Let g be a simple Lie algebra, h a Cartan subalgebra, and $\mathrm{\&gfr;}\mathit{ }\mathit{\&equals;}\mathit{ }\mathrm{\&hfr;}\mathit{ }{\oplus }_{\mathrm{\&alpha;} \in \mathrm{\&Delta;}} R_{\mathrm{\&alpha;} }$the root space decomposition of g with respect to h. Let <⋅,⋅> be the Killing form of g. For each root $\mathrm{\&alpha;} \in \mathrm{\&Delta;}$, there are vectors $X_{\mathrm{\&alpha;}} \in R_{\mathrm{\&alpha;}} \&comma; X_{-\mathrm{\&alpha;}} \in R_{-\mathrm{\&alpha;}}$and $H_{\mathrm{\&alpha;}}\in \mathrm{\&hfr;}$ such that $\&lsqb;H_{\mathrm{\alpha }} \&comma;$$X_{\mathrm{\&alpha;}}\&rsqb; \&equals; 2 X_{\mathrm{\&alpha;}}\&comma;$$\&lsqb;H_{\mathrm{\alpha }} \&comma;$$X_{-\mathrm{\&alpha;}}\&rsqb; \&equals; -2 X_{-\mathrm{\&alpha;}}$and  $\left[X_{\mathrm{\&alpha;}} \&comma; X_{-\mathrm{\&alpha;}}\right] \&equals; H_{\mathrm{\&alpha;}} \&period;$These conditions uniquely determine $H_{}\&period;$The vector $H_{\mathrm{\alpha }}$ can be computed using the command RootToCartanSubalgebraElementH.

Let ${\mathrm{\&Delta;}}_0 \&equals; \left\{{\mathrm{\&alpha;}}_1 \&comma; {\mathrm{\&alpha;}}_2\&comma; .... {\mathrm{\&alpha;}}_m\right\}\subseteq \mathrm{\&Delta;}$be a set of simple roots for g. Then the associated Cartan matrix is the $m\times m$ matrix with entries $C_{\mathrm{ij}}\&equals;$$2$<$H_{{\mathrm{\&alpha;}}_i}$ , $H_{{\mathrm{\&alpha;}}_j}$$\&gt;$/ <$H_{{\mathrm{\&alpha;}}_i}$, $H_{{\mathrm{\&alpha;}}_i}$ >. 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 ${\mathrm{\&Delta;}}_0 \&period;$

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.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\&colon;$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\&colon;$

$\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left(''su(4)''\&comma;\mathrm{su}\right)\&colon;$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; su}$

$\mathrm{StandardRepresentation}\left(\mathrm{su}\right)$

$\mathrm{CSA}≔\left[\mathrm{e1}\&comma;\mathrm{e2}\&comma;\mathrm{e3}\right]$

$\mathrm{CSA}:=\left[\mathrm{e1}\&comma;\mathrm{e2}\&comma;\mathrm{e3}\right]$

$\mathrm{Query}\left(\mathrm{CSA}\&comma;''CartanSubalgebra''\right)$

$\mathrm{true}$

$\mathrm{RSD}≔\mathrm{RootSpaceDecomposition}\left(\mathrm{CSA}\right)$

$\mathrm{RSD}:=\mathrm{table}\left(\left[\left[\mathrm{I}\&comma;2\mathrm{I}\&comma;\mathrm{I}\right]\&equals;\mathrm{e8}-\mathrm{I}\mathrm{e14}\&comma;\left[2\mathrm{I}\&comma;\mathrm{I}\&comma;\mathrm{I}\right]\&equals;\mathrm{e9}-\mathrm{I}\mathrm{e15}\&comma;\left[\mathrm{I}\&comma;\mathrm{I}\&comma;2\mathrm{I}\right]\&equals;\mathrm{e6}-\mathrm{I}\mathrm{e12}\&comma;\left[-\mathrm{I}\&comma;-2\mathrm{I}\&comma;-\mathrm{I}\right]\&equals;\mathrm{e8}\&plus;\mathrm{I}\mathrm{e14}\&comma;\left[-2\mathrm{I}\&comma;-\mathrm{I}\&comma;-\mathrm{I}\right]\&equals;\mathrm{e9}\&plus;\mathrm{I}\mathrm{e15}\&comma;\left[0\&comma;\mathrm{I}\&comma;-\mathrm{I}\right]\&equals;\mathrm{e5}-\mathrm{I}\mathrm{e11}\&comma;\left[-\mathrm{I}\&comma;\mathrm{I}\&comma;0\right]\&equals;\mathrm{e4}\&plus;\mathrm{I}\mathrm{e10}\&comma;\left[\mathrm{I}\&comma;0\&comma;-\mathrm{I}\right]\&equals;\mathrm{e7}-\mathrm{I}\mathrm{e13}\&comma;\left[-\mathrm{I}\&comma;0\&comma;\mathrm{I}\right]\&equals;\mathrm{e7}\&plus;\mathrm{I}\mathrm{e13}\&comma;\left[-\mathrm{I}\&comma;-\mathrm{I}\&comma;-2\mathrm{I}\right]\&equals;\mathrm{e6}\&plus;\mathrm{I}\mathrm{e12}\&comma;\left[0\&comma;-\mathrm{I}\&comma;\mathrm{I}\right]\&equals;\mathrm{e5}\&plus;\mathrm{I}\mathrm{e11}\&comma;\left[\mathrm{I}\&comma;-\mathrm{I}\&comma;0\right]\&equals;\mathrm{e4}-\mathrm{I}\mathrm{e10}\right]\right)$

$\mathrm{\&Delta;0}≔\left[⟨\mathrm{I}\&comma;\mathrm{I}\&comma;2\mathrm{I}⟩\&comma;⟨0\&comma;\mathrm{I}\&comma;-\mathrm{I}⟩\&comma;⟨\mathrm{I}\&comma;-\mathrm{I}\&comma;0⟩\right]$

$\mathrm{CM}≔\mathrm{CartanMatrix}\left(\mathrm{\&Delta;0}\&comma;\mathrm{RSD}\right)$

$\mathrm{CartanMatrix}\left(''A''\&comma;3\right)$

$\mathrm{\&Delta;1}≔\left[\mathrm{\&Delta;0}\left[3\right]\&comma;\mathrm{\&Delta;0}\left[1\right]\&comma;\mathrm{\&Delta;0}\left[2\right]\right]$

$\mathrm{CartanMatrix}\left(\mathrm{\&Delta;1}\&comma;\mathrm{RSD}\right)$

$\mathrm{CartanMatrix}\left(''E''\&comma;6\&comma;\mathrm{version}=''I''\right),\mathrm{CartanMatrix}\left(''E''\&comma;6\&comma;\mathrm{version}=''II''\right)$