KostantCodifferential - Maple Help

LieAlgebras[KostantCodifferential] - calculate the Kostant co-differential of a p-form or a list of p-forms defined on a nilpotent Lie algebra with coefficients in a representation

LieAlgebras[KostantLaplacian] - calculate the Kostant Laplacian of a form defined on a nilpotent Lie algebra with coefficients in a representation

Calling Sequences

KonstantCodifferential(${\mathbf{ω}}$, B$,$invB)

KostantLaplacian(${\mathbf{ω}}$$,$B$,$invB$,$ method)

Parameters

$\mathrm{ω}$         - a form defined on a nilpotent Lie algebra with coefficients in a representation $V$

B         - (optional) the Killing form for the associated semi-simple Lie algebra

invB      - (optional) the inverse of the Killing form

method    - with the keyword argument method = "LieDerivative", the Kostant Laplacian is computed in terms of the Lie derivative operator

Description

 • Let $\mathrm{𝔤}$ be a semi-simple Lie algebra and let $V$ be a representation space for $\mathrm{𝔤}$. Fix a grading of :

and let  and .

Such gradings can be systematically constructed using the command GradeSemiSimpleLieAlgebra. The Lie algebra is nilpotent and the representation for restricts to give a representation space for ${\mathrm{𝔤}}_{-}$. The commands Representation and RestrictedRepresentation can be used to create these representations for and ${\mathrm{𝔤}}_{-}$.

 • Now let denote the space of $p$-forms on with coefficients in $V$. If and ${X}_{1}$, ...are vectors in ${𝔤}_{-}$, then . The spaces can be constructed using the command DGsetup and RelativeChains. The exterior derivative defines a mapping while the Kostant co-differential is a map .
 • The Kostant co-differential is easily defined in terms of the general Codifferentialoperator. Let  and let $X={\mathrm{α}}^{+}$be the degree $p$ multi-vector obtained by raising the indices of the form using the inverse of the Killing metric for $\mathrm{𝔤}$. Take the co-differential of $X$ to obtain the multi-vector $\partial$X of degree $p-1$ and use the Killing form to lower the indices of $\partial$X to obtain a form. This is the Kostant co-differential of that is,

 • The command PositiveDefiniteMetricOnRepresentationSpace can be to used to construct positive definite inner products on $\mathrm{𝔤}$ and on such that the Kostant co-differential is the adjoint of the exterior derivative operator with respect to the induced inner product on , that is,

for all and .

 • The Kostant Laplacian is the map defined by

.

 • The Kostant co-differential and Kostant Laplacian are very useful for the explicit calculation of the Lie algebra cohomology of the complex as

This is generally a much faster way to calculate these cohomology spaces than with a direct calculation using the command Cohomology. See also the command DGNullSpace.

 • The optional arguments (Killing form and inverse Killing form) for KostantCodifferential and KostantLaplacian dramatically improve the computational efficiency of these commands.
 • The formula for the Kostant Laplacian in terms of the Lie derivative is as follows. First pick a basis for $\mathrm{𝔤}$ which is adapted to the decomposition . Let ${\mathrm{η}}_{k}$ be the dual basis for defined with respect to the Killing form $B$, that is, $B($${\mathrm{η}}_{k}$ , . . Then

While this formula has important theoretical implications, it is used here primary as a consistency check on the software implementation of the Kostant Laplacian.

Examples

 > ${\mathrm{restart}}{:}$${\mathrm{with}}{}\left({\mathrm{DifferentialGeometry}}\right){:}$${\mathrm{with}}{}\left({\mathrm{Tensor}}\right){:}$${\mathrm{with}}{}\left({\mathrm{LieAlgebras}}\right){:}$

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.

 1 Use the command SimpleLieAlgebraData and DGsetup to initialize a simple Lie algebra $\mathrm{𝔤}$. 
 2 Use the command SimpleLieAlgebraProperties to retrieve the simple roots of $\mathrm{𝔤}$ . Use the command GradeSemiSimpleAlgebra to construct a gradation of $\mathrm{𝔤}$.
 3 Initialize the Lie algebras ${𝔤}_{}$ and ${\mathrm{𝔤}}_{-}$.
 4 Use the commands StandardRepresentation, Adjoint, Representation etc. to make a representation of . Use the RestrictedRepresentation command to restrict the representation of on $V$ to ${\mathrm{𝔤}}_{-}$
 5 Initialize the Lie algebra of ${𝔤}_{}$ with coefficients in $V$. Initialize the Lie algebra of ${{\mathrm{𝔤}}_{-}}_{}$ with coefficients in.

Step 1. Use the command SimpleLieAlgebraData to retrieve the structure equations for . Initialize this Lie algebra.

