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LieAlgebras[SymbolAlgebra] - find the symbol algebra for a distribution

Calling Sequences

     SymbolAlgebra(D, pt, alg, option)

Parameters

   D       - a list of vectors on a manifold M

   pt      - a list of equations specifying a point on M

   alg     - a name or string, the name to be assigned to the symbol algebra

   option  - (optional keyword arguments) output = "WeakDerivedFlag" , order = "Increasing", order = "Decreasing",

Description

• 

 In differential geometry a distribution is a set of vector fields 𝒟 defined on a manifold M. The sequence of distributions defined inductively by

𝒟0 = 𝒟 ,  𝒟1= [𝒟0 , 𝒟0] + 𝒟0 , ...,  𝒟i+1= [𝒟0 , 𝒟i] + 𝒟i 

is called the weak derived flag of the distribution. These distributions satisfy 𝒟i ⊂ 𝒟i+1 and [𝒟i , 𝒟j]⊂𝒟i+j. The symbol algebra 𝔪x𝒟 for the distribution 𝒟 at the point x ∈M is the graded nilpotent Lie algebra defined by

𝔪x𝒟= ⨁p= −1p = − μ 𝔤px   where  𝔤−1x = 𝒟x0,   𝔤−2x = 𝒟x−1/𝒟x0 , ... , 𝔤−px = 𝒟x−p +1/𝒟x−p+2, ...

The grading weight of 𝔤px is p.

• 

The command SymbolAlgebra((D, pt, alg) returns the structure equations for the symbol algebra of 𝒟 at the point specified by the second argument. These structure equations can be initialized with DGsetup. The command DGinfo can be used to view the grading of the symbol algebra.

• 

With the keyword argument output = "WeakDerivedFlag", the structure equations for the symbol algebra and the vector fields defining the weak derived flag [𝒟0, 𝒟1, 𝒟2, ...] are returned.

• 

The basis used to define the symbol algebra is given in terms of decreasing weights. For example, if 𝔪 = 𝔤−1⊕ 𝔤−2⊕𝔤−3 with dim 𝔤−1= 4, dim 𝔤−2= 2 and dim 𝔤−3= 1, then 𝔤−1 =e1, e2, e3, e4, 𝔤−2 =e5, e6 and 𝔤−3 =e7. With the keyword argument order = "Increasing", the basis used to define the symbol algebra is given in terms of increasing weights so that, for example, if 𝔪= 𝔤−3⊕ 𝔤−2⊕𝔤−1, then 𝔤−3 =e1, 𝔤−2 =e2, e3 and 𝔤−1 =e4, e5, e6 ,e7.

See Also

DifferentialGeometry

LieAlgebras

DGinfo

Query

TanakaProlongation

Examples

> 

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

In this example we find the symbol algebra for the Hilbert-Cartan equation z' = y''2.

 

First create a 5-dimensional manifold.

> 

DGsetup([x, y, z, y1, y2], M);

frame name: M

(1)

 

Define the rank 3 distribution for the Hilbert-Cartan equation z''=y''2.

M > 

Delta := [D_x + y1*D_y + y2*D_y1 + y2^2*D_z, D_y2];

Δ:=D_x+y1⁢D_y+y2⁢D_y1+y22⁢D_z,D_y2

(2)

 

Calculate the symbol algebra and initialize.

M > 

LD := SymbolAlgebra(Delta, [x = 0, y = 0, y1 = 0, y2 = 0, z = 0], alg);

LD:=e1,e2=−e3,e1,e3=e4,e2,e3=e5

(3)
M > 

DGsetup(LD);

Lie algebra: alg

(4)

 

Here is the multiplication table.

M > 

MultiplicationTable("LieTable");

 

The algebra is nilpotent

alg > 

Query("Nilpotent");

true

(5)

 

Here is the grading of the algebra as a list and in table format.

alg > 

Gr := Tools:-DGinfo("Grading");

Gr:=−1,−1,−2,−3,−3

(6)
alg > 

convert(Gr, DGgrading, "table", [e1, e2, e3, e4, e5]);

table−3=e4,e5,−2=e3,−1=e1,e2

(7)

 

Example 2.

We continue with the distribution given in Example 1. We use the keyword argument output = "WeakDerivedFlag" to display the weak derived flag used to calculate the symbol algebra.

alg > 

SymbolAlgebra(Delta, [x = 0, y = 0, y1 = 0, y2 = 0, z = 0], alg, output = "WeakDerivedFlag");

e1,e2=−e3,e1,e3=e4,e2,e3=e5,D_x+y1⁢D_y+y22⁢D_z+y2⁢D_y1,D_y2,−2⁢y2⁢D_z−D_y1,D_y,−2⁢D_z

(8)

 

We use the second calling sequence to give the symbol algebra in ascending grading order.

M > 

LD2 := SymbolAlgebra(Delta, [x = 0, y = 0, y1 = 0, y2 = 0, z = 0], alg2, order = "Increasing");

LD2:=e3,e4=−e1,e3,e5=−e2,e4,e5=e3

(9)
M > 

DGsetup(LD2);

Lie algebra: alg2

(10)

 

Here is the multiplication table and the grading.

alg2 > 

MultiplicationTable("LieTable"), Tools:-DGinfo("Grading");

 

Example 3.

In this example we find the symbol algebra for the jet space J2R2,R. First create an 8 dimensional manifold.

alg2 > 

DGsetup([x, y, z, p, q, r, s, t], M3);

frame name: M3

(11)

 

Define the rank 3 distribution for the canonical system on jet space.

M3 > 

Delta3 := evalDG([D_x + p*D_z + r*D_p + s*D_q, D_y + q*D_z + s*D_p + t*D_q, D_r, D_s, D_t]);

Δ3:=D_x+p⁢D_z+r⁢D_p+s⁢D_q,D_y+q⁢D_z+s⁢D_p+t⁢D_q,D_r,D_s,D_t

(12)

 

Calculate the symbol algebra and initialize it.

M3 > 

LD3 := SymbolAlgebra(Delta3, [x = 0, y = 0, z = 0, p =0, q = 0, r = 0, s = 0, t = 0], alg3);

LD3:=e1,e4=−e6,e2,e4=−e7,e2,e5=−e6,e3,e5=−e7,e4,e6=e8,e5,e7=e8

(13)
M > 

DGsetup(LD3);

Lie algebra: alg3

(14)

 

Here is the multiplication table and the grading.

alg2 > 

MultiplicationTable("LieTable"), Tools:-DGinfo("Grading");


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