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GroupActions[LiesThirdTheorem] - find a Lie algebra of pointwise independent vector fields with prescribed structure equations (solvable algebras only)

Calling Sequences

     LiesThirdTheorem(Alg, M, option)

     LiesThirdTheorem(A, M)

Parameters

     Alg       - a Maple name or string, the name of an initialized Lie algebra 𝔤

     M         - a Maple name or string, the name of an initialized manifold with the same dimension as that of 𝔤

     option    - with output = "forms" the dual 1-forms (Maurer-Cartan forms) are returned

     A         - a  list of square matrices, defining a matrix Lie algebra

 

Description

Examples

Description

• 

Let g be an ndimensional Lie algebra with structure constants C. Then Lie's Third Theorem (see, for example, Flanders, page 108) asserts that there is, at least locally, a Lie algebra of n pointwise independent vector fields Γ on an n-dimensional manifold M with structure constants C.

• 

The command LiesThirdTheorem(Alg, M) produces a globally defined Lie algebra of vector fields Γ in the special case that 𝔤 is solvable. More general cases will be handled in subsequent versions of the DifferentialGeometry package.

• 

The command LiesThirdTheorem(A, M) produces a globally defined matrix of 1-forms (Maurer-Cartan forms) in the special case that the list of matrices A defines a solvable Lie algebra.

• 

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

Examples

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

 

Example 1.

We obtain a Lie algebra from the DifferentialGeometry library using the Retrieve command and initialize it.

L := Retrieve("Winternitz", 1, [4, 4], Alg1);

L:=e1,e4=e1,e2,e4=e2+e1,e3,e4=e3+e2

(2.1)

DGsetup(L):

 

We define a manifold M of dimension 4 (the same dimension as the Lie algebra).

Alg1 > 

DGsetup([x, y, z, w], M1);

frame name: M1

(2.2)
M1 > 

Gamma1 := LiesThirdTheorem(Alg1, M1);

Γ1:=D_x,D_y,D_z,x+yD_x+y+zD_y+zD_z+D_w

(2.3)
M1 > 

Omega1 := LiesThirdTheorem(Alg1, M1, output = "forms");

Ω1:=dxx+ydw,dyy+zdw,dwz+dz,dw

(2.4)

 

We calculate the structure equations for the Lie algebra of vector fields Gamma1 and check that these structure equations coincide with those for Alg1.

M1 > 

LieAlgebraData(Gamma1, Alg1a);

e1,e4=e1,e2,e4=e2+e1,e3,e4=e3+e2

(2.5)

 

Example 2.

We re-work the previous example in a more complicated basis. In this basis the adjoint representation is not upper triangular, in which case LiesThirdTheorem first calls the program SolvableRepresentation to find a basis for the algebra in which the adjoint representation is upper triangular. (Remark: It is almost always useful, when working with solvable algebras, to transform to a basis where the adjoint representation is upper triangular.)

M1 > 

L2 := LieAlgebraData([e4, e2 - e4, e3, e1 + e3], Alg2);

L2:=e1,e2=e4+e3e2e1,e1,e3=e3e2e1,e1,e4=e4e2e1,e2,e3=e3+e2+e1,e2,e4=e4+e2+e1

(2.6)
Alg2 > 

DGsetup(L2);

Lie algebra: Alg2

(2.7)
Alg2 > 

Gamma2 := LiesThirdTheorem(Alg2, M1);

Γ2:=xyD_x+y+zD_y+zD_z+D_w,xyD_x1+y+zD_yzD_zD_w,D_z,D_x+D_z

(2.8)
M1 > 

LieAlgebraData(Gamma2, Alg2a);

e1,e2=e4+e3e2e1,e1,e3=e3e2e1,e1,e4=e4e2e1,e2,e3=e3+e2+e1,e2,e4=e4+e2+e1

(2.9)

 

Example 3.

Here is an example where one of the adjoint matrices has complex eigenvalues. The Lie algebra contains parameters p and b.

M1 > 

L3 := Retrieve("Winternitz", 1, [5, 25], Alg3);

L3:=e1,e5=2_pe1,e2,e3=e1,e2,e5=_pe2+e3,e3,e5=_pe3e2,e4,e5=_be4

(2.10)
M1 > 

DGsetup(L3):

Alg3 > 

Adjoint(e5);

2_p00000_p10001_p00000_b000000

(2.11)
Alg3 > 

DGsetup([x, y, z, u ,v], M3);

frame name: M3

(2.12)
M3 > 

Gamma3 := LiesThirdTheorem(Alg3, M3);

Γ3:=D_x,D_xz+D_y,D_z,D_u,12z212y2+2_pxD_x+_pyzD_y+_pz+yD_z+_buD_u+D_v

(2.13)
M3 > 

LieAlgebraData(Gamma3, Alg3a);

e1,e5=2_pe1,e2,e3=e1,e2,e5=_pe2+e3,e3,e5=_pe3e2,e4,e5=_be4

(2.14)

 

Example 4.

We calculate the Maurer-Cartan matrix Ω of 1-forms for a solvable matrix algebra, namely the matrices defining the adjoint representation for Alg1 from Example 1.

A := Adjoint(Alg1);

A:=0001000000000000,0001000100000000,0000000100010000,1100011000100000

(2.15)

MaurerCartan := LiesThirdTheorem(A, M1);

MaurerCartan:=dwdw0dxdx+dy2y+z+xdw0dxdwdwdy+dz2z+ydw0dx0dxdwdwz+dz0dx0dx0dx0dx

(2.16)

 

Note that the elements of this matrix   coincide with the appropriate linear combinations of the forms in the list Ω1 from Example 1.

Alg1 > 

MaurerCartan[1,4], Omega1[1] &plus Omega1[2];

dx+dy2y+z+xdw,dx+dy2y+z+xdw

(2.17)
Alg1 > 

MaurerCartan[2,4], Omega1[2] &plus Omega1[3];

dy+dz2z+ydw,dy+dz2z+ydw

(2.18)
Alg1 > 

MaurerCartan[3,4], Omega1[3];

dwz+dz,dwz+dz

(2.19)
Alg1 > 

MaurerCartan[1,1], (-1) &mult Omega1[4];

dw,dw

(2.20)

See Also

DifferentialGeometry

GroupActions

Library

LieAlgebras

Adjoint

LieAlgebraData

Representation

Retrieve

SolvableRepresentation