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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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![L1 := _DG([["LieAlgebra", Alg1, [5]], [[[2, 3, 1], 1], [[2, 5, 3], 1], [[4, 5, 4], 1]]])](/support/helpjp/helpview.aspx?si=6610/file05820/math93.png)
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Display the Lie bracket multiplication table.
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Alg1 >
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![[[e2, e3] = e1, [e2, e5] = e3, [e4, e5] = e4]](/support/helpjp/helpview.aspx?si=6610/file05820/math108.png)
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Display the exterior derivatives of the dual 1-forms.
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Alg1 >
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Display the Lie derivatives of the dual 1-forms.
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Alg1 >
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![Matrix([[``, `|`, theta1, theta2, theta3, theta4, theta5], [``, `---`, `----`, `----`, `----`, `----`, `----`], [e1, `|`, 0, 0, 0, 0, 0], [e2, `|`, -theta3, 0, -theta5, 0, 0], [e3, `|`, theta2, 0, 0, 0, 0], [e4, `|`, 0, 0, 0, -theta5, 0], [e5, `|`, 0, 0, theta2, theta4, 0]])](/support/helpjp/helpview.aspx?si=6610/file05820/math138.png)
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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].
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Alg1 >
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![L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 3, 1], 1], [[2, 4, 3], 1], [[4, 2, 4], 1]]])](/support/helpjp/helpview.aspx?si=6610/file05820/math152.png)
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Alg1 >
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![DGsetup(L1, [X, Y, U, V], [alpha, beta, sigma, tau])](/support/helpjp/helpview.aspx?si=6610/file05820/math156.png)
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Display the Lie bracket multiplication table.
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Alg1 >
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![[[Y, U] = X, [Y, V] = U-V]](/support/helpjp/helpview.aspx?si=6610/file05820/math167.png)
| (2.4) |
Display the exterior derivatives of the dual 1-forms.
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Alg1 >
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Display the Lie derivatives of the dual 1-forms.
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Alg1 >
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![Matrix([[``, `|`, alpha, beta, sigma, tau], [``, `---`, `----`, `----`, `----`, `----`], [X, `|`, 0, 0, 0, 0], [Y, `|`, -sigma, 0, -tau, tau], [U, `|`, beta, 0, 0, 0], [V, `|`, 0, 0, beta, -beta]])](/support/helpjp/helpview.aspx?si=6610/file05820/math195.png)
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