 > ${\mathrm{LD}}{≔}{\mathrm{SimpleLieAlgebraData}}{}\left({"sp\left(4, R\right)"}{,}{\mathrm{alg}}\right)$
 ${\mathrm{LD}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e4}}\right]$ (2.1)
 > ${\mathrm{DGsetup}}{}\left({\mathrm{LD}}{,}\left[{U}\right]{,}\left[{\mathrm{\upsilon }}\right]\right)$
 ${\mathrm{Lie algebra: alg}}$ (2.2)

Step 2. Various properties of the classical Lie algebras are available with the command SimpleLieAlgebraProperties. We need the simple roots here.

 alg > ${\mathrm{Properties}}{≔}{\mathrm{SimpleLieAlgebraProperties}}{}\left({\mathrm{alg}}\right){:}$
 alg > ${\mathrm{Δ0}}{≔}{\mathrm{Properties}}\left[{"SimpleRoots"}\right]$
 ${\mathrm{Δ0}}{≔}\left[\left[\begin{array}{c}{1}\\ {-1}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {2}\end{array}\right]\right]$ (2.3)

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.

 alg > ${G}{≔}{\mathrm{GradeSemiSimpleLieAlgebra}}{}\left({\mathrm{Δ0}}{,}{\mathrm{Properties}}\right)$
 ${G}{≔}{table}{}\left(\left[{-1}{=}\left[{\mathrm{U3}}{,}{\mathrm{U10}}\right]{,}{0}{=}\left[{\mathrm{U1}}{,}{\mathrm{U4}}\right]{,}{-2}{=}\left[{\mathrm{U9}}\right]{,}{1}{=}\left[{\mathrm{U2}}{,}{\mathrm{U7}}\right]{,}{-3}{=}\left[{\mathrm{U8}}\right]{,}{2}{=}\left[{\mathrm{U6}}\right]{,}{3}{=}\left[{\mathrm{U5}}\right]\right]\right)$ (2.4)

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 $\mathrm{alg}$ adapted to the grading. We call  in this new basis and we call the negatively graded part $N.$

The negatively graded component $N$ is always nilpotent. With the following calling sequence LieAlgebraData returns the structure equations for $\mathrm{sp4}$, the structure equations for $N$, and the basis of our original algebra, adapted to the grading, and a basis for $N$.

 alg > ${\mathrm{LD1}}{,}{\mathrm{LD2}}{,}{\mathrm{B1}}{,}{\mathrm{B2}}{≔}{\mathrm{LieAlgebraData}}{}\left({G}{,}{\mathrm{sp4}}{,}{"negative"}{,}{N}{,}{\mathrm{output}}{=}{"basis"}\right)$
 ${\mathrm{LD1}}{,}{\mathrm{LD2}}{,}{\mathrm{B1}}{,}{\mathrm{B2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e6}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e10}}\right]{,}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}\right]{,}\left[{\mathrm{U8}}{,}{\mathrm{U9}}{,}{\mathrm{U3}}{,}{\mathrm{U10}}{,}{\mathrm{U1}}{,}{\mathrm{U4}}{,}{\mathrm{U2}}{,}{\mathrm{U7}}{,}{\mathrm{U6}}{,}{\mathrm{U5}}\right]{,}\left[{\mathrm{U8}}{,}{\mathrm{U9}}{,}{\mathrm{U3}}{,}{\mathrm{U10}}\right]$ (2.5)



Initialize the Lie algebras $\mathrm{sp4}$ and $N$.



 alg > ${\mathrm{DGsetup}}{}\left({\mathrm{LD1}}\right)$
 ${\mathrm{Lie algebra: sp4}}$ (2.6)
 sp4 > ${\mathrm{DGsetup}}{}\left({\mathrm{LD2}}\right)$
 ${\mathrm{Lie algebra: N}}$ (2.7)

Step 4. Now we are ready to define the adjoint representation for and its restriction to $N$. Since is 10-dimensional, we need a 10-dimensional representation space. Call it $V$.

