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()
CartanMatrix()
Parameters
SR - 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
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Description
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The Cartan matrix is the fundamental invariant for semi-simple Lie algebras over C -- two complex semi-simple Lie algebras are isomophic 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.
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The Cartan matrix encodes the re-construction of the root system of the Lie algebra from its simple roots. See PositiveRoots .
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The information contained in the Cartan matrix is also encoded in the Dynkin diagram of the Lie algebra.
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The first calling sequence calculates the Cartan matrix of a Lie algebra from a set of simple roots and a root space decomposition.
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The second calling sequence displays the standard form of the Cartan matrix for each possible root type of a simple Lie algebra.
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Examples
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Example 1.
We use the command SimpleLieAlgebraData to obtain the Lie algebra data for the Lie algebra . This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices
We surpress the output of this command which is a lengthy list of structure equations.
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Initialize this Lie algebra -- the basis elements are given the default labels
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We remark that the command StandardRepresentation can be used to explicitly display the matrices defining .
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The first 3 matrices define a Cartan subalgebra. We can use the Query command to check this
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We use the command RootSpaceDecomposition to find the root space decomposition for with respect to this Cartan subalgebra.
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A choice of simple roots for this root space decomposition is:
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| (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 :
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| (2.8) |
We easily identify this as the standard Cartan matrix for
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| (2.9) |
Notice that a permutation of the simple roots gives a permuted Cartan matrix.
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| (2.10) |
su >
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Example 2.
For the exceptional Lie algebras , and there are two different conventions for the Cartan matrix. For these are:
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| (2.12) |
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