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=_DGvector,pralg1,,4,1,_DGvector,pralg1,,5,1,0=_DGvector,pralg1,,6,1,_DGvector,pralg1,,7,1,_DGvector,pralg1,,8,1,_DGvector,pralg1,,9,1,−2=_DGvector,pralg1,,3,1,1=_DGvector,pralg1,,10,1,_DGvector,pralg1,,11,1,−3=_DGvector,pralg1,,1,1,_DGvector,pralg1,,2,1,2=_DGvector,pralg1,,12,1,3=_DGvector,pralg1,,13,1,_DGvector,pralg1,,14,1
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=−