 N > ${\mathrm{DGsetup}}{}\left(\left[{\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}{,}{\mathrm{x5}}{,}{\mathrm{x6}}{,}{\mathrm{x7}}{,}{\mathrm{x8}}{,}{\mathrm{x9}}{,}{\mathrm{x10}}\right]{,}{V}{,}{\mathrm{grading}}{=}\left[{-}{3}{,}{-}{2}{,}{-}{1}{,}{-}{1}{,}{0}{,}{0}{,}{1}{,}{1}{,}{2}{,}{3}\right]\right)$
 ${\mathrm{frame name: V}}$ (2.8)

The command Adjoint gives the adjoint representation for $\mathrm{sp4}.$

 V > ${\mathrm{ρ1}}{≔}{\mathrm{Adjoint}}{}\left({\mathrm{sp4}}{,}{\mathrm{representationspace}}{=}{V}\right)$
 ${\mathrm{ρ1}}{≔}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {0}& {0}& {2}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {2}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {2}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{cccccccccc}{0}& {-2}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {-1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {2}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {2}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e5}}{,}\left[\begin{array}{cccccccccc}{-2}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {2}\end{array}\right]\right]{,}\left[{\mathrm{e6}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-2}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {2}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e7}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {-1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-2}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-1}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {2}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e8}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-2}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e9}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-2}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-1}& {-1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-2}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e10}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-2}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.9)

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 $N$. )

 sp4 > ${\mathrm{ρ2}}{≔}{\mathrm{RestrictedRepresentation}}{}\left({\mathrm{ρ1}}{,}{N}\right)$
 ${\mathrm{ρ2}}{≔}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {0}& {0}& {2}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {2}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {2}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{cccccccccc}{0}& {-2}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {-1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {2}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{cccccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {2}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.10)

Step 5. Initialize the Lie algebra with coefficients in the adjoint representation. Call it $\mathrm{sp4V}$ and label the basis vectors  and the basis 1-forms .

 N > ${\mathrm{DGsetup}}{}\left({\mathrm{ρ1}}{,}{\mathrm{sp4V}}{,}\left['{X}'\right]{,}\left['{\mathrm{\xi }}'\right]\right)$
 ${\mathrm{Lie algebra with coefficients: sp4V}}$ (2.11)

Initialize the Lie algebra $N$ 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 $N$ in This information is provided by the keyword argument ambientalgebra.

 sp4V > ${\mathrm{DGsetup}}{}\left({\mathrm{ρ2}}{,}{\mathrm{NV}}{,}\left['{\mathrm{O}}'\right]{,}\left['{o}'\right]{,}{\mathrm{ambientalgebra}}{=}{\mathrm{sp4V}}\right)$
 ${\mathrm{Lie algebra with coefficients: NV}}$ (2.12)

Example 1.

Here are some sample calculations of the Kostant co-differential.

 NV > ${\mathrm{\alpha }}{≔}{\mathrm{evalDG}}{}\left({\mathrm{x1}}{}{\mathrm{o2}}\right)$
 ${\mathrm{\alpha }}{≔}{\mathrm{x1}}{}{\mathrm{o2}}$ (2.13)
 NV > ${\mathrm{KostantCodifferential}}{}\left({\mathrm{\alpha }}\right)$
 $\frac{{\mathrm{x3}}}{{12}}$ (2.14)
 NV > ${\mathrm{\beta }}{≔}{\mathrm{evalDG}}{}\left({\mathrm{x3}}{}{\mathrm{o2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o3}}\right)$
 ${\mathrm{\beta }}{≔}\left({\mathrm{x3}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}$ (2.15)
 NV > ${\mathrm{KostantCodifferential}}{}\left({\mathrm{\beta }}\right)$
 ${-}\left(\frac{{\mathrm{x3}}}{{12}}{}{\mathrm{o1}}\right){-}\left(\left(\frac{{\mathrm{x5}}}{{12}}{-}\frac{{\mathrm{x6}}}{{12}}\right){}{\mathrm{o2}}\right){-}\left(\frac{{\mathrm{x8}}}{{6}}{}{\mathrm{o3}}\right)$ (2.16)

Let's do this last calculation directly from the definition. For this we need the Killing form and its inverse.

 NV > ${B}{≔}{\mathrm{KillingForm}}{}\left({\mathrm{sp4V}}\right)$
 ${B}{≔}\left({6}{}{\mathrm{ξ1}}\right){}{\mathrm{ξ10}}{+}\left({12}{}{\mathrm{ξ2}}\right){}{\mathrm{ξ9}}{+}\left({12}{}{\mathrm{ξ3}}\right){}{\mathrm{ξ7}}{+}\left({6}{}{\mathrm{ξ4}}\right){}{\mathrm{ξ8}}{+}\left({12}{}{\mathrm{ξ5}}\right){}{\mathrm{ξ5}}{+}\left({12}{}{\mathrm{ξ6}}\right){}{\mathrm{ξ6}}{+}\left({12}{}{\mathrm{ξ7}}\right){}{\mathrm{ξ3}}{+}\left({6}{}{\mathrm{ξ8}}\right){}{\mathrm{ξ4}}{+}\left({12}{}{\mathrm{ξ9}}\right){}{\mathrm{ξ2}}{+}\left({6}{}{\mathrm{ξ10}}\right){}{\mathrm{ξ1}}$ (2.17)
 NV > ${\mathrm{invB}}{≔}{\mathrm{InverseMetric}}{}\left({B}\right)$
 ${\mathrm{invB}}{≔}\left(\frac{{1}}{{6}}{}{\mathrm{X1}}\right){}{\mathrm{X10}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X2}}\right){}{\mathrm{X9}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X3}}\right){}{\mathrm{X7}}{+}\left(\frac{{1}}{{6}}{}{\mathrm{X4}}\right){}{\mathrm{X8}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X5}}\right){}{\mathrm{X5}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X6}}\right){}{\mathrm{X6}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X7}}\right){}{\mathrm{X3}}{+}\left(\frac{{1}}{{6}}{}{\mathrm{X8}}\right){}{\mathrm{X4}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X9}}\right){}{\mathrm{X2}}{+}\left(\frac{{1}}{{6}}{}{\mathrm{X10}}\right){}{\mathrm{X1}}$ (2.18)

