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 𝔤
k - a positive integer, the number of times the Lie algebra 𝔤 is to be prolonged
pralg - an unassigned name or string, the name to be given to the Tanaka prolongation of 𝔤
Description
See Also
Let 𝔪 be a negatively graded Lie algebra, 𝔪 = 𝔤−μ ⊕ 𝔤−μ+1⊕ ⋅⋅⋅⊕𝔤−2⊕𝔤−1. The Tanaka prolongation of 𝔪 is a graded (possibly infinite dimensional) Lie algebra
𝔤𝔪 = ⨁p≥ −μp =∞𝔤p , with 𝔤p , 𝔤q ⊂ 𝔤p+q .
The Tanaka prolongation is computed inductively in terms of the partial prolongations
𝔤{ℓ)𝔪=⨁p≥ −μp =ℓ𝔤p 𝔪 = 𝔤−p 𝔪⊕ ⋅⋅⋅⊕𝔤−1𝔪⊕ 𝔤0𝔪⊕𝔤1𝔪⊕ ⋅⋅⋅⊕ 𝔤ℓ𝔪.
Here 𝔤ℓ 𝔪 = 𝔤ℓ for ℓ <0 and 𝔤ℓ , for ℓ≥0, is defined as the derivations of the Lie algebra 𝔪 which shift the grading degree by ℓ. In particular, the weight 0 component 𝔤0 is precisely the gradation preserving derivations (or infinitesimal automorphisms) of 𝔪. If 𝔤q+1 = 0 for some q≥0, then all subsequent components 𝔤p =0 for p >q and the process of Tanaka prolongation is said to terminate at order q.
The command TanakaProlongation requires that the basis e1, e2, ..., en for the Lie algebra 𝔪 be adapted to the grading in the sense that
𝔤−p=e1, ..., enp, 𝔤−p+1=enp+1 , ... , enp−1, ... , 𝔤−1=en2 ,..., en.
The command TanakaProlongation(alg, k, pralg) returns the structure equations for the ℓ-th prolongation
𝔤{ℓ)𝔪 = 𝔤−p ⊕ 𝔤−p+1⊕ ⋅⋅⋅ ⊕𝔤−2⊕𝔤−1⊕ 𝔤0⊕𝔤1 ⊕⋅⋅⋅⊕ 𝔤ℓ
where ℓ = min(k, q) and where q is the smallest non-negative integer such that 𝔤q+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 𝒟 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 sym𝒟of 𝒟 is the Lie algebra of vector fields Z such that Z, 𝒟 ⊂𝒟.
Let G be the nilpotent Lie group with Lie algebra 𝔪. Let X1, X2, ... , Xn 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 e1, e2, ..., en for the Lie algebra 𝔪 . Because the algebra 𝔪 is a nilpotent algebra, the Lie group G and the vector fields X1, X2, ..., Xn can be explicitly constructed using the LieGroup and InvariantGeometricObjectFields in the GroupActions package. Set 𝒟=Xn2 , e2, ..., Xn . This is the distribution corresponding to the 𝔤−1 component of 𝔪. This distribution has 2 remarkable properties: (1) its symbol algebra σ𝒟 coincides with 𝔪. and (2) the symmetry algebra sym𝒟 is isomorphic, as an abstract Lie algebra, to the Tanaka prolongation 𝔤𝔪.
There is an important criterion which implies that the prolongation 𝔤𝔪 is infinite dimensional. Calculate the 0-th order prolongation
𝔤{0)𝔪 = 𝔤−p ⊕ 𝔤−p+1⊕ ⋅⋅⋅ ⊕𝔤−2⊕𝔤−1⊕ 𝔤0 = 𝔪 ⊕ 𝔤0.
Let 𝒜 be the linear span of the adjoint matrices adx for x ∈ 𝔤0, restricted to 𝔪. If 𝒜 contains a rank 1 matrix then the Tanaka prolongation is infinite. This test can be implemented with the command Rank1Elements.
