TanakaProlongation - Maple Help

LieAlgebras[TanakaProlongation] - calculate the Tanaka prolongation, to a specified order, of a graded nilpotent Lie algebra

Calling Sequences

TanakaProlongation(alg, k,pralg)

Parameters

alg    - a name or string, the name of an initialized graded, nilpotent Lie algebra $\mathrm{𝔤}$

k      - a positive integer, the number of times the Lie algebra $\mathrm{𝔤}$ is to be prolonged

pralg  -  an unassigned name or string, the name to be given to the Tanaka prolongation of $\mathrm{𝔤}$

Description

 • Let be a negatively graded Lie algebra, . The Tanaka prolongation of $𝔪$ is a graded (possibly infinite dimensional) Lie algebra

with

The Tanaka prolongation is computed inductively in terms of the partial prolongations

.

Here  = ${\mathrm{𝔤}}_{\ell }$ for 0 and for $\mathrm{ℓ}\ge 0$, is defined as the derivations of the Lie algebra $\mathrm{𝔪}$which shift the grading degree by In particular, the weight 0 component is precisely the gradation preserving derivations (or infinitesimal automorphisms) of $\mathrm{𝔪}$. If for some then all subsequent components forand the process of Tanaka prolongation is said to terminate at order $q$.

 • The command TanakaProlongation requires that the basis for the Lie algebra be adapted to the grading in the sense that

 • The command TanakaProlongation(alg, k, pralg) returns the structure equations for the $\mathrm{ℓ}$-th prolongation

where $\mathrm{ℓ}$ = min$($ and where $q$ is the smallest non-negative integer such that

 • With infolevel[TanakaProlongation] = 2, information on the sequential partial prolongations of the algebra is displayed.
 • We note 3 important properties of this prolongation procedure. First, let be a distribution on a manifold about each point of $M$, $\mathrm{𝒟}$ can be described as the span of a finite number of vector fields. We recall that the infinitesimal symmetry algebra sym$\left(\mathrm{𝒟}\right)$of $𝒟$ is the Lie algebra of vector fields such that .
 1 Let $G$ be the nilpotent Lie group with Lie algebra $\mathrm{𝔪}\mathit{.}$ Let  be the left (or right) invariant vector fields on $G$. The structure equations for these vector fields coincide with the structure equations for the basis for the Lie algebra . Because the algebra $\mathrm{𝔪}$ is a nilpotent algebra, the Lie group and the vector fields can be explicitly constructed using the LieGroup and InvariantGeometricObjectFields in the GroupActions package. Set $\mathrm{𝒟}\mathit{=}$This is the distribution corresponding to the ${𝔤}_{-1}$ component of $𝔪$. This distribution has 2 remarkable properties: (1) its symbol algebra coincides with $\mathrm{𝔪}\mathit{.}$ and (2) the symmetry algebra sym$\left(\mathrm{𝒟}\right)$ is isomorphic, as an abstract Lie algebra, to the Tanaka prolongation $𝔤\left(𝔪\right)$.
 2 There is an important criterion which implies that the prolongation $𝔤\left(𝔪\right)$ is infinite dimensional. Calculate the 0-th order prolongation 

.

Let be the linear span of the adjoint matrices $\mathrm{ad}\left(x\right)$ for , restricted to $𝔪$. If contains a rank 1 matrix then the Tanaka prolongation is infinite. This test can be implemented with the command Rank1Elements.

 3 Let be a semi-simple Lie algebra with roots $\mathrm{Δ}$ and positive roots ${\mathrm{Δ}}^{+}$. For the sake of simplicity, let us assume that ${\mathrm{𝔤}}_{\mathrm{ss}}$ is a split real form so that the roots are all real vectors and the corresponding root space decomposition is real. Then every subset of ${\mathrm{Δ}}^{+}$defines a (symmetric) grading of say

These gradations can be constructed with the GradeSemiSimpleLieAlgebra command. Let  be the negatively graded part of this decomposition of ${𝔤}_{\mathrm{ss}}.$ Then, with the exception of a few well-noted cases, the Tanaka prolongation of reproduces the semi-simple Lie algebra ${𝔤}_{\mathrm{ss}}$, that is, ${𝔤}_{\mathrm{ss}}={𝔤}^{{\mathrm{μ})}\left(𝔪\right)$ and .