Re-define as a form on sp4R.

 sp4V > ${\mathrm{β1}}{≔}{\mathrm{evalDG}}{}\left({\mathrm{x3}}{}{\mathrm{ξ2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{ξ3}}\right)$
 ${\mathrm{β1}}{≔}\left({\mathrm{x3}}{}{\mathrm{ξ2}}\right){}{\bigwedge }{}{\mathrm{ξ3}}$ (2.19)

Use the inverse of the Killing form to convert to a multi-vector $X$:

 sp4V > ${X}{≔}{\mathrm{RaiseLowerIndices}}{}\left({\mathrm{invB}}{,}{\mathrm{β1}}{,}\left[{1}{,}{2}\right]\right)$
 ${X}{≔}{-}\left(\left(\frac{{\mathrm{x3}}}{{144}}{}{\mathrm{X7}}\right){}{\bigwedge }{}{\mathrm{X9}}\right)$ (2.20)

Take the co-differential of $X$.

 sp4V > ${Y}{≔}{\mathrm{Codifferential}}{}\left({X}\right)$
 ${Y}{≔}{-}\left(\frac{{\mathrm{x8}}}{{72}}{}{\mathrm{X7}}\right){-}\left(\left(\frac{{\mathrm{x5}}}{{144}}{-}\frac{{\mathrm{x6}}}{{144}}\right){}{\mathrm{X9}}\right){-}\left(\frac{{\mathrm{x3}}}{{72}}{}{\mathrm{X10}}\right)$ (2.21)

Lower the indices of $Y$ with the Killing form.

 sp4V > ${\mathrm{RaiseLowerIndices}}{}\left({B}{,}{Y}{,}\left[{1}\right]\right)$
 ${-}\left(\frac{{\mathrm{x3}}}{{12}}{}{\mathrm{ξ1}}\right){-}\left(\left(\frac{{\mathrm{x5}}}{{12}}{-}\frac{{\mathrm{x6}}}{{12}}\right){}{\mathrm{ξ2}}\right){-}\left(\frac{{\mathrm{x8}}}{{6}}{}{\mathrm{ξ3}}\right)$ (2.22)

Example 2.

The square of the Kostant co-differential vanishes.

 NV > ${\mathrm{α2}}{≔}{\mathrm{evalDG}}{}\left({\mathrm{x3}}{}\left({\mathrm{o1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o3}}\right)$
 ${\mathrm{α2}}{≔}\left(\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}$ (2.23)
 NV > ${\mathrm{β2}}{≔}{\mathrm{KostantCodifferential}}{}\left({\mathrm{\beta }}\right)$
 ${\mathrm{β2}}{≔}{-}\left(\frac{{\mathrm{x3}}}{{12}}{}{\mathrm{o1}}\right){-}\left(\left(\frac{{\mathrm{x5}}}{{12}}{-}\frac{{\mathrm{x6}}}{{12}}\right){}{\mathrm{o2}}\right){-}\left(\frac{{\mathrm{x8}}}{{6}}{}{\mathrm{o3}}\right)$ (2.24)
 sp4V > ${\mathrm{KostantCodifferential}}{}\left({\mathrm{β2}}\right)$
 ${0}$ (2.25)

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) .