Let 𝔤ss be a semi-simple Lie algebra with roots Δ and positive roots Δ+. For the sake of simplicity, let us assume that 𝔤ss 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 𝔤ss, say
𝔤ss = ⨁p= −μp =μ𝔤ss,p= 𝔤ss,−μ ⊕ ⋅⋅⋅⊕𝔤ss,−1⊕𝔤ss,0⊕𝔤ss,1 ⊕ ⋅⋅⋅⊕𝔤ss,μ (*)
These gradations can be constructed with the GradeSemiSimpleLieAlgebra command. Let 𝔪 = 𝔤ss,−μ ⊕ ⋅⋅⋅⊕𝔤ss,−1 be the negatively graded part of this decomposition of 𝔤ss. Then, with the exception of a few well-noted cases, the Tanaka prolongation of 𝔪 reproduces the semi-simple Lie algebra 𝔤ss, that is, 𝔤ss=𝔤{μ)𝔪 and 𝔤ss,p= 𝔤p𝔪.
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 𝔤 = alg1 with grading 𝔤 =𝔤−3⊕ 𝔤−2 ⊕ 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
LD := LieAlgebraData(StrEq, alg1, grading = [-3, -3, -2, -1, -1]):
DGsetup(LD);
Lie algebra: alg1
Calculate the Tanaka prolongation for alg1. With infolevel[TanakaProlongation] = 2, information on the sequential partial prolongations is displayed.
infolevel[TanakaProlongation] := 2:
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 e6, e7, e8, e9 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.
DGsetup(prLD);
Lie algebra: pralg1
We can use the command DGinfo to display the grading of this algebra and the Query command to verify it is a valid gradation.
G := Tools:-DGinfo( "table", "Grading");
G≔table−1=e4,e5,0=e6,e7,e8,e9,−2=e3,1=e10,e11,−3=e1,e2,2=e12,3=e13,e14
Query(G, "Gradation");
true
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 g2.
Before continuing to the next example, reset the infolevel.
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.
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=−e33,e3,e9=−e33,e3,e10=−4e53,e3,e11=4e43,e4,e5=e3,e4,e6=−2e43,e4,e8=−e5,e4,e9=e43,e4,e10=e6,e4,e11=e7,e5,e6=e53,e5,e7=−e4,e5,e9=−2e53,e5,e10=e8,e5,e11=e9,e6,e7=e7,e6,e8=−e8,e6,e10=−2e103,e6,e11=e113,e7,e8=e6−e9,e7,e9=e7,e7,e10=−e11,e8,e9=−e8,e8,e11=−e10,e9,e10=e103,e9,e11=−2e113
DGsetup(LD2a);
Lie algebra: P1
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.
LD2b := TanakaProlongation(P1, 4, P2);
LD2b≔e1,e6=−e1,e1,e8=−e2,e1,e10=−e3,e1,e12=−e4,e1,e13=−e92−e6,e1,e14=−e7,e2,e7=−e1,e2,e9=−e2,e2,e11=−e3,e2,e12=−e5,e2,e13=−e82,e2,e14=−2e9−e6,e3,e4=−e1,e3,e5=−e2,e3,e6=−e33,e3,e9=−e33,e3,e10=−4e53,e3,e11=4e43,e3,e12=−e9−e6,e3,e13=−e102,e3,e14=−e11,e4,e5=e3,e4,e6=−2e43,e4,e8=−e5,e4,e9=e43,e4,e10=e6,e4,e11=e7,e4,e12=e11,e4,e13=−e122,e5,e6=e53,e5,e7=−e4,e5,e9=−2e53,e5,e10=e8,e5,e11=e9,e5,e12=−e10,e5,e14=−e12,e6,e7=e7,e6,e8=−e8,e6,e10=−2e103,e6,e11=e113,e6,e12=−e123,e6,e13=−e13,e7,e8=e6−e9,e7,e9=e7,e7,e10=−e11,e7,e13=−e142,e8,e9=−e8,e8,e11=−e10,e8,e14=−2e13,e9,e10=e103,e9,e11=−2e113,e9,e12=−e123,e9,e14=−e14,e10,e11=4e123,e10,e12=−2e13,e11,e12=−e14
DGsetup(LD2b);
Lie algebra: P2
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.
phi := Transformation(pralg1, P2, LinearAlgebra:-IdentityMatrix(14));
φ≔e1,e1,e2,e2,e3,e3,e4,e4,e5,e5,e6,e6,e7,e7,e8,e8,e9,e9,e10,e10,e11,e11,e12,e12,e13,e13,e14,e14
Query(phi,"Homomorphism");
Example 3.
