Early in the theoretical development of simple Lie algebras, it was conjectured by Killing that there exists a certain small number of exceptional Lie algebras not belonging to the families of simple Lie algebras, which were wellknown at the time. It was an important problem to explicitly construct these exceptional algebras. In 1908, E. Cartan proved that the smallest of these exceptional Lie algebras, the 14dimensional Lie algebra denoted by , is the algebra of derivations of the 8dimensional algebra of octonions . With Maple 17, it is easy to verify Cartan's computations.
First, use the new AlgebraLibraryData command to retrieve the structure equations for the octonions:
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Initialize this algebra:
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A derivation of the octonions is a linear transformation such that for all . The command Derivations will compute a list of matrices which define a basis for the Lie algebra of derivations of the any given algebra. (In prior releases of Maple, the derivation command only applied to Lie algebras and not more general algebras like the octonions).
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You can see that the dimension of the space of derivations is 14.
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It remains then to verify that this 14 dimensional Lie algebra is in fact the exceptional Lie algebra , as claimed by Cartan. To do this, one needs to apply the general structure theory for the classification of Lie algebras. This can be easily done with commands from the LieAlgebras package.
First, calculate the structure equations for the Lie algebra defined by the above 14 matrices using LieAlgebraData:
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Initialize this Lie algebra:
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Use Query to check that you have a semisimple Lie algebra:
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To check that our algebra is use the commands CartanSubalgebra, RootSpaceDecomposition, SimpleRoots, CartanMatrix and CartanMatrixToStandardForm to find the Cartan matrix. This matrix uniquely identifies the Lie algebra as .
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In this way you have proven that the algebra of derivations of the octonions is the exceptional Lie algebra . 