 sp4 > ${g}{≔}{\mathrm{evalDG}}{}\left({6}{}{\mathrm{o1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o1}}{+}{12}{}{\mathrm{o2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o2}}{+}{12}{}{\mathrm{o3}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o3}}{+}{6}{}{\mathrm{o4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o4}}\right)$
 ${g}{≔}\left({6}{}{\mathrm{o1}}\right){}{\mathrm{o1}}{+}\left({12}{}{\mathrm{o2}}\right){}{\mathrm{o2}}{+}\left({12}{}{\mathrm{o3}}\right){}{\mathrm{o3}}{+}\left({6}{}{\mathrm{o4}}\right){}{\mathrm{o4}}$ (2.26)
 sp4 > ${\mathrm{gV}}{≔}{\mathrm{evalDG}}{}\left({\mathrm{dx1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx1}}{+}{2}{}{\mathrm{dx2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx2}}{+}{2}{}{\mathrm{dx3}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx3}}{+}{\mathrm{dx4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx4}}{+}{2}{}{\mathrm{dx5}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx5}}{+}{2}{}{\mathrm{dx6}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx6}}{+}{2}{}{\mathrm{dx7}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx7}}{+}{\mathrm{dx8}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx8}}{+}{2}{}{\mathrm{dx9}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx9}}{+}{\mathrm{dx10}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{dx10}}\right)$
 ${\mathrm{gV}}{≔}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}\left({2}{}{\mathrm{dx2}}\right){}{\mathrm{dx2}}{+}\left({2}{}{\mathrm{dx3}}\right){}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}{+}\left({2}{}{\mathrm{dx5}}\right){}{\mathrm{dx5}}{+}\left({2}{}{\mathrm{dx6}}\right){}{\mathrm{dx6}}{+}\left({2}{}{\mathrm{dx7}}\right){}{\mathrm{dx7}}{+}{\mathrm{dx8}}{}{\mathrm{dx8}}{+}\left({2}{}{\mathrm{dx9}}\right){}{\mathrm{dx9}}{+}{\mathrm{dx10}}{}{\mathrm{dx10}}$ (2.27)
 sp4 > ${\mathrm{α3}}{≔}{\mathrm{evalDG}}{}\left({\mathrm{x3}}{}{\mathrm{o1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o2}}\right)$
 ${\mathrm{α3}}{≔}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}$ (2.28)
 NV > ${\mathrm{β3}}{≔}{\mathrm{evalDG}}{}\left({\mathrm{x8}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&wedge}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o1}}\right)$
 ${\mathrm{β3}}{≔}{\mathrm{x8}}{}{\mathrm{o1}}$ (2.29)

Here is the left-hand side of the adjoint equation.

 NV > ${\mathrm{FormInnerProduct}}{}\left({g}{,}{\mathrm{gV}}{,}{\mathrm{KostantCodifferential}}{}\left({\mathrm{α3}}\right){,}{\mathrm{β3}}\right)$
 $\frac{{1}}{{36}}$ (2.30)

Here is the right-hand side of the adjoint equation.

 NV > ${\mathrm{FormInnerProduct}}{}\left({g}{,}{\mathrm{gV}}{,}{\mathrm{α3}}{,}{\mathrm{ExteriorDerivative}}{}\left({\mathrm{β3}}\right)\right)$
 $\frac{{1}}{{36}}$ (2.31)

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.