In this example we define a gradation of the 15-dimensional Lie algebra sl4. We calculate the Tanaka prolongation of the negatively graded part and show that the prolongation is isomorphic to sl4. The command SimpleLieAlgebraData is used to retrieve the structure equations for sl4.
LD3a := SimpleLieAlgebraData("sl(4)", sl4a):
DGsetup(LD3a);
Lie algebra: sl4a
Here is the grading we shall use ( It was constructed with the commands GradeSemiSimpleLieAlgebra and SimpleLieAlgebraProperties).
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−1=e7,e11,e15,0=e1,e2,e3,−2=e10,e14,1=e4,e8,e12,−3=e13,2=e5,e9,3=e6
Here is the Lie algebra sl4a but now in the basis adapted to this grading.
LD3b := LieAlgebraData(G, sl4);
LD3b≔e1,e7=2e1,e1,e8=e1,e1,e9=e1,e1,e10=e3,e1,e12=−e2,e1,e13=e6,e1,e14=−e4,e1,e15=−e7,e2,e6=−e1,e2,e7=e2,e2,e9=−e2,e2,e10=e5,e2,e11=−e4,e2,e13=e9−e7,e2,e15=e12,e3,e4=e1,e3,e7=e3,e3,e8=2e3,e3,e9=e3,e3,e11=e6,e3,e12=−e5,e3,e14=−e8,e3,e15=−e10,e4,e5=−e2,e4,e7=e4,e4,e8=−e4,e4,e10=e8−e7,e4,e13=e11,e4,e15=e14,e5,e6=−e3,e5,e8=e5,e5,e9=−e5,e5,e11=e9−e8,e5,e13=−e10,e5,e14=e12,e6,e7=e6,e6,e8=e6,e6,e9=2e6,e6,e12=−e9,e6,e14=−e11,e6,e15=−e13,e7,e10=e10,e7,e12=e12,e7,e13=e13,e7,e14=e14,e7,e15=2e15,e8,e10=−e10,e8,e11=e11,e8,e12=e12,e8,e14=2e14,e8,e15=e15,e9,e11=−e11,e9,e12=2e12,e9,e13=−e13,e9,e14=e14,e9,e15=e15,e10,e11=e13,e10,e14=e15,e11,e12=e14,e12,e13=−e15
DGsetup(LD3b);
Lie algebra: sl4
Tools:-DGinfo(sl4, "Grading");
−3,−2,−2,−1,−1,−1,0,0,0,1,1,1,2,2,3
Now we calculate the structure equations for the negatively graded part. We initialize this nilpotent graded Lie algebra with the name M.
LD3b := LieAlgebraData([e1, e2, e3, e4, e5, e6], M, grading = [-3, -2, -2, -1, -1, -1]);
LD3b≔e2,e6=−e1,e3,e4=e1,e4,e5=−e2,e5,e6=−e3
Lie algebra: M
Calculate the prolongation of M and initialize the result.
LD3c := TanakaProlongation(M, 4, prM);
LD3c≔e1,e7=−e1,e1,e10=−e2,e1,e11=−e3,e1,e13=−e4,e1,e14=−e6,e1,e15=−e92−e82−e7,e2,e6=−e1,e2,e8=−e2,e2,e11=−e5,e2,e12=−e4,e2,e14=−2e8−e7,e2,e15=−e102,e3,e4=e1,e3,e9=−e3,e3,e10=−e5,e3,e12=e6,e3,e13=e7+2e9,e3,e15=−e112,e4,e5=−e2,e4,e7=−e4,e4,e9=e4,e4,e11=e9−e8−e7,e4,e14=−e12,e4,e15=−e132,e5,e6=−e3,e5,e7=e5,e5,e8=−e5,e5,e9=−e5,e5,e12=e8+e9,e5,e13=e10,e5,e14=−e11,e6,e7=−e6,e6,e8=e6,e6,e10=e9+e7−e8,e6,e13=−e12,e6,e15=−e142,e7,e10=−e10,e7,e11=−e11,e7,e12=e12,e7,e15=−e15,e8,e10=e10,e8,e12=−e12,e8,e14=−e14,e9,e11=e11,e9,e12=−e12,e9,e13=−e13,e10,e12=−e13,e10,e14=−2e15,e11,e12=e14,e11,e13=2e15
DGsetup(LD3c);
Lie algebra: prM
We see that the prM is a 15 dimensional Lie algebra with the same grading as the one assigned to sl4.