 • An excellent reference on the Tanaka prolongation of a Lie algebra is K. Yamaguchi, Differential Systems associated with Simple Graded Lie Algebras, Advanced Studies in Pure Mathematics, 22, 413-294 (1993).

.Examples

 > with(DifferentialGeometry): with(LieAlgebras):
 > interface(rtablesize = 15):

Example 1.

Define a 5-dimensional graded nilpotent Lie algebra = alg1 with grading and dim dimdim The keyword argument grading = [-3,-3,-2,-1,-1] is used to specify the grading.

Here are the structure equations for this Lie algebra.

 > StrEq := [[x3, x4] = -x1, [x3, x5] = -x2, [x4, x5] = x3], [x1, x2, x3, x4 ,x5];
 ${\mathrm{StrEq}}{≔}\left[\left[{\mathrm{x3}}{,}{\mathrm{x4}}\right]{=}{-}{\mathrm{x1}}{,}\left[{\mathrm{x3}}{,}{\mathrm{x5}}\right]{=}{-}{\mathrm{x2}}{,}\left[{\mathrm{x4}}{,}{\mathrm{x5}}\right]{=}{\mathrm{x3}}\right]{,}\left[{\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}{,}{\mathrm{x5}}\right]$ (1)
 > LD := LieAlgebraData(StrEq, alg1, grading = [-3, -3, -2, -1, -1]):
 > DGsetup(LD);
 ${\mathrm{Lie algebra: alg1}}$ (2)

Calculate the Tanaka prolongation for alg1. With infolevel[TanakaProlongation] = 2, information on the sequential partial prolongations is displayed.

 alg1 > infolevel[TanakaProlongation] := 2:
 alg1 > prLD := TanakaProlongation(alg1, 5, pralg1):
 m:    [[e1, e2], [e3], [e4, e5]]    [-3, -2, -1] g^(0):    [[e1, e2], [e3], [e4, e5], [e6, e7, e8, e9]]    [-3, -2, -1, 0] g^(1):    [[e1, e2], [e3], [e4, e5], [e6, e7, e8, e9], [e10, e11]]    [-3, -2, -1, 0, 1] g^(2):    [[e1, e2], [e3], [e4, e5], [e6, e7, e8, e9], [e10, e11], [e12]]    [-3, -2, -1, 0, 1, 2] g^(3):    [[e1, e2], [e3], [e4, e5], [e6, e7, e8, e9], [e10, e11], [e12], [e13, e14]]    [-3, -2, -1, 0, 1, 2, 3]

The first 3 lines produced by the infolevel command display the gradation of the original algebra (the first argument in the calling sequence). We see from the next 3 lines that the 0-th order prolongation is a 9 dimensional Lie algebra and that the vectors define the weight 0 vectors. The next 3 lines describe the 1st prolongation and so on. Finally we asked for the 5th prolongation of the algebra (with the second argument in the calling sequence set to 5) but we see that the Tanaka prolongation terminated at order 3. Thus the Tanaka prolongation of the nilpotent algebra alg1 is 14-dimensional.

