calculate a basis of left and right invariant vector fields and differential 1-forms on a Lie group - Maple Help

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GroupActions[InvariantVectorsAndForms] - calculate a basis of left and right invariant vector fields and differential 1-forms on a Lie group

Calling Sequences

     InvariantVectorsAndForms(LG, options)

Parameters

     LG        - a module defining a Lie group

     options   - output = O, where O is a list of keywords chosen from: "LeftVectors", "LeftForms", "RightVectors", "RightForms"

 

Description

Examples

Description

• 

Let G be a Lie group with multiplication * and define diffeomorphisms La:GG and Ra:G G by Lax = a*g and Rag = x*a. A vector field X on G is left invariant if La*Xb = Xab and right invariant if Ra*Xb = Xba for all b G. A differential form ω on G  is left invariant if La*ωab = ωb and right invariant if Ra*ωba = ωb . Every Lie group admits a set of n= dimG pointwise linearly independent left or right invariant vector fields (an invariant frame) and a set of n= dimG pointwise linearly independent left or right invariant 1-forms (an invariant coframe). Indeed, the infinitesimal generators for the group action defined by Ra give a left invariant frame while the infinitesimal generators for the group action defined by La give a right invariant frame.

• 

The command InvariantVectorsAndForms(LG) returns up to a sequence of four lists XL, OmegaL, XR, OmegaR, where XL is a frame of left invariant vector fields, OmegaL is a coframe of left invariant 1-forms, XR is a frame of right invariant vector fields, and OmegaR is a frame of right invariant 1-forms.

• 

The output option allows the user to dictate precisely which lists of invariant vector fields and forms are returned and the order in which they are returned. The default is output = ["LeftVectors", "LeftForms", "RightVectors", "RightForms"].

• 

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

Examples

withDifferentialGeometry:withGroupActions:withLieAlgebras:

 

Example 1.

We calculate a basis for the invariant vector fields and forms for the 4-dimensional matrix group defined by the matrix M.

MMatrixx,y,z,0,w,0,0,0,1

M:=xyz0w0001

(2.1)

 

Create a local system of coordinates for the Lie group.

DGsetupx,y,z,w,G

frame name: G

(2.2)

 

Create the Lie group module for the matrix group M using the LieGroup command.

G > 

LGLieGroupM,G

LG:=moduleexportFrame,Identity,LeftMultiplication,RightMultiplication,Inverse;end module

(2.3)

 

Find a basis of left invariant vector fields and differential 1-forms.

G > 

XL,OmegaLInvariantVectorsAndFormsLG,output=LeftVectors,LeftForms

XL,OmegaL:=xD_x,xD_y,xD_z,yD_y+wD_w,dxx,dyxydwxw,dzx,dww

(2.4)

 

Find a basis of right invariant vector fields.

G > 

XRInvariantVectorsAndFormsLG,output=RightVectors

XR:=xD_x+yD_y+zD_z,wD_y,D_z,wD_w

(2.5)

 

Details for Example 1

We check various properties of these invariant bases of vector fields and forms. First note that the structure constants for the right invariant vector fields are the negatives of those for the left invariant vector fields.

G > 

LieAlgebraDataXL,Alg2

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

(2.1.1)
G > 

LieAlgebraDataXR,Alg3

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

(2.1.2)

 

The Lie derivatives of XL and OmegaL with respect to XR vanish:

G > 

Matrix4,4,i,j→LieDerivativeXRi,XLj

0D_x0D_x0D_x0D_x0D_x0D_x0D_x0D_x0D_x0D_x0D_x0D_x0D_x0D_x0D_x0D_x

(2.1.3)
G > 

Matrix4,4,i,j→LieDerivativeXRi,OmegaLj

0dx0dx0dx0dx0dx0dx0dx0dx0dx0dx0dx0dx0dx0dx0dx0dx

(2.1.4)

 

Let us check explicitly the left invariance of the vector field XL[4].

G > 

XXL4

X:=yD_y+wD_w

(2.1.5)

 

Define points a and b and compute c = a*b.

G > 

ax1,y1,z1,w1

a:=x1,y1,z1,w1

(2.1.6)
G > 

bx=x2,y=y2,z=z2,w=w2

b:=x=x2,y=y2,z=z2,w=w2

(2.1.7)
G > 

muLLG:-LeftMultiplicationa

muL:=x=x1x,y=x1y+y1w,z=x1z+z1,w=w1w

(2.1.8)
G > 

cApplyTransformationmuL,b

c:=x=x1x2,y=x1y2+y1w2,z=x1z2+z1,w=w1w2

(2.1.9)

 

Evaluate X at b and at c.

G > 

X_bevalX,b

X_b:=y2D_y+w2D_w

(2.1.10)
G > 

X_cevalX,c

X_c:=x1y2+y1w2D_y+w1w2D_w

(2.1.11)

 

Pushforward X by μL. Since Y= Xc, the vector field X is left invariant.

G > 

YPushforwardmuL,X_b

Y:=x1y2+y1w2D_y+w1w2D_w

(2.1.12)

 

Alternatively, we can verify the left invariance of X using the second calling sequence for Pushforward to see that X is unchanged.

G > 

PushforwardmuL,InverseTransformationmuL,X

yD_y+wD_w

(2.1.13)

 

The left invariance of the form OmegaL[2] is similarly verified (by observing that θ = ωc).

G > 

ωOmegaL2

ω:=dyxydwxw

(2.1.14)
G > 

omega_cevalω,c

omega_c:=dyx1x2x1y2+y1w2dwx1x2w1w2

(2.1.15)
G > 

omega_bevalω,b

omega_b:=dyx2y2dwx2w2

(2.1.16)
G > 

θPullbackmuL,omega_c

θ:=dyx2y2dwx2w2

(2.1.17)
G > 

PullbackmuL,ω

dyxydwxw

(2.1.18)

See Also

DifferentialGeometry

GroupActions

LieAlgebras

LieGroup

LieDerivative

Pushforward

Pullback

 


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