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LieAlgebras[LieAlgebraWithCoefficientsData] - calculate the structure equations for a Lie algebra with coefficients in a representation
Calling Sequences
LieAlgebraWithCoefficientsData(rhoalgC)
LieAlgebraWithCoefficientsData(algVM, algC
LieAlgebraWithCoefficientsData(algC)
Parameters
- a representation of a Lie algebra
algC - name or a string, the name to be assigned to the Lie algebra with coefficients
V - name of the representation space used to define the Lie algebra with coefficients
M - a list of square matrices which form a Lie algebra
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Description
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Let be a linear space with basis ; let be a Lie algebra with basis and dual basis ,; and let be a representation of . The representation defines the multiplication = Let be the vector space of -forms with coefficients in the representation space . A form if for all vectors , For example, the general 1-formand 2-form with coefficients in can be written as sums
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and ,
where the coefficients and are constants. The spaces play an important role in a number of constructions in Lie theory (See, for example, Cohomology, Deformation, MasseyProduct, KostantLaplacian). To work with forms defined on Lie algebras with coefficients in a representation, one first uses the commands LieAlgebraWithCoefficientsData and DGsetup -- in much the same way that one uses LieAlgebraData and AlgebraData to calculate the structure equations for a Lie algebra or a general non-commutative algebra.
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The output of the LieAlgebraWithCoefficientsData is a data structure which can be passed to the command DGsetup.The structure equations are displayed.
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Examples
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>
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with(DifferentialGeometry): with(LieAlgebras):
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Example 1.
We use the 6 dimensional Lie algebra and its standard representation by skew-symmetric matrices to illustrate the 3 calling sequences for LieAlgebraWithCoefficientsData. First, use the command SimpleLieAlgebraData to retrieve the structure equations for .
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LD := SimpleLieAlgebraData("so(4)", so4);
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Use the command StandardRepresentation to retrieve the matrices for the standard representation.
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so4 >
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M := StandardRepresentation(so4);
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Define a 4-dimensional representation space and the representation
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so4 >
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DGsetup([x1, x2,x3,x4], V);
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V >
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rho := Representation(so4, V, M);
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Use the first calling sequence to calculate the structure equations for with coefficients in the representation .
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so4 >
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LC1 := LieAlgebraWithCoefficientsData(rho, algC1);
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Initialize.
Here is a sample calculation using a 2-form form on with coefficients in .
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algC >
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alpha := evalDG(x3*theta1 &w theta2);
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algC >
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ExteriorDerivative(alpha);
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The second calling sequence simply allows one to calculate the structure equations (4) directly from the matrices without having to first define the representation .
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so4 >
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LieAlgebraWithCoefficientsData(so4, V, M, algC2);
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The third calling sequence retrieves the structure equations of a previously defined Lie algebra with coefficients in a representation.
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algC >
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LieAlgebraWithCoefficientsData(algC1);
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