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Example 1.
We define 4 different Cartan matrices and calculate their standard forms and root type.
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Here are the standard forms, permutation matrices and root types.
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For each example the second output is a permutation matrix which transforms the given input Cartan matrix to its standard form.
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Example 2.
We define a 21-dimensional simple Lie algebra and calculate its root type.
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Initialize this Lie algebra.
Find a Cartan subalgebra.
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Find the root space decomposition.
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Find the roots, positive roots and a choice of simple roots.
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Find the Cartan matrix.
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Transform the Cartan matrix to standard form. Here we use the second calling sequence. The command CartanMatrixToStandardForm now returns a permuted set of simple roots for which the Cartan matrix will be in standard form.
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Check the result by re-calculating the Cartan matrix with respect to the permuted set of roots. We get the standard form immediately.
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The root type of our 21-dimensional Lie algebra is