InvariantVectorsAndForms - Maple Help

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

 • Let be a Lie group with multiplication $*$ and define diffeomorphisms ${L}_{a}:G\to G$ and by  and . A vector field $X$ on $G$ is left invariant if and right invariant if for all . A differential form $\mathrm{ω}$ on  is left invariant if ${L}_{a}^{*}\left({\mathrm{ω}}_{\mathrm{ab}}\right)$ = and right invariant if ${R}_{a}^{*}\left({\mathrm{ω}}_{\mathrm{ba}}\right)$ = . Every Lie group admits a set of dimpointwise linearly independent left or right invariant vector fields (an invariant frame) and a set of dimpointwise linearly independent left or right invariant 1-forms (an invariant coframe). Indeed, the infinitesimal generators for the group action defined by give a left invariant frame while the infinitesimal generators for the group action defined by 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{GroupActions}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

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

 > $M≔\mathrm{Matrix}\left(\left[\left[x,y,z\right],\left[0,w,0\right],\left[0,0,1\right]\right]\right)$
 ${M}{:=}\left[\begin{array}{ccc}{x}& {y}& {z}\\ {0}& {w}& {0}\\ {0}& {0}& {1}\end{array}\right]$ (2.1)

Create a local system of coordinates for the Lie group.

 > $\mathrm{DGsetup}\left(\left[x,y,z,w\right],G\right)$
 ${\mathrm{frame name: G}}$ (2.2)

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

 G > $\mathrm{LG}≔\mathrm{LieGroup}\left(M,G\right)$
 ${\mathrm{LG}}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{export}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{Frame}}{,}{\mathrm{Identity}}{,}{\mathrm{LeftMultiplication}}{,}{\mathrm{RightMultiplication}}{,}{\mathrm{Inverse}}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (2.3)

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

 G > $\mathrm{XL},\mathrm{OmegaL}≔\mathrm{InvariantVectorsAndForms}\left(\mathrm{LG},\mathrm{output}=\left["LeftVectors","LeftForms"\right]\right)$
 ${\mathrm{XL}}{,}{\mathrm{OmegaL}}{:=}\left[{x}{}{\mathrm{D_x}}{,}{x}{}{\mathrm{D_y}}{,}{x}{}{\mathrm{D_z}}{,}{y}{}{\mathrm{D_y}}{+}{w}{}{\mathrm{D_w}}\right]{,}\left[\frac{{\mathrm{dx}}}{{x}}{,}\frac{{\mathrm{dy}}}{{x}}{-}\frac{{y}{}{\mathrm{dw}}}{{x}{}{w}}{,}\frac{{\mathrm{dz}}}{{x}}{,}\frac{{\mathrm{dw}}}{{w}}\right]$ (2.4)

Find a basis of right invariant vector fields.

 G > $\mathrm{XR}≔\mathrm{InvariantVectorsAndForms}\left(\mathrm{LG},\mathrm{output}=\left["RightVectors"\right]\right)$
 ${\mathrm{XR}}{:=}\left[{x}{}{\mathrm{D_x}}{+}{y}{}{\mathrm{D_y}}{+}{z}{}{\mathrm{D_z}}{,}{w}{}{\mathrm{D_y}}{,}{\mathrm{D_z}}{,}{w}{}{\mathrm{D_w}}\right]$ (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 > $\mathrm{LieAlgebraData}\left(\mathrm{XL},\mathrm{Alg2}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.1.1)
 G > $\mathrm{LieAlgebraData}\left(\mathrm{XR},\mathrm{Alg3}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}\right]$ (2.1.2)

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

 G > $\mathrm{Matrix}\left(4,4,\left(i,j\right)↦\mathrm{LieDerivative}\left(\mathrm{XR}\left[i\right],\mathrm{XL}\left[j\right]\right)\right)$
 $\left[\begin{array}{cccc}{0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}\\ {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}\\ {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}\\ {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}& {0}{}{\mathrm{D_x}}\end{array}\right]$ (2.1.3)
 G > $\mathrm{Matrix}\left(4,4,\left(i,j\right)↦\mathrm{LieDerivative}\left(\mathrm{XR}\left[i\right],\mathrm{OmegaL}\left[j\right]\right)\right)$
 $\left[\begin{array}{cccc}{0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}\\ {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}\\ {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}\\ {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}\end{array}\right]$ (2.1.4)

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

 G > $X≔\mathrm{XL}\left[4\right]$
 ${X}{:=}{y}{}{\mathrm{D_y}}{+}{w}{}{\mathrm{D_w}}$ (2.1.5)

Define points $a$ and $b$ and compute .

