CartanMatrix - Maple Help

LieAlgebras[CartanMatrix] - find the Cartan matrix for a simple Lie algebra from a root space decomposition, display the Cartan matrix for a given root type

Calling Sequences

CartanMatrix(SimRts, RSD)

CartanMatrix(RT, m)

Parameters

SimRts   - a list of column vectors, defining the simple roots of a simple Lie algebra

RSD      - a table, defining the root space decomposition of an initialized Lie algebra

RT       - a string, the root type of a simple Lie algebra "A", "B", "C", "D", "E", "F", "G"

m        - a positive integer, the dimension of the Cartan matrix

Description

 • Let g be a simple Lie algebra, h a Cartan subalgebra, and the root space decomposition of g with respect to h. Let <⋅,⋅> be the Killing form of g. For each root , there are vectors and  such that and  These conditions uniquely determine The vector ${H}_{\mathrm{\alpha }}$ can be computed using the command RootToCartanSubalgebraElementH.
 • Let be a set of simple roots for g. Then the associated Cartan matrix is the $m×m$ matrix with entries $2$<${H}_{{\mathrm{α}}_{i}}$ , ${H}_{{\mathrm{α}}_{j}}$$>$/ <${H}_{{\mathrm{α}}_{i}}$, ${H}_{{\mathrm{α}}_{i}}$ >. The entries of the Cartan matrix are 0, 1, -1 or 2. The Cartan matrix is independent of the choice of Cartan subalgebra h but is dependent upon the ordering of the simple roots in
 • The Cartan matrix is the fundamental invariant for semi-simple Lie algebras over C -- two complex semi-simple Lie algebras are isomorphic if and only if their Cartan matrices are the same, modulo a permutation of the vectors in the Cartan subalgebra. The command CartanMatrixToStandardForm will transform a given Cartan matrix to a standard form.
 • The Cartan matrix encodes the re-construction of the root system of the Lie algebra from its simple roots. See PositiveRoots .
 • The information contained in the Cartan matrix is also encoded in the Dynkin diagram of the Lie algebra.
 • The first calling sequence calculates the Cartan matrix of a Lie algebra from a set of simple roots and a root space decomposition.
 • The second calling sequence displays the standard form of the Cartan matrix for each possible root type of a simple Lie algebra.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

We use the command SimpleLieAlgebraData to obtain the Lie algebra data for the Lie algebra $\mathrm{su}\left(4\right)$. This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices

We suppress the output of this command which is a lengthy list of structure equations.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("su\left(4\right)",\mathrm{su}\right):$

Initialize this Lie algebra -- the basis elements are given the default labels

 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: su}}$ (2.1)

We remark that the command StandardRepresentation can be used to explicitly display the matrices defining $\mathrm{su}\left(4\right)$.

 su > $\mathrm{StandardRepresentation}\left(\mathrm{su}\right)$
 $\left[\left[\begin{array}{cccc}{-}{I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {I}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {-}{I}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {I}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{I}& {0}\\ {0}& {0}& {0}& {I}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {1}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {I}& {0}& {0}\\ {I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {I}& {0}\\ {0}& {I}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {I}\\ {0}& {0}& {I}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {I}& {0}\\ {0}& {0}& {0}& {0}\\ {I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}\\ {0}& {I}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {I}& {0}& {0}& {0}\end{array}\right]\right]$ (2.2)
 $\left[\left[\begin{array}{cccc}{I}& {0}& {0}& {0}\\ {0}& {-}{I}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {I}& {0}& {0}\\ {0}& {0}& {-}{I}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {I}& {0}\\ {0}& {0}& {0}& {-}{I}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {1}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {I}& {0}& {0}\\ {I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {I}& {0}\\ {0}& {I}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {I}\\ {0}& {0}& {I}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {I}& {0}\\ {0}& {0}& {0}& {0}\\ {I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}\\ {0}& {I}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {I}& {0}& {0}& {0}\end{array}\right]\right]$ (2.3)

The first 3 matrices define a Cartan subalgebra. We can use the Query command to check this

 su > $\mathrm{CSA}≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right]$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]$ (2.4)
 su > $\mathrm{Query}\left(\mathrm{CSA},"CartanSubalgebra"\right)$
 ${\mathrm{true}}$ (2.5)

We use the command RootSpaceDecomposition to find the root space decomposition for with respect to this Cartan subalgebra.