Now initialize the prolonged algebra.

 alg1 > DGsetup(prLD);
 ${\mathrm{Lie algebra: pralg1}}$ (3)

We can use the command DGinfo to display the grading of this algebra and the Query command to verify it is a valid gradation.

 pralg1 > G := Tools:-DGinfo( "table", "Grading");
 ${G}{≔}{table}{}\left(\left[{-1}{=}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{,}{0}{=}\left[{\mathrm{e6}}{,}{\mathrm{e7}}{,}{\mathrm{e8}}{,}{\mathrm{e9}}\right]{,}{-2}{=}\left[{\mathrm{e3}}\right]{,}{1}{=}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{,}{-3}{=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{,}{2}{=}\left[{\mathrm{e12}}\right]{,}{3}{=}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]\right]\right)$ (4)
 ${\mathrm{true}}$ (5)

This prolongation algebra is semi-simple and, indeed, one can use the commands CartanSubalgebra, RootSpaceDecomposition, PositiveRoots, SimpleRoots, CartanMatrix, CartanMatrixToStandardForm to identify this Lie algebra as the split real form of the exceptional Lie algebra ${g}_{2}$.

Before continuing to the next example, reset the infolevel.

 newalg > infolevel[TanakaProlongation] := 0:

Example 2.

We use the Lie algebra from Example 1 to show that the Tanaka prolongations can be computed one order at a time.

Calculate the prolongation of alg1 to order 1 and initialize.

 alg1 > LD2a := TanakaProlongation(alg1, 2, P1);
 ${\mathrm{LD2a}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}\frac{{\mathrm{e3}}}{{3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}\frac{{\mathrm{e3}}}{{3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}\frac{{4}{}{\mathrm{e5}}}{{3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e11}}\right]{=}\frac{{4}{}{\mathrm{e4}}}{{3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}\frac{{2}{}{\mathrm{e4}}}{{3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}\frac{{\mathrm{e4}}}{{3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}\frac{{\mathrm{e5}}}{{3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}\frac{{2}{}{\mathrm{e5}}}{{3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}\frac{{2}{}{\mathrm{e10}}}{{3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}\frac{{\mathrm{e11}}}{{3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}\frac{{\mathrm{e10}}}{{3}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{-}\frac{{2}{}{\mathrm{e11}}}{{3}}\right]$ (6)
 alg1 > DGsetup(LD2a);
 ${\mathrm{Lie algebra: P1}}$ (7)

At this point the prolongation has dimension 11. To prolong further, it is not necessary to begin the calculation anew. Instead one can continue the prolongation using P1.

 pr0 > LD2b := TanakaProlongation(P1, 4, P2);
 ${\mathrm{LD2b}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{-}\frac{{\mathrm{e9}}}{{2}}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{-}\frac{{\mathrm{e8}}}{{2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e9}}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}\frac{{\mathrm{e3}}}{{3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}\frac{{\mathrm{e3}}}{{3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}\frac{{4}{}{\mathrm{e5}}}{{3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e11}}\right]{=}\frac{{4}{}{\mathrm{e4}}}{{3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e9}}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{-}\frac{{\mathrm{e10}}}{{2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}\frac{{2}{}{\mathrm{e4}}}{{3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}\frac{{\mathrm{e4}}}{{3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{-}\frac{{\mathrm{e12}}}{{2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}\frac{{\mathrm{e5}}}{{3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}\frac{{2}{}{\mathrm{e5}}}{{3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}\frac{{2}{}{\mathrm{e10}}}{{3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}\frac{{\mathrm{e11}}}{{3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e12}}\right]{=}{-}\frac{{\mathrm{e12}}}{{3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{-}\frac{{\mathrm{e14}}}{{2}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e13}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}\frac{{\mathrm{e10}}}{{3}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{-}\frac{{2}{}{\mathrm{e11}}}{{3}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}\frac{{\mathrm{e12}}}{{3}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}\frac{{4}{}{\mathrm{e12}}}{{3}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e13}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e14}}\right]$ (8)
 P1 > DGsetup(LD2b);
 ${\mathrm{Lie algebra: P2}}$ (9)

We check that the two Tanaka prolongations -- pralg1 (which was calculated all in one go) and P2 (which was calculated in two steps) coincide. We do this by showing that the identity matrix defines a Lie algebra homomorphism.

