CartanMatrix - Maple Help
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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)


     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







 Let g be a simple Lie algebra, h a Cartan subalgebra, and &gfr; &equals; &hfr; &alpha;  &Delta; R&alpha; the root space decomposition of g with respect to h. Let <⋅,⋅> be the Killing form of g. For each root &alpha; &Delta;, there are vectors X&alpha; R&alpha; &comma; X&alpha; R&alpha; and H&alpha; &hfr; such that  &lsqb;Hα &comma; X&alpha;&rsqb;  &equals; 2 X&alpha;&comma;  &lsqb;Hα &comma;  X&alpha;&rsqb;  &equals; 2 X&alpha; and  X&alpha; &comma; X&alpha; &equals; H&alpha; &period;  These conditions uniquely determine H&period;  The vector Hα can be computed using the command RootToCartanSubalgebraElementH.


Let &Delta;0 &equals; &alpha;1 &comma; &alpha;2&comma; .... &alpha;m &Delta; be a set of simple roots for g. Then the associated Cartan matrix is the m×m matrix with entries Cij&equals; 2<H&alpha;i , H&alpha;j&gt;/ <H&alpha;i, H&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 &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.




Example 1.

We use the command SimpleLieAlgebraData to obtain the Lie algebra data for the Lie algebra su4. 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.




Initialize this Lie algebra -- the basis elements are given the default labels e1&comma; e2&comma; ..&period; &comma;e15 &period;


Lie algebra: su



We remark that the command StandardRepresentation can be used to explicitly display the matrices defining su4.

su > 



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

su > 



su > 





We use the command RootSpaceDecomposition to find the root space decomposition for su4 with respect to this Cartan subalgebra.

su > 





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

su > 



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 > 



We easily identify this as the standard Cartan matrix for A3 &period;

su > 



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

su > 


su > 




Example 2.

For the exceptional Lie algebras E6, E7 and E8 there are two different conventions for the Cartan matrix. For E6 these are:

su > 



See Also