 0 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 1 \\ 1 & 4 & 0 \\ 1 & 0 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \end{array}]$
$D_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & - 1 \\ 0 & - 1 & 2 & 0 \\ 0 & - 1 & 0 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 \\ 1 & 4 & 1 & 1 \\ 0 & 1 & 4 & 0 \\ 0 & 1 & 0 & 4 \end{array}],$	$[\begin{array}{r} 1 \\ 1 \\ 1 \end{array}]$
$D_{5}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & - 1 \\ 0 & 0 & - 1 & 2 & 0 \\ 0 & 0 & - 1 & 0 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 1 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 1 & 0 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \end{array}]$
$D_{6}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & - 1 \\ 0 & 0 & 0 & - 1 & 2 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 1 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 1 & 0 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}]$

Here is the Cartan matrix for $D_{4}$ and the corresponding edge matrix and root length vector.

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

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

Re -construct the $D_{4}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

Exceptional Root Types

Root Type	Cartan Matrix	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$E_{6}$	$[\begin{array}{r} 2 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & - 1 & 0 & 0 \\ - 1 & 0 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 0 & 1 & 0 & 0 & 0 \\ 0 & 4 & 0 & 1 & 0 & 0 \\ 1 & 0 & 4 & 1 & 0 & 0 \\ 0 & 1 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 0 & 1 & 4 \end{array}]$	--
$E_{7}$	$[\begin{array}{r} 2 & 0 & - 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & - 1 & 0 & 0 & 0 \\ - 1 & 0 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 4 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 4 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 4 \end{array}]$	--
$E_{8}$	$[\begin{array}{r} 2 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 2 & - 1 & 0 & 0 & 0 & 0 \\ 0 & - 1 & - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 4 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 4 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 4 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 \end{array}]$
$F_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 2 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 \\ 1 & 4 & 2 & 0 \\ 0 & 2 & 4 & 1 \\ 0 & 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 2 \\ 1 \end{array}]$
$G_{2}$	$[\begin{array}{r} 2 & - 1 \\ - 3 & 2 \end{array}]$	$[\begin{array}{r} 4 & 3 \\ 3 & 4 \end{array}] \begin{array}{r} \end{array}$	$[\begin{array}{c} \frac{1}{3} \end{array}]$

Here is the Cartan matrix for $E_{6}$ and the corresponding edge matrix and root length vector.

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

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

Re -construct the $E_{6}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

Maple

Erweiterungen für Maple

Student Success Platform

Verbesserung der Studienerfolgsquote

MapleSim

Erweiterungen für MapleSim

Systementwicklung

Beratende Dienstleistungen

Produkte zur Online-Ausbildung

Ausbildung

Industrie

Automobil und Luftfahrt

Robotik

Maschinendesign & industrielle Automation

Andere

Lösungen für die Industrie

Kaufen: Einzelheiten und Preise

Kaufen

Institutionelle Studentenlizenzierung

Elite-Wartungsprogramm

Support

Produktschulung

Online Hilfe zu Produkten

Webinare und Veranstaltungen

Veröffentlichungen

Content Hubs

Anwendungsbeispiele

Community

Über Maplesoft

Pressezentrum

User Community

Kontakt

Online Help

All Products Maple MapleSim

Maple

Leistungsfähige intuitive Mathematiksoftware

Erweiterungen für Maple

Student Success Platform

Verbesserung der Studienerfolgsquote

MapleSim

Modellierung auf Systemebene

Erweiterungen für MapleSim

Systementwicklung

Beratende Dienstleistungen

Produkte zur Online-Ausbildung

Ausbildung

Industrie

Automobil und Luftfahrt

Robotik

Maschinendesign & industrielle Automation

Andere

Lösungen für die Industrie

Kaufen: Einzelheiten und Preise

Kaufen

Institutionelle Studentenlizenzierung

Elite-Wartungsprogramm

Support

Produktschulung

Online Hilfe zu Produkten

Webinare und Veranstaltungen

Veröffentlichungen

Content Hubs

Anwendungsbeispiele

Community

Über Maplesoft

Pressezentrum

User Community

Kontakt

Online Help

All Products Maple MapleSim