 pr3 > phi := Transformation(pralg1, P2, LinearAlgebra:-IdentityMatrix(14));
 ${\mathrm{\phi }}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e1}}\right]{,}\left[{\mathrm{e2}}{,}{\mathrm{e2}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e3}}\right]{,}\left[{\mathrm{e4}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e5}}{,}{\mathrm{e5}}\right]{,}\left[{\mathrm{e6}}{,}{\mathrm{e6}}\right]{,}\left[{\mathrm{e7}}{,}{\mathrm{e7}}\right]{,}\left[{\mathrm{e8}}{,}{\mathrm{e8}}\right]{,}\left[{\mathrm{e9}}{,}{\mathrm{e9}}\right]{,}\left[{\mathrm{e10}}{,}{\mathrm{e10}}\right]{,}\left[{\mathrm{e11}}{,}{\mathrm{e11}}\right]{,}\left[{\mathrm{e12}}{,}{\mathrm{e12}}\right]{,}\left[{\mathrm{e13}}{,}{\mathrm{e13}}\right]{,}\left[{\mathrm{e14}}{,}{\mathrm{e14}}\right]\right]$ (10)
 pr3 > Query(phi,"Homomorphism");
 ${\mathrm{true}}$ (11)

Example 3.

In this example we define a gradation of the 15-dimensional Lie algebra We calculate the Tanaka prolongation of the negatively graded part and show that the prolongation is isomorphic to The command SimpleLieAlgebraData is used to retrieve the structure equations for

 P2 > LD3a := SimpleLieAlgebraData("sl(4)", sl4a):
 P2 > DGsetup(LD3a);
 ${\mathrm{Lie algebra: sl4a}}$ (12)

Here is the grading we shall use ( It was constructed with the commands GradeSemiSimpleLieAlgebra and SimpleLieAlgebraProperties).

 sl4 > G := table([0 = [e1, e2, e3], 1 = [e4, e8, e12], 2 = [e5, e9], 3 = [e6], -3 = [e13], -2 = [e10, e14], -1 = [e7, e11, e15]]);
 ${G}{≔}{table}{}\left(\left[{-1}{=}\left[{\mathrm{e7}}{,}{\mathrm{e11}}{,}{\mathrm{e15}}\right]{,}{0}{=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]{,}{-2}{=}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{,}{1}{=}\left[{\mathrm{e4}}{,}{\mathrm{e8}}{,}{\mathrm{e12}}\right]{,}{-3}{=}\left[{\mathrm{e13}}\right]{,}{2}{=}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{,}{3}{=}\left[{\mathrm{e6}}\right]\right]\right)$ (13)

Here is the Lie algebra sl4a but now in the basis adapted to this grading.

 sl4 > LD3b := LieAlgebraData(G, sl4);
 ${\mathrm{LD3b}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e9}}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e8}}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e9}}{-}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e15}}\right]{=}{2}{}{\mathrm{e15}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{2}{}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e15}}\right]$ (14)
 sl4 > DGsetup(LD3b);
 ${\mathrm{Lie algebra: sl4}}$ (15)
 $\left[{-3}{,}{-2}{,}{-2}{,}{-1}{,}{-1}{,}{-1}{,}{0}{,}{0}{,}{0}{,}{1}{,}{1}{,}{1}{,}{2}{,}{2}{,}{3}\right]$ (16)

Now we calculate the structure equations for the negatively graded part. We initialize this nilpotent graded Lie algebra with the name M.

 sl4 > LD3b := LieAlgebraData([e1, e2, e3, e4, e5, e6], M, grading = [-3, -2, -2, -1, -1, -1]);
 ${\mathrm{LD3b}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}\right]$ (17)
 sl4 > DGsetup(LD3b);
 ${\mathrm{Lie algebra: M}}$ (18)

Calculate the prolongation of M and initialize the result.

