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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 $&gfr;$

k - a positive integer, the number of times the Lie algebra $&gfr;$ is to be prolonged

pralg - an unassigned name or string, the name to be given to the Tanaka prolongation of $&gfr;$

Description

See Also

Description

•	Let $&mfr;$ be a negatively graded Lie algebra, $&mfr; = {&gfr;}_{- μ} \oplus {&gfr;}_{- μ + 1} \oplus \cdot \cdot \cdot \oplus {&gfr;}_{- 2} \oplus {&gfr;}_{- 1}$ . The Tanaka prolongation of $𝔪$ is a graded (possibly infinite dimensional) Lie algebra

$&gfr; (&mfr;) = ⨁_{p \geq - μ}^{p = \infty} {&gfr;}_{p},$ with $[{&gfr;}_{p}, {&gfr;}_{q}] \subset {&gfr;}_{p + q} &period;$

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

${&gfr;}^{{&ell;)} (&mfr;) = ⨁_{p \geq - μ}^{p = &ell;} {&gfr;}_{p} (𝔪) = {&gfr;}_{- p} (𝔪) \oplus \cdot \cdot \cdot \oplus {&gfr;}_{- 1} (𝔪) \oplus {&gfr;}_{0} (𝔪) \oplus {&gfr;}_{1} (𝔪) \oplus \cdot \cdot \cdot \oplus {&gfr;}_{&ell;} (𝔪)$ .

Here ${&gfr;}_{&ell;} (𝔪)$ = $𝔤_{ℓ}$ for $&ell; <$ 0 and $𝔤_{ℓ},$ for $&ell; \geq 0$ , is defined as the derivations of the Lie algebra $&mfr;$ which shift the grading degree by $&ell; &period;$ In particular, the weight 0 component ${&gfr;}_{0}$ is precisely the gradation preserving derivations (or infinitesimal automorphisms) of $&mfr;$ . If ${&gfr;}_{q + 1} = 0$ for some $q \geq 0,$ then all subsequent components $𝔤_{p} = 0$ for $p > q$ and the process of Tanaka prolongation is said to terminate at order $q$ .

•	The command TanakaProlongation requires that the basis $\{e_{1}, e_{2}, .. &period;, e_{n}\}$ for the Lie algebra $&mfr;$ be adapted to the grading in the sense that

$𝔤_{- p} = \{e_{1}, .. &period;, e_{n_{p}}\}, 𝔤_{- p + 1} = \{e_{n_{p} + 1}, .. &period;, e_{n_{p - 1}}\}, .. &period;, {&gfr;}_{- 1} = \{e_{n_{2}}, .. &period;, e_{n}\} &period;$

•	The command TanakaProlongation(alg, k, pralg) returns the structure equations for the $&ell;$ -th prolongation

${&gfr;}^{{&ell;)} (&mfr;) = {&gfr;}_{- p} \oplus {&gfr;}_{- p + 1} \oplus \cdot \cdot \cdot \oplus {&gfr;}_{- 2} \oplus {&gfr;}_{- 1} \oplus {&gfr;}_{0} \oplus {&gfr;}_{1} \oplus \cdot \cdot \cdot \oplus {&gfr;}_{&ell;}$

where $&ell;$ = min $($ $k, q)$ and where $q$ is the smallest non-negative integer such that ${&gfr;}_{q + 1} = 0.$

•	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 $&Dscr;$ be a distribution on a manifold $M -$ about each point of $M$ , $𝒟$ can be described as the span of a finite number of vector fields. We recall that the infinitesimal symmetry algebra sym $(&Dscr;)$ of $𝒟$ is the Lie algebra of vector fields $Z$ such that $[Z, &Dscr;] \subset &Dscr;$ .

Let $G$ be the nilpotent Lie group with Lie algebra $&mfr; &period;$ Let $\{X_{1}, X_{2}, .. &period;, X_{n}\}$ 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 $\{e_{1}, e_{2}, .. &period;, e_{n}\}$ for the Lie algebra $&mfr;$ . Because the algebra $&mfr;$ is a nilpotent algebra, the Lie group $G$ and the vector fields $\{X_{1}, X_{2}, .. &period;, X_{n}\}$ can be explicitly constructed using the LieGroup and InvariantGeometricObjectFields in the GroupActions package. Set $&Dscr; =$ $\{X_{n_{2}}, e_{2}, .. &period;, X_{n}\} &period;$ This is the distribution corresponding to the $𝔤_{- 1}$ component of $𝔪$ . This distribution has 2 remarkable properties: (1) its symbol algebra $σ (&Dscr;)$ coincides with $&mfr; &period;$ and (2) the symmetry algebra sym $(&Dscr;)$ is isomorphic, as an abstract Lie algebra, to the Tanaka prolongation $𝔤 (𝔪)$ .

2.	There is an important criterion which implies that the prolongation $𝔤 (𝔪)$ is infinite dimensional. Calculate the 0-th order prolongation

$𝔤^{{0)} (𝔪) = 𝔤_{- p} \oplus {&gfr;}_{- p + 1} \oplus \cdot \cdot \cdot \oplus 𝔤_{- 2} \oplus 𝔤_{- 1} \oplus 𝔤_{0} = 𝔪 \oplus 𝔤_{0}$ .

Let $&Ascr;$ be the linear span of the adjoint matrices $ad (x)$ for $x \in {&gfr;}_{0}$ , restricted to $𝔪$ . If $&Ascr;$ contains a rank 1 matrix then the Tanaka prolongation is infinite. This test can be implemented with the command Rank1Elements.

