In this series of examples and applications we shall work with a 3-step graduation of the 10-dimensional real symplectic Lie algebra with coefficients in the adjoint representation.The following steps are needed to create the environment for defining the Kostant codifferential and the Kostant Laplacian.
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Initialize the Lie algebras and .
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Initialize the Lie algebra of with coefficients in . Initialize the Lie algebra of with coefficients in.
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Step 1. Use the command SimpleLieAlgebraData to retrieve the structure equations for . Initialize this Lie algebra.
Step 2. Various properties of the classical Lie algebras are available with the command SimpleLieAlgebraProperties. We need the simple roots here.
Every subset of the simple roots of a Lie algebra defines a grading of that algebra. Here we use all the roots of to obtain a 2-step gradation with the command GradeSemiSimpleLieAlgebra.
Step 3. Note that the vectors define the negative part of with respect to the chosen grading. The next step is to introduce a new basis for adapted to the grading. We call in this new basis and we call the negatively graded part
The negatively graded component is always nilpotent. With the following calling sequence LieAlgebraData returns the structure equations for , the structure equations for , and the basis of our original algebra, adapted to the grading, and a basis for .
Initialize the Lie algebras and .
Step 4. Now we are ready to define the adjoint representation for and its restriction to . Since is 10-dimensional, we need a 10-dimensional representation space. Call it .
The command Adjoint gives the adjoint representation for
The command RestrictedRepresentation gives the restriction of the adjoint representation for to the subalgebra. (The following calling sequence assumes that the first 4 vectors in the given basis define the subalgebra . )
Step 5. Initialize the Lie algebra with coefficients in the adjoint representation. Call it and label the basis vectors and the basis 1-forms .
Initialize the Lie algebra with coefficients in the adjoint representation of . Call it and label the basis vectors and the basis 1-forms . As described above the Kostant codifferential uses the embedding of in This information is provided by the keyword argument ambientalgebra.
Example 1.
Here are some sample calculations of the Kostant co-differential.
Let's do this last calculation directly from the definition. For this we need the Killing form and its inverse.
Re-define as a form on sp4R.
Use the inverse of the Killing form to convert to a multi-vector :
Take the co-differential of .
Lower the indices of with the Killing form.
Example 2.
The square of the Kostant co-differential vanishes.
Example 3.
We check that the Kostant co-differential is the adjoint of the exterior derivative. Here are the inner products we need (See PositiveDefiniteMetricOnRepresentationSpace) .
Here is the left-hand side of the adjoint equation.
Here is the right-hand side of the adjoint equation.
We can easily check the adjoint equation for lists of forms. We use the command RelativeChains to generate lists of forms. For this example, we specify the weight of the forms to keep the lists small.
The equality of these matrices verifies the adjoint equation for given lists of forms.
Example 4.
Here are some sample calculations of the Kostant Laplacian.
A scalar:
A 2-form:
A 3-form:
Example 5.
We calculate the second cohomology
Here are the 2-chains
For these cohomology calculations it is preferable to specify the Killing form and its inverse in the arguments for KostantCodifferential and KostantLaplacian. Otherwise the Killing form and its inverse will be re-calculated for each 2 form in
Method 1. Here we shall calculate using the DGNullSpace and IntersectSubspaces commands.
Here is.
Here is First we make a small procedure which fixes the Killing form and its inverse in the arguments for KostantCodifferential.
Now calculate the intersection .
Method 2. Here we shall calculate . Again we make a small procedure which fixes the Killing form and its inverse in the arguments for KostantLaplacian.