 M > LD3c := TanakaProlongation(M, 4, prM);
 ${\mathrm{LD3c}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e15}}\right]{=}{-}\frac{{\mathrm{e9}}}{{2}}{-}\frac{{\mathrm{e8}}}{{2}}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e8}}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e15}}\right]{=}{-}\frac{{\mathrm{e10}}}{{2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e7}}{+}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e15}}\right]{=}{-}\frac{{\mathrm{e11}}}{{2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e9}}{-}{\mathrm{e8}}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e15}}\right]{=}{-}\frac{{\mathrm{e13}}}{{2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e8}}{+}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e9}}{+}{\mathrm{e7}}{-}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e15}}\right]{=}{-}\frac{{\mathrm{e14}}}{{2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e15}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e13}}\right]{=}{2}{}{\mathrm{e15}}\right]$ (19)
 M > DGsetup(LD3c);
 ${\mathrm{Lie algebra: prM}}$ (20)

We see that the prM is a 15 dimensional Lie algebra with the same grading as the one assigned to sl4.

 $\left[{-3}{,}{-2}{,}{-2}{,}{-1}{,}{-1}{,}{-1}{,}{0}{,}{0}{,}{0}{,}{1}{,}{1}{,}{1}{,}{2}{,}{2}{,}{3}\right]$ (21)

To complete this example we explicitly construct a Lie algebra isomorphism between sl4 and prM. The following matrix defines the most general Lie transformation between these two Lie algebras which preserves the grading.

 M > A := LinearAlgebra:-DiagonalMatrix([ a1, Matrix([[a2, a3], [a4, a5]]), Matrix([[a6, a7, a8], [a9, a10, a11], [a12, a13,a14]]), Matrix([[a15, a16, a17], [a18, a19, a20], [a21, a22,a23]]), Matrix([[a24,a25, a26], [a27, a28, a29], [a30, a31,a32]]), Matrix([[a33, a34], [a35, a36]]), a37]);
 ${A}{≔}\left[\begin{array}{ccccccccccccccc}{\mathrm{a1}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {\mathrm{a2}}& {\mathrm{a3}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {\mathrm{a4}}& {\mathrm{a5}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {\mathrm{a6}}& {\mathrm{a7}}& {\mathrm{a8}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {\mathrm{a9}}& {\mathrm{a10}}& {\mathrm{a11}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {\mathrm{a12}}& {\mathrm{a13}}& {\mathrm{a14}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a15}}& {\mathrm{a16}}& {\mathrm{a17}}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a18}}& {\mathrm{a19}}& {\mathrm{a20}}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a21}}& {\mathrm{a22}}& {\mathrm{a23}}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a24}}& {\mathrm{a25}}& {\mathrm{a26}}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a27}}& {\mathrm{a28}}& {\mathrm{a29}}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a30}}& {\mathrm{a31}}& {\mathrm{a32}}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a33}}& {\mathrm{a34}}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a35}}& {\mathrm{a36}}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a37}}\end{array}\right]$ (22)

We find the parameters for which this matrix defines a homomorphism.

 prM > TF, EQ, Soln, B := Query(sl4, prM, A, {seq(a||i, i = 1..37)}, "Homomorphism"):

One choice is:

 prM > B[1];
 $\left[\begin{array}{ccccccccccccccc}{\mathrm{a1}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& \frac{{\mathrm{a1}}}{{\mathrm{a5}}{}{\mathrm{a31}}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {\mathrm{a5}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& \frac{{\mathrm{a1}}}{{\mathrm{a5}}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& \frac{{1}}{{\mathrm{a31}}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {\mathrm{a5}}{}{\mathrm{a31}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-2}& {-1}& {-1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-1}& {-2}& {-1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& \frac{{1}}{{\mathrm{a5}}{}{\mathrm{a31}}}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-}\frac{{\mathrm{a5}}}{{\mathrm{a1}}}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{a31}}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& \frac{{1}}{{\mathrm{a5}}}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-}\frac{{\mathrm{a5}}{}{\mathrm{a31}}}{{\mathrm{a1}}}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-}\frac{{2}}{{\mathrm{a1}}}\end{array}\right]$ (23)

We have illustrated one of the remarkable properties of the Tanaka prolongation procedure, namely, that the prolongation of the negatively graded part $\mathrm{𝔪}$ of a simple Lie algebra ${\mathrm{𝔤}}_{\mathrm{ss}}$ is the simple Lie algebra ${\mathrm{𝔤}}_{\mathrm{ss}}$ .