 su > $\mathrm{RSD}≔\mathrm{RootSpaceDecomposition}\left(\mathrm{CSA}\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{I}{,}{2}{}{I}{,}{I}\right]{=}{\mathrm{e8}}{-}{I}{}{\mathrm{e14}}{,}\left[{2}{}{I}{,}{I}{,}{I}\right]{=}{\mathrm{e9}}{-}{I}{}{\mathrm{e15}}{,}\left[{I}{,}{I}{,}{2}{}{I}\right]{=}{\mathrm{e6}}{-}{I}{}{\mathrm{e12}}{,}\left[{-}{I}{,}{-}{2}{}{I}{,}{-}{I}\right]{=}{\mathrm{e8}}{+}{I}{}{\mathrm{e14}}{,}\left[{-}{2}{}{I}{,}{-}{I}{,}{-}{I}\right]{=}{\mathrm{e9}}{+}{I}{}{\mathrm{e15}}{,}\left[{0}{,}{I}{,}{-}{I}\right]{=}{\mathrm{e5}}{-}{I}{}{\mathrm{e11}}{,}\left[{-}{I}{,}{I}{,}{0}\right]{=}{\mathrm{e4}}{+}{I}{}{\mathrm{e10}}{,}\left[{I}{,}{0}{,}{-}{I}\right]{=}{\mathrm{e7}}{-}{I}{}{\mathrm{e13}}{,}\left[{-}{I}{,}{0}{,}{I}\right]{=}{\mathrm{e7}}{+}{I}{}{\mathrm{e13}}{,}\left[{-}{I}{,}{-}{I}{,}{-}{2}{}{I}\right]{=}{\mathrm{e6}}{+}{I}{}{\mathrm{e12}}{,}\left[{0}{,}{-}{I}{,}{I}\right]{=}{\mathrm{e5}}{+}{I}{}{\mathrm{e11}}{,}\left[{I}{,}{-}{I}{,}{0}\right]{=}{\mathrm{e4}}{-}{I}{}{\mathrm{e10}}\right]\right)$ (2.6)

A choice of simple roots for this root space decomposition is:

 su > $\mathrm{Δ0}≔\left[⟨I,I,2I⟩,⟨0,I,-I⟩,⟨I,-I,0⟩\right]$
 ${\mathrm{Δ0}}{:=}\left[\left[\begin{array}{c}{I}\\ {I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {-}{I}\\ {0}\end{array}\right]\right]$ (2.7)

This set of simple roots can be determined by the command SimpleRoots. The Cartan matrix for this root space decomposition and choice of simple roots is :

 su > $\mathrm{CM}≔\mathrm{CartanMatrix}\left(\mathrm{Δ0},\mathrm{RSD}\right)$
 ${\mathrm{CM}}{:=}\left[\begin{array}{rrr}{2}& {-}{1}& {0}\\ {-}{1}& {2}& {-}{1}\\ {0}& {-}{1}& {2}\end{array}\right]$ (2.8)

We easily identify this as the standard Cartan matrix for

 su > $\mathrm{CartanMatrix}\left("A",3\right)$
 $\left[\begin{array}{rrr}{2}& {-}{1}& {0}\\ {-}{1}& {2}& {-}{1}\\ {0}& {-}{1}& {2}\end{array}\right]$ (2.9)

Notice that a permutation of the simple roots gives a permuted Cartan matrix.

 su > $\mathrm{Δ1}≔\left[\mathrm{Δ0}\left[3\right],\mathrm{Δ0}\left[1\right],\mathrm{Δ0}\left[2\right]\right]$
 ${\mathrm{Δ1}}{:=}\left[\left[\begin{array}{c}{I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {I}\\ {-}{I}\end{array}\right]\right]$ (2.10)
 su > $\mathrm{CartanMatrix}\left(\mathrm{Δ1},\mathrm{RSD}\right)$
 $\left[\begin{array}{rrr}{2}& {0}& {-}{1}\\ {0}& {2}& {-}{1}\\ {-}{1}& {-}{1}& {2}\end{array}\right]$ (2.11)

Example 2.

For the exceptional Lie algebras ${E}_{6}$, and there are two different conventions for the Cartan matrix. For ${E}_{6}$ these are:

 su > $\mathrm{CartanMatrix}\left("E",6,\mathrm{version}="I"\right),\mathrm{CartanMatrix}\left("E",6,\mathrm{version}="II"\right)$
 $\left[\begin{array}{rrrrrr}{2}& {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {2}& {0}& {-}{1}& {0}& {0}\\ {-}{1}& {0}& {2}& {-}{1}& {0}& {0}\\ {0}& {-}{1}& {-}{1}& {2}& {-}{1}& {0}\\ {0}& {0}& {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {0}& {0}& {-}{1}& {2}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{2}& {-}{1}& {0}& {0}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}& {0}& {0}\\ {0}& {-}{1}& {2}& {-}{1}& {0}& {-}{1}\\ {0}& {0}& {-}{1}& {2}& {-}{1}& {0}\\ {0}& {0}& {0}& {-}{1}& {2}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {2}\end{array}\right]$ (2.12)