Tools:-DGinfo(prM, "Grading");
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.
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≔a1000000000000000a2a30000000000000a4a5000000000000000a6a7a8000000000000a9a10a11000000000000a12a13a14000000000000000a15a16a17000000000000a18a19a20000000000000a21a22a23000000000000000a24a25a26000000000000a27a28a29000000000000a30a31a32000000000000000a33a340000000000000a35a36000000000000000a37
We find the parameters for which this matrix defines a homomorphism.
TF, EQ, Soln, B := Query(sl4, prM, A, {seq(a||i, i = 1..37)}, "Homomorphism"):
One choice is:
B[1];
a1000000000000000a1a5a31000000000000000a5000000000000000a1a50000000000000001a31000000000000000a5a31000000000000000−2−1−1000000000000−101000000000000−1−2−1000000000000000001a5a31000000000000−a5a1000000000000000a31000000000000000001a50000000000000−a5a31a10000000000000000−2a1
We have illustrated one of the remarkable properties of the Tanaka prolongation procedure, namely, that the prolongation of the negatively graded part 𝔪 of a simple Lie algebra 𝔤ss is the simple Lie algebra 𝔤ss .
Example 4.
In this example we consider a negatively graded Lie algebra whose prolongation is infinite.
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]);
LD4≔e2,e5=−e1,e3,e5=−e2,e4,e5=−e3,e5,e6=e4
DGsetup(LD4);
Lie algebra: alg4
Calculate the first 7 prolongations of this Lie algebra.
T0 := TanakaProlongation(alg4, 1, pr1alg4):
T1 := TanakaProlongation(alg4, 2, pr2alg4):
T2 := TanakaProlongation(alg4, 3, pr3alg4):
T3 := TanakaProlongation(alg4, 4, pr4alg4):
T4 := TanakaProlongation(alg4, 5, pr5alg4):
T5 := TanakaProlongation(alg4, 6, pr6alg4):
T6 := TanakaProlongation(alg4, 7, pr7alg4):
We see that the dimensions of the prolongations grow by 2 at each order.
map(Tools:-DGinfo, [T0, T1, T2, T3, T4, T5, T6], "LieAlgebraDimension");
10,12,14,16,18,20,22
We use the command Rank1Elements to show that there are elements of 𝔤0 = e7,e8, e9, e10 whose adjoint matrices, restricted to 𝔪 = e1, e2, e3, e4, e5, e6,have rank 1. This will prove that the Tanaka prolongation of alg4 is infinite. First, initialize the 0-th prolongation
DGsetup(T0);
Lie algebra: pr1alg4
E := Rank1Elements([e7, e8, e9, e10], [e1, e2, e3, e4, e5, e6]);
E≔_t3e9+_t4e10
We can see by inspection that the rank of the adjoint matrix for _t3e9+_t4e10 has rank 1.
Adjoint(E[1], [e1, e2, e3, e4, e5, e6]);
0000000000000000000000_t300000000000_t40
Finally, if we use the command ChangeGradedComponent to remove the vectors e9, e10 from 𝔤0 = e7,e8, e9, e10 we obtain a Lie algebra newalg with finite Tanaka prolongation - in fact, in this simple example the prolongation is just newalg itself.
LD4a := ChangeGradedComponent(pr1alg4,[ 0 = [e7, e8]], newalg);
LD4a≔e1,e7=−e1,e2,e5=−e1,e2,e8=−e2,e3,e5=−e2,e3,e7=e3,e3,e8=−2e3,e4,e5=−e3,e4,e7=2e4,e4,e8=−3e4,e5,e6=e4,e5,e7=−e5,e5,e8=e5,e6,e7=3e6,e6,e8=−4e6
DGsetup(LD4a);
Lie algebra: newalg
TanakaProlongation(newalg, 6, prnewalg);
e1,e7=−e1,e2,e5=−e1,e2,e8=−e2,e3,e5=−e2,e3,e7=e3,e3,e8=−2e3,e4,e5=−e3,e4,e7=2e4,e4,e8=−3e4,e5,e6=e4,e5,e7=−e5,e5,e8=e5,e6,e7=3e6,e6,e8=−4e6
DifferentialGeometry, LieAlgebras, ChangeGradedComponent, DGinfo, LieAlgebraData, Query, SimpleLieAlgebraData, SimpleLieAlgebraProperties, Rank1Element
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