Example 4.

In this example we consider a negatively graded Lie algebra whose prolongation is infinite.

 alg3 > LD4 := LieAlgebraData([ [x2, x5] = -x1, [x3, x5] = -x2, [x4, x5] = -x3, [x5, x6] = x4 ], [x1, x2, x3, x4, x5, x6], alg4, grading = [-3, -3, -2, -1, -1, -1]);
 ${\mathrm{LD4}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}\right]$ (24)
 alg3 > DGsetup(LD4);
 ${\mathrm{Lie algebra: alg4}}$ (25)

Calculate the first 7 prolongations of this Lie algebra.

 alg4 > T0 := TanakaProlongation(alg4, 1, pr1alg4):
 alg3 > T1 := TanakaProlongation(alg4, 2, pr2alg4):
 alg4 > T2 := TanakaProlongation(alg4, 3, pr3alg4):
 alg4 > T3 := TanakaProlongation(alg4, 4, pr4alg4):
 alg4 > T4 := TanakaProlongation(alg4, 5, pr5alg4):
 alg4 > T5 := TanakaProlongation(alg4, 6, pr6alg4):
 alg4 > T6 := TanakaProlongation(alg4, 7, pr7alg4):

We see that the dimensions of the prolongations grow by 2 at each order.

 alg4 > map(Tools:-DGinfo, [T0, T1, T2, T3, T4, T5, T6], "LieAlgebraDimension");
 $\left[{10}{,}{12}{,}{14}{,}{16}{,}{18}{,}{20}{,}{22}\right]$ (26)

We use the command Rank1Elements to show that there are elements of  whose adjoint matrices, restricted to have rank 1. This will prove that the Tanaka prolongation of alg4 is infinite. First, initialize the 0-th prolongation

 alg4 > DGsetup(T0);
 ${\mathrm{Lie algebra: pr1alg4}}$ (27)
 pr1alg4 > E := Rank1Elements([e7, e8, e9, e10], [e1, e2, e3, e4, e5, e6]);
 ${E}{≔}\left[{\mathrm{_t3}}{}{\mathrm{e9}}{+}{\mathrm{_t4}}{}{\mathrm{e10}}\right]$ (28)

We can see by inspection that the rank of the adjoint matrix for ${{\mathrm{_t}}_{3}e}_{9}+{{\mathrm{_t}}_{4}e}_{10}$ has rank 1.



 pr1alg4 > Adjoint(E[1], [e1, e2, e3, e4, e5, e6]);
 $\left[\begin{array}{cccccc}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {\mathrm{_t3}}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {\mathrm{_t4}}& {0}\end{array}\right]$ (29)

Finally, if we use the command ChangeGradedComponent to remove the vectors  from  we obtain a Lie algebra newalg with finite Tanaka prolongation - in fact, in this simple example the prolongation is just newalg itself.

 pr1alg4 > LD4a := ChangeGradedComponent(pr1alg4,[ 0 = [e7, e8]], newalg);
 ${\mathrm{LD4a}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{3}{}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{3}{}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{4}{}{\mathrm{e6}}\right]$ (30)
 pr0alg4 > DGsetup(LD4a);
 ${\mathrm{Lie algebra: newalg}}$ (31)
 newalg > TanakaProlongation(newalg, 6, prnewalg);
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{3}{}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{3}{}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{4}{}{\mathrm{e6}}\right]$ (32)
 newalg >