Let ${&gfr;}_{ss}$ be a semi-simple Lie algebra with roots $Δ$ and positive roots $Δ^{+}$ . For the sake of simplicity, let us assume that ${&gfr;}_{ss}$ 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 $Δ^{+}$ defines a (symmetric) grading of $𝔤_{ss},$ say

${&gfr;}_{ss} = ⨁_{p = - μ}^{p = μ} 𝔤_{ss, p} =$ $𝔤_{ss, - μ} \oplus \cdot \cdot \cdot \oplus 𝔤_{ss, - 1} \oplus 𝔤_{ss, 0} \oplus {&gfr;}_{ss, 1} \oplus \cdot \cdot \cdot \oplus {&gfr;}_{ss, μ} (*)$

These gradations can be constructed with the GradeSemiSimpleLieAlgebra command. Let $&mfr; = {&gfr;}_{ss, - μ} \oplus \cdot \cdot \cdot \oplus 𝔤_{ss, - 1}$ be the negatively graded part of this decomposition of $𝔤_{ss} &period;$ Then, with the exception of a few well-noted cases, the Tanaka prolongation of $&mfr;$ reproduces the semi-simple Lie algebra $𝔤_{ss}$ , that is, $𝔤_{ss} = 𝔤^{{μ)} (𝔪)$ and $𝔤_{ss, p} = {&gfr;}_{p} (&mfr;)$ .

•	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 $&gfr;$ = alg1 with grading $&gfr; = {&gfr;}_{- 3} \oplus {&gfr;}_{- 2} \oplus 𝔤_{- 1}$ and dim $𝔤_{- 3} = 2,$ dim $𝔤_{- 2} = 1,$ dim $𝔤_{- 1} = 2.$ 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];

$StrEq ≔ [[x3, x4] = - x1, [x3, x5] = - x2, [x4, x5] = x3], [x1, x2, x3, x4, x5]$

(1)

>	LD := LieAlgebraData(StrEq, alg1, grading = [-3, -3, -2, -1, -1]):

>	DGsetup(LD);

$Lie algebra: alg1$

(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):

   [[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 $\{e_{6}, e_{7}, e_{8}, e_{9}\}$ 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);

$Lie algebra: pralg1$

(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");

$G ≔ table ([−1 = [_DG ([[vector, pralg1, []], [[[4], 1]]]),_DG ([[vector, pralg1, []], [[[5], 1]]])], 0 = [_DG ([[vector, pralg1, []], [[[6], 1]]]),_DG ([[vector, pralg1, []], [[[7], 1]]]),_DG ([[vector, pralg1, []], [[[8], 1]]]),_DG ([[vector, pralg1, []], [[[9], 1]]])], −2 = [_DG ([[vector, pralg1, []], [[[3], 1]]])], 1 = [_DG ([[vector, pralg1, []], [[[10], 1]]]),_DG ([[vector, pralg1, []], [[[11], 1]]])], −3 = [_DG ([[vector, pralg1, []], [[[1], 1]]]),_DG ([[vector, pralg1, []], [[[2], 1]]])], 2 = [_DG ([[vector, pralg1, []], [[[12], 1]]])], 3 = [_DG ([[vector, pralg1, []], [[[13], 1]]]),_DG ([[vector, pralg1, []], [[[14], 1]]])]])$

(4)

pralg1 >

Query(G, "Gradation");

$true$

(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);

$LD2a ≔ [[e1, e6] = - e1, [e1, e8] = - e2, [e1, e10] = - e3, [e2, e7] = - e1, [e2, e9] = - e2, [e2, e11] = - e3, [e3, e4] = - e1, [e3, e5] = - e2, [e3, e6] = - \frac{e3}{3}, [e3, e9] = - \frac{e3}{3}, [e3, e10] = - \frac{4 e5}{3}, [e3, e11] = \frac{4 e4}{3}, [e4, e5] = e3, [e4, e6] = - \frac{2 e4}{3}, [e4, e8] = - e5, [e4, e9] = \frac{e4}{3}, [e4, e10] = e6, [e4, e11] = e7, [e5, e6] = \frac{e5}{3}, [e5, e7] = - e4, [e5, e9] = - \frac{2 e5}{3}, [e5, e10] = e8, [e5, e11] = e9, [e6, e7] = e7, [e6, e8] = - e8, [e6, e10] = - \frac{2 e10}{3}, [e6, e11] = \frac{e11}{3}, [e7, e8] = e6 - e9, [e7, e9] = e7, [e7, e10] = - e11, [e8, e9] = - e8, [e8, e11] = - e10, [e9, e10] = \frac{e10}{3}, [e9, e11] = - \frac{2 e11}{3}]$

(6)

alg1 >

DGsetup(LD2a);

$Lie algebra: P1$

(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);

$LD2b ≔ [[e1, e6] = - e1, [e1, e8] = - e2, [e1, e10] = - e3, [e1, e12] = - e4, [e1, e13] = - \frac{e9}{2} - e6, [e1, e14] = - e7, [e2, e7] = - e1, [e2, e9] = - e2, [e2, e11] = - e3, [e2, e12] = - e5, [e2, e13] = - \frac{e8}{2}, [e2, e14] = - 2 e9 - e6, [e3, e4] = - e1, [e3, e5] = - e2, [e3, e6] = - \frac{e3}{3}, [e3,]]$

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