 V > ${\mathrm{ChangeFrame}}{}\left({\mathrm{NV}}\right)$
 ${V}$ (2.32)
 V > ${\mathrm{Λ1}}{≔}{\mathrm{RelativeChains}}{}\left(\left[{}\right]{,}{1}{,}{\mathrm{weight}}{=}{1}\right)$
 ${\mathrm{Λ1}}{≔}\left[{\mathrm{x2}}{}{\mathrm{o1}}{,}{\mathrm{x3}}{}{\mathrm{o2}}{,}{\mathrm{x4}}{}{\mathrm{o2}}{,}{\mathrm{x5}}{}{\mathrm{o3}}{,}{\mathrm{x5}}{}{\mathrm{o4}}{,}{\mathrm{x6}}{}{\mathrm{o3}}{,}{\mathrm{x6}}{}{\mathrm{o4}}\right]$ (2.33)
 V > ${\mathrm{Λ2}}{≔}{\mathrm{RelativeChains}}{}\left(\left[{}\right]{,}{2}{,}{\mathrm{weight}}{=}{1}\right)$
 ${\mathrm{Λ2}}{≔}\left[\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x2}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x2}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x3}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x4}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right]$ (2.34)
 V > ${\mathrm{LHS}}{≔}{\mathrm{FormInnerProduct}}{}\left({g}{,}{\mathrm{gV}}{,}{\mathrm{KostantCodifferential}}{}\left({\mathrm{Λ2}}\right){,}{\mathrm{Λ1}}\right)$
 ${\mathrm{LHS}}{≔}\left[\begin{array}{ccccccc}\frac{{1}}{{36}}& {0}& {0}& \frac{{1}}{{36}}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& \frac{{1}}{{18}}& {0}& {0}\\ {-}\frac{{1}}{{36}}& {0}& \frac{{1}}{{72}}& \frac{{1}}{{72}}& {0}& \frac{{1}}{{72}}& {0}\\ {0}& {-}\frac{{1}}{{36}}& {0}& {0}& \frac{{1}}{{36}}& {0}& \frac{{1}}{{36}}\\ {0}& \frac{{1}}{{36}}& {0}& {0}& \frac{{1}}{{36}}& {0}& {-}\frac{{1}}{{36}}\\ {0}& {0}& \frac{{1}}{{72}}& {0}& {0}& {-}\frac{{1}}{{36}}& {0}\end{array}\right]$ (2.35)
 NV > ${\mathrm{RHS}}{≔}{\mathrm{FormInnerProduct}}{}\left({g}{,}{\mathrm{gV}}{,}{\mathrm{Λ2}}{,}{\mathrm{ExteriorDerivative}}{}\left({\mathrm{Λ1}}\right)\right)$
 ${\mathrm{RHS}}{≔}\left[\begin{array}{ccccccc}\frac{{1}}{{36}}& {0}& {0}& \frac{{1}}{{36}}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& \frac{{1}}{{18}}& {0}& {0}\\ {-}\frac{{1}}{{36}}& {0}& \frac{{1}}{{72}}& \frac{{1}}{{72}}& {0}& \frac{{1}}{{72}}& {0}\\ {0}& {-}\frac{{1}}{{36}}& {0}& {0}& \frac{{1}}{{36}}& {0}& \frac{{1}}{{36}}\\ {0}& \frac{{1}}{{36}}& {0}& {0}& \frac{{1}}{{36}}& {0}& {-}\frac{{1}}{{36}}\\ {0}& {0}& \frac{{1}}{{72}}& {0}& {0}& {-}\frac{{1}}{{36}}& {0}\end{array}\right]$ (2.36)

The equality of these matrices verifies the adjoint equation for given lists of forms.

 NV > ${\mathrm{LHS}}{-}{\mathrm{RHS}}$
 $\left[\begin{array}{ccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]$ (2.37)

Example 4.

Here are some sample calculations of the Kostant Laplacian.

A scalar:

 NV > ${\mathrm{ChangeFrame}}{}\left({\mathrm{NV}}\right)$
 ${V}$ (2.38)
 NV > ${\mathrm{α4}}{≔}{\mathrm{x6}}$
 ${\mathrm{α4}}{≔}{\mathrm{x6}}$ (2.39)
 NV > ${\mathrm{KostantLaplacian}}{}\left({\mathrm{α4}}\right)$
 $\frac{{\mathrm{x6}}}{{2}}$ (2.40)

A 2-form:

 NV > ${\mathrm{β4}}{≔}{\mathrm{evalDG}}{}\left({\mathrm{x3}}{}{\mathrm{o1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o4}}\right)$
 ${\mathrm{β4}}{≔}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}$ (2.41)
 NV > ${\mathrm{KostantLaplacian}}{}\left({\mathrm{β4}}\right)$
 $\left(\frac{{\mathrm{x3}}}{{2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}$ (2.42)

A 3-form:

 NV > ${\mathrm{δ4}}{≔}{\mathrm{evalDG}}{}\left({\mathrm{x3}}{}\left({\mathrm{o1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o3}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{o4}}\right)$
 ${\mathrm{δ4}}{≔}\left(\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}$ (2.43)
 NV > ${\mathrm{KostantLaplacian}}{}\left({\mathrm{δ4}}\right)$
 $\left(\left(\frac{{2}{}{\mathrm{x3}}}{{3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}$ (2.44)

Example 5.

We calculate the second cohomology

 V > ${\mathrm{ChangeFrame}}{}\left({\mathrm{NV}}\right)$
 ${\mathrm{NV}}$ (2.45)

Here are the 2-chains

 V > ${\mathrm{Λ2}}{≔}{\mathrm{RelativeChains}}{}\left(\left[{}\right]{,}{2}\right)$
 ${\mathrm{Λ2}}{≔}\left[\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x1}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x1}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x1}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x2}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x2}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x2}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x3}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x3}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x3}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x4}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x4}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x4}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x5}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x5}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x5}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x5}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x5}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x5}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x6}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x6}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x6}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x6}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x6}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x6}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x7}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x7}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x7}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x7}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x8}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x8}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x8}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x8}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x9}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x9}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x9}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x9}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x9}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x9}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x10}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x10}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x10}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x10}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x10}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x10}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right]$ (2.46)

