find the infinitesimal isotropy subalgebra of a Lie algebra of vector fields and the representation of the isotropy subalgebra on the tangent space - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : DifferentialGeometry : Group Actions : DifferentialGeometry/GroupActions/IsotropySubalgebra

GroupActions[IsotropySubalgebra] - find the infinitesimal isotropy subalgebra of a Lie algebra of vector fields and the representation of the isotropy subalgebra on the tangent space

Calling Sequences

     IsotropySubalgebra(Gamma, p, option)

Parameters

     Gamma     - a list of vector fields on a manifold M

     p         - a list of equations x1 = p1, x2 = p2, ... specifying the coordinates of point p  M

     option    - the optional argument output = O, where O is a list containing the keywords "Vector", "Representation", and/or the name of an initialized abstract algebra for the Lie algebra of vector fields Gamma. 

 

Description

Examples

Description

• 

Let Γ be a Lie algebra of vector fields on a manifold M and letp M. The isotropy subalgebra  Γp of the Lie algebra of vector fields Γ at the point p is defined by Γp = {X Γ |Xp= 0}. The Lie bracket of vector fields defines a natural representation ρ of Γp on the tangent space TpM  by ρXY = X, Y for X Γp ,Y TpM and Y any vector field on M such that Yp =Y. The representation ρ is called the linear isotropy representation.

• 

IsotropySubalgebra(Gamma, p) returns a list of vectors whose span defines the isotropy subalgebra Γpas a subalgebra of  Γ.

• 

With output = ["Vector", "Representation"], two lists are returned. The first is a list of vectors giving the isotropy subalgebra Γpas a subalgebra of  Γ and the second is the list of matrices defining the linear isotropy representation with respect to the standard basis for TpM.

• 

Let algname be the name of the abstract Lie algebra 𝔤 created from Γ. With output = ["Vector", algname], the second list returned gives the isotropy subalgebra as a subalgebra of the abstract Lie algebra 𝔤.

• 

The command IsotropySubalgebra is part of the DifferentialGeometry:-GroupActions package.  It can be used in the form IsotropySubalgebra(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-IsotropySubalgebra(...).

Examples

with(DifferentialGeometry): with(GroupActions): with(Library): with(LieAlgebras):

 

Example 1.

We use the Retrieve command to obtain a Lie algebra of vector fields in the paper by Gonzalez-Lopez, Kamran, and Olver from the DifferentialGeometry Library. We compute the isotropy subalgebra and isotropy representation at the points x =0, y = 0 and x =1, y =1. 

DGsetup([x, y], M):

M > 

G := Retrieve("Gonzalez-Lopez", 1, [5], manifold = M);

G:=D_x,D_y,D_xxD_yy,yD_x,xD_y

(2.1)
M > 

L := LieAlgebraData(G, Alg1);

L:=e1,e3=e1,e1,e5=e2,e2,e3=e2,e2,e4=e1,e3,e4=2e4,e3,e5=2e5,e4,e5=e3

(2.2)
M > 

DGsetup(L);

Lie algebra: Alg1

(2.3)
Alg1 > 

MultiplicationTable("LieTable");

| e1e2e3e4e5------------------------e1| 00e10e2e2| 00e2e10e3| e1e202e42e5e4| 0e12e40e3e5| e202e5e30

(2.4)

 

We illustrate some different possible outputs from the IsotropySubalgebra program.

Alg1 > 

Iso1 := IsotropySubalgebra(G, [x = 0, y = 0]);

Iso1:=D_xxD_yy,yD_x,xD_y

(2.5)
M > 

Iso1, A1 := IsotropySubalgebra(G, [x = 0, y = 0], output = ["Vector", Alg1]);

Iso1,A1:=D_xxD_yy,yD_x,xD_y,e3,e4,e5

(2.6)
Alg1 > 

Iso1, A1, S1 := IsotropySubalgebra(G, [x = 0, y = 0], output = ["Vector", Alg1,"Representation"]);

Iso1,A1,S1:=D_xxD_yy,yD_x,xD_y,e3,e4,e5,1001,0100,0010

(2.7)
Alg1 > 

A1 := IsotropySubalgebra(G, [x = 0, y = 0], output = [Alg1]);

A1:=e3,e4,e5

(2.8)
Alg1 > 

Iso2, A2, S2 := IsotropySubalgebra(G, [x = 1, y = 1], output = ["Vector", Alg1,"Representation"]);

Iso2,A2,S2:=y1D_x,x1D_y,y+xD_x+y+xD_y,e1e4,e2e5,e3e4+e5,0100,0010,1111

(2.9)

Note that the vectors in Iso2 all vanish at x =1, y =1. 

 

It is apparent from the multiplication table that the pair Alg1, S1 is a symmetric pair with respect to the complementary subspace T = e1, e2. We can check this with the command Query/"SymmetricPair".

Alg1 > 

Query(A1, [e1, e2], "SymmetricPair");

true

(2.10)

 

The linear isotropy representation can be converted to a representation.

Alg1 > 

L2 := LieAlgebraData(A1, iso1);

L2:=e1,e2=2e2,e1,e3=2e3,e2,e3=e1

(2.11)
Alg1 > 

DGsetup(L2);

Lie algebra: iso1

(2.12)
iso1 > 

rho := Representation(iso1, M, S1);

ρ:=e1,1001,e2,0100,e3,0010

(2.13)
iso1 > 

Query(rho, "Representation");

true

(2.14)

See Also

DifferentialGeometry

GroupActions

Library

LieAlgebras

LieAlgebraData

MultiplicationTable

Query

Representation

Retrieve