 G > $a≔\left[\mathrm{x1},\mathrm{y1},\mathrm{z1},\mathrm{w1}\right]$
 ${a}{:=}\left[{\mathrm{x1}}{,}{\mathrm{y1}}{,}{\mathrm{z1}}{,}{\mathrm{w1}}\right]$ (2.1.6)
 G > $b≔\left[x=\mathrm{x2},y=\mathrm{y2},z=\mathrm{z2},w=\mathrm{w2}\right]$
 ${b}{:=}\left[{x}{=}{\mathrm{x2}}{,}{y}{=}{\mathrm{y2}}{,}{z}{=}{\mathrm{z2}}{,}{w}{=}{\mathrm{w2}}\right]$ (2.1.7)
 G > $\mathrm{muL}≔\mathrm{LG}:-\mathrm{LeftMultiplication}\left(a\right)$
 ${\mathrm{muL}}{:=}\left[{x}{=}{\mathrm{x1}}{}{x}{,}{y}{=}{\mathrm{x1}}{}{y}{+}{\mathrm{y1}}{}{w}{,}{z}{=}{\mathrm{x1}}{}{z}{+}{\mathrm{z1}}{,}{w}{=}{\mathrm{w1}}{}{w}\right]$ (2.1.8)
 G > $c≔\mathrm{ApplyTransformation}\left(\mathrm{muL},b\right)$
 ${c}{:=}\left[{x}{=}{\mathrm{x1}}{}{\mathrm{x2}}{,}{y}{=}{\mathrm{x1}}{}{\mathrm{y2}}{+}{\mathrm{y1}}{}{\mathrm{w2}}{,}{z}{=}{\mathrm{x1}}{}{\mathrm{z2}}{+}{\mathrm{z1}}{,}{w}{=}{\mathrm{w1}}{}{\mathrm{w2}}\right]$ (2.1.9)

Evaluate $X$ at $b$ and at $c$.

 G > $\mathrm{X_b}≔\mathrm{eval}\left(X,b\right)$
 ${\mathrm{X_b}}{:=}{\mathrm{y2}}{}{\mathrm{D_y}}{+}{\mathrm{w2}}{}{\mathrm{D_w}}$ (2.1.10)
 G > $\mathrm{X_c}≔\mathrm{eval}\left(X,c\right)$
 ${\mathrm{X_c}}{:=}\left({\mathrm{x1}}{}{\mathrm{y2}}{+}{\mathrm{y1}}{}{\mathrm{w2}}\right){}{\mathrm{D_y}}{+}{\mathrm{w1}}{}{\mathrm{w2}}{}{\mathrm{D_w}}$ (2.1.11)

Pushforward $X$ by ${\mathrm{μ}}_{L}$. Since Y the vector field X is left invariant.

 G > $Y≔\mathrm{Pushforward}\left(\mathrm{muL},\mathrm{X_b}\right)$
 ${Y}{:=}\left({\mathrm{x1}}{}{\mathrm{y2}}{+}{\mathrm{y1}}{}{\mathrm{w2}}\right){}{\mathrm{D_y}}{+}{\mathrm{w1}}{}{\mathrm{w2}}{}{\mathrm{D_w}}$ (2.1.12)

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

 G > $\mathrm{Pushforward}\left(\mathrm{muL},\mathrm{InverseTransformation}\left(\mathrm{muL}\right),X\right)$
 ${y}{}{\mathrm{D_y}}{+}{w}{}{\mathrm{D_w}}$ (2.1.13)

The left invariance of the form OmegaL[2] is similarly verified (by observing that

 G > $\mathrm{\omega }≔\mathrm{OmegaL}\left[2\right]$
 ${\mathrm{ω}}{:=}\frac{{\mathrm{dy}}}{{x}}{-}\frac{{y}{}{\mathrm{dw}}}{{x}{}{w}}$ (2.1.14)
 G > $\mathrm{omega_c}≔\mathrm{eval}\left(\mathrm{\omega },c\right)$
 ${\mathrm{omega_c}}{:=}\frac{{\mathrm{dy}}}{{\mathrm{x1}}{}{\mathrm{x2}}}{-}\frac{\left({\mathrm{x1}}{}{\mathrm{y2}}{+}{\mathrm{y1}}{}{\mathrm{w2}}\right){}{\mathrm{dw}}}{{\mathrm{x1}}{}{\mathrm{x2}}{}{\mathrm{w1}}{}{\mathrm{w2}}}$ (2.1.15)
 G > $\mathrm{omega_b}≔\mathrm{eval}\left(\mathrm{\omega },b\right)$
 ${\mathrm{omega_b}}{:=}\frac{{\mathrm{dy}}}{{\mathrm{x2}}}{-}\frac{{\mathrm{y2}}{}{\mathrm{dw}}}{{\mathrm{x2}}{}{\mathrm{w2}}}$ (2.1.16)
 G > $\mathrm{\theta }≔\mathrm{Pullback}\left(\mathrm{muL},\mathrm{omega_c}\right)$
 ${\mathrm{θ}}{:=}\frac{{\mathrm{dy}}}{{\mathrm{x2}}}{-}\frac{{\mathrm{y2}}{}{\mathrm{dw}}}{{\mathrm{x2}}{}{\mathrm{w2}}}$ (2.1.17)
 G > $\mathrm{Pullback}\left(\mathrm{muL},\mathrm{\omega }\right)$
 $\frac{{\mathrm{dy}}}{{x}}{-}\frac{{y}{}{\mathrm{dw}}}{{x}{}{w}}$ (2.1.18)