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

 V > ${B}{≔}{\mathrm{KillingForm}}{}\left({\mathrm{sp4V}}\right)$
 ${B}{≔}\left({6}{}{\mathrm{ξ1}}\right){}{\mathrm{ξ10}}{+}\left({12}{}{\mathrm{ξ2}}\right){}{\mathrm{ξ9}}{+}\left({12}{}{\mathrm{ξ3}}\right){}{\mathrm{ξ7}}{+}\left({6}{}{\mathrm{ξ4}}\right){}{\mathrm{ξ8}}{+}\left({12}{}{\mathrm{ξ5}}\right){}{\mathrm{ξ5}}{+}\left({12}{}{\mathrm{ξ6}}\right){}{\mathrm{ξ6}}{+}\left({12}{}{\mathrm{ξ7}}\right){}{\mathrm{ξ3}}{+}\left({6}{}{\mathrm{ξ8}}\right){}{\mathrm{ξ4}}{+}\left({12}{}{\mathrm{ξ9}}\right){}{\mathrm{ξ2}}{+}\left({6}{}{\mathrm{ξ10}}\right){}{\mathrm{ξ1}}$ (2.47)
 sp4V > ${\mathrm{invB}}{≔}{\mathrm{InverseMetric}}{}\left({B}\right)$
 ${\mathrm{invB}}{≔}\left(\frac{{1}}{{6}}{}{\mathrm{X1}}\right){}{\mathrm{X10}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X2}}\right){}{\mathrm{X9}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X3}}\right){}{\mathrm{X7}}{+}\left(\frac{{1}}{{6}}{}{\mathrm{X4}}\right){}{\mathrm{X8}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X5}}\right){}{\mathrm{X5}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X6}}\right){}{\mathrm{X6}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X7}}\right){}{\mathrm{X3}}{+}\left(\frac{{1}}{{6}}{}{\mathrm{X8}}\right){}{\mathrm{X4}}{+}\left(\frac{{1}}{{12}}{}{\mathrm{X9}}\right){}{\mathrm{X2}}{+}\left(\frac{{1}}{{6}}{}{\mathrm{X10}}\right){}{\mathrm{X1}}$ (2.48)

Method 1. Here we shall calculate  using the DGNullSpace and IntersectSubspaces commands.

Here is.

 NV > ${\mathrm{KerD}}{≔}{\mathrm{DGNullSpace}}{}\left({\mathrm{ExteriorDerivative}}{,}{\mathrm{Λ2}}{,}{\mathrm{method}}{=}{"real"}\right)$
 ${\mathrm{KerD}}{≔}\left[\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{+}\left(\left({-}\frac{{\mathrm{x6}}}{{2}}{+}\frac{{\mathrm{x5}}}{{2}}\right){}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({\mathrm{x3}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x1}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x1}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x1}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{+}\left(\left({\mathrm{x5}}{-}{\mathrm{x6}}\right){}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{+}\left({\mathrm{x7}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left(\left({-}{3}{}{\mathrm{x6}}{+}{\mathrm{x5}}\right){}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x2}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x2}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({\mathrm{x3}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x2}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{+}\left({2}{}{\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{-}\left(\left({\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){-}\left(\left({\mathrm{x9}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{+}\left(\left({\mathrm{x5}}{+}{\mathrm{x6}}\right){}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{-}\left(\left({\mathrm{x7}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({\mathrm{x5}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({3}{}{\mathrm{x8}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x3}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{-}\left(\left({\mathrm{x6}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x3}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{+}\left(\frac{{\mathrm{x6}}}{{2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{+}\left(\frac{{3}{}{\mathrm{x7}}}{{2}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({2}{}{\mathrm{x6}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{-}\left(\left({\mathrm{x7}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x4}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x4}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({\mathrm{x6}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x4}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x5}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{+}\left({\mathrm{x9}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{-}\left(\left({\mathrm{x10}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x5}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{+}\left({\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x5}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({\mathrm{x9}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x6}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{-}\left(\left({2}{}{\mathrm{x8}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){+}\left({\mathrm{x7}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({3}{}{\mathrm{x9}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){+}\left({\mathrm{x10}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x6}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({4}{}{\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){+}\left({\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({\mathrm{x9}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x6}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({\mathrm{x8}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x7}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{-}\left(\left({\mathrm{x9}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){-}\left(\left({2}{}{\mathrm{x10}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){,}\left({\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right]$ (2.49)

Here is First we make a small procedure which fixes the Killing form and its inverse in the arguments for KostantCodifferential.

