LieAlgebras[MultiplicationTable] - display the multiplication table of a Lie algebra
Calling Sequences
MultiplicationTable(LieAlgebraName, keyword)
Parameters
LieAlgebraName - (optional) name or string, the name assigned to a Lie algebra
keyword - keyword string, one of "LieBracket", "ExteriorDerivative", "LieDerivative"
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Description
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MultiplicationTable(LieAlgebraName, keyword) displays the form of structure equations for the Lie algebra dictated by the keyword.
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If the keyword is "LieBracket", then the Lie brackets [ei, ej] of the basis elements [e1, e2, ... eN] are displayed in a two dimensional array.
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If the keyword is "ExteriorDerivative", then the exterior derivatives d(thetai) of the dual basis elements [theta1, theta2, ... thetaN] are printed.
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If the keyword is "LieDerivative", then the Lie derivatives L_(ei) thetaj of the dual 1-forms [theta1, theta2, ... thetaN] with respect to the basis vectors [e1, e2, ...eN] are displayed in a two dimensional array.
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If LieAlgebraName is omitted, then the appropriate multiplication table of the current Lie algebra is displayed.
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The command MultiplicationTable is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form MultiplicationTable(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MultiplicationTable(...).
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Examples
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Example 1.
First we initialize a 5 dimensional Lie algebra.
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Display the Lie bracket multiplication table.
Alg1 >
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Display the exterior derivatives of the dual 1-forms.
Alg1 >
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Display the Lie derivatives of the dual 1-forms.
Alg1 >
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Example 2.
We initialize a 4 dimensional Lie algebra. Instead of using the standard default labels for the basis vectors we use [X, Y, U, V] and for the dual 1-forms we use [alpha, beta, sigma, tau].
Alg1 >
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Alg1 >
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Display the Lie bracket multiplication table.
Alg1 >
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Display the exterior derivatives of the dual 1-forms.
Alg1 >
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Display the Lie derivatives of the dual 1-forms.
Alg1 >
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