 sp4V > ${\mathrm{KCoD}}{≔}{\mathrm{\theta }}{↦}{\mathrm{KostantCodifferential}}{}\left({\mathrm{\theta }}{,}{B}{,}{\mathrm{invB}}\right)$
 ${\mathrm{KCoD}}{≔}{\mathrm{\theta }}{↦}{\mathrm{LieAlgebras}}{:-}{\mathrm{KostantCodifferential}}{}\left({\mathrm{\theta }}{,}{B}{,}{\mathrm{invB}}\right)$ (2.50)
 sp4V > ${\mathrm{KerKCoD}}{≔}{\mathrm{DGNullSpace}}{}\left({\mathrm{KCoD}}{,}{\mathrm{Λ2}}{,}{\mathrm{method}}{=}{"real"}\right)$
 ${\mathrm{KerKCoD}}{≔}\left[\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{-}\left(\left({\mathrm{x3}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){-}\left(\left({\mathrm{x4}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){-}\left(\left(\left(\frac{{\mathrm{x6}}}{{2}}{+}\frac{{3}{}{\mathrm{x5}}}{{2}}\right){}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x1}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({\mathrm{x2}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){-}\left(\left({\mathrm{x3}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{-}\left(\left(\left({\mathrm{x5}}{+}{\mathrm{x6}}\right){}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){-}\left(\left({\mathrm{x7}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x2}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({2}{}{\mathrm{x3}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){-}\left(\left({2}{}{\mathrm{x4}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){-}\left(\left(\left({\mathrm{x5}}{+}{\mathrm{x6}}\right){}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{+}\left({2}{}{\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{+}\left({\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({\mathrm{x9}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{-}\left(\left(\left({\mathrm{x5}}{-}{\mathrm{x6}}\right){}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){+}\left({\mathrm{x7}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x3}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left(\frac{{2}{}{\mathrm{x6}}}{{3}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){+}\left(\frac{{4}{}{\mathrm{x8}}}{{3}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{-}\left(\left({\mathrm{x6}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){,}\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({2}{}{\mathrm{x6}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){-}\left(\left({\mathrm{x7}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x5}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{+}\left({\mathrm{x9}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{+}\left({3}{}{\mathrm{x10}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left(\left({3}{}{\mathrm{x6}}{+}{\mathrm{x5}}\right){}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{-}\left(\left({2}{}{\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){,}\left({\mathrm{x5}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{-}\left(\left({\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){+}\left({\mathrm{x9}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left(\left({-}\frac{{\mathrm{x6}}}{{3}}{+}{\mathrm{x5}}\right){}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left(\frac{{2}{}{\mathrm{x8}}}{{3}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x6}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{+}\left({\mathrm{x9}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{+}\left({\mathrm{x10}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x6}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({4}{}{\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{+}\left({\mathrm{x7}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({\mathrm{x9}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x7}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{+}\left({2}{}{\mathrm{x10}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x7}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x7}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{+}\left({2}{}{\mathrm{x9}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}{+}\left({2}{}{\mathrm{x10}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x8}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x8}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{-}\left(\left({\mathrm{x9}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){-}\left(\left({\mathrm{x10}}{}{\mathrm{o3}}\right){}{\bigwedge }{}{\mathrm{o4}}\right){,}\left({\mathrm{x8}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x9}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x9}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{-}\left(\left({2}{}{\mathrm{x10}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o3}}\right){,}\left({\mathrm{x9}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x9}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x10}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o2}}{,}\left({\mathrm{x10}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x10}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o4}}{,}\left({\mathrm{x10}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right]$ (2.51)

Now calculate the intersection .

 NV > ${\mathrm{IntersectSubspaces}}{}\left(\left[{\mathrm{KerD}}{,}{\mathrm{KerKCoD}}\right]{,}{\mathrm{method}}{=}{"real"}\right)$
 $\left[\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right]$ (2.52)

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.

 NV > ${\mathrm{KBox}}{≔}{\mathrm{\theta }}{↦}{\mathrm{KostantLaplacian}}{}\left({\mathrm{\theta }}{,}{B}{,}{\mathrm{invB}}\right)$
 ${\mathrm{KBox}}{≔}{\mathrm{\theta }}{↦}{\mathrm{LieAlgebras}}{:-}{\mathrm{KostantLaplacian}}{}\left({\mathrm{\theta }}{,}{B}{,}{\mathrm{invB}}\right)$ (2.53)
 NV > ${\mathrm{DGNullSpace}}{}\left({\mathrm{KBox}}{,}{\mathrm{Λ2}}{,}{\mathrm{method}}{=}{"real"}\right)$
 $\left[\left({\mathrm{x4}}{}{\mathrm{o1}}\right){}{\bigwedge }{}{\mathrm{o3}}{,}\left({\mathrm{x8}}{}{\mathrm{o2}}\right){}{\bigwedge }{}{\mathrm{o4}}\right]$ (2.54)
 NV > ${}$