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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"
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Description
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Let G be a Lie group with multiplication * and define diffeomorphisms L_a: G -> G and R_a: G -> G by L_a(g) = a*g and R_a(g) = g*a. A vector field X on G is left invariant if (L_a)_*(X_b) = X_(a*b) and right invariant if (R_a)_*(X_b) = X_(b*a). A differential form omega on G is left invariant if (L_a)^*(omega_(a*b)) = omega_(b) and right invariant if (R_a)^*(omega_(b*a)) = omega_(b).
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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.
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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"].
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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(...).
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Examples
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Example 1.
We calculate a basis for the invariant vector fields and forms for the 4 dimensional matrix group defined by the matrix M.
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![M := Matrix([[x, y, z], [0, w, 0], [0, 0, 1]])](/support/helpjp/helpview.aspx?si=6643/file05744/math121.png)
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| (2.1) |
Create a local system of coordinates for the Lie group.
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![DGsetup([x, y, z, w], G)](/support/helpjp/helpview.aspx?si=6643/file05744/math132.png)
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| (2.2) |
Create the Lie group module for the matrix group M using the LieGroup command.
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| (2.3) |
Find a basis of left invariant vector fields and differential 1-forms.
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![XL, OmegaL := InvariantVectorsAndForms(LG, output = ["LeftVectors", "LeftForms"])](/support/helpjp/helpview.aspx?si=6643/file05744/math160.png)
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![[_DG([["vector", G, []], [[[1], x]]]), _DG([["vector", G, []], [[[2], x]]]), _DG([["vector", G, []], [[[3], x]]]), _DG([["vector", G, []], [[[2], y], [[4], w]]])], [_DG([["form", G, 1], [[[1], 1/x]]]), _DG([["form", G, 1], [[[2], 1/x], [[4], -y/(x*w)]]]), _DG([["form", G, 1], [[[3], 1/x]]]), _DG([["form", G, 1], [[[4], 1/w]]])]](/support/helpjp/helpview.aspx?si=6643/file05744/math163.png)
| (2.4) |
Find a basis of right invariant vector fields.
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![XR := InvariantVectorsAndForms(LG, output = ["RightVectors"])](/support/helpjp/helpview.aspx?si=6643/file05744/math172.png)
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![[_DG([["vector", G, []], [[[1], x], [[2], y], [[3], z]]]), _DG([["vector", G, []], [[[2], w]]]), _DG([["vector", G, []], [[[3], 1]]]), _DG([["vector", G, []], [[[4], w]]])]](/support/helpjp/helpview.aspx?si=6643/file05744/math175.png)
| (2.5) |
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Details for Example 1
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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.
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![_DG([["LieAlgebra", Alg2, [4]], [[[1, 2, 2], 1], [[1, 3, 3], 1], [[2, 4, 2], 1]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math190.png)
| (2.1.1) |
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![_DG([["LieAlgebra", Alg3, [4]], [[[1, 2, 2], -1], [[1, 3, 3], -1], [[2, 4, 2], -1]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math197.png)
| (2.1.2) |
The Lie derivatives of XL and OmegaL with respect to XR vanish:
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![Matrix(4, 4, proc (i, j) options operator, arrow; LieDerivative(XR[i], XL[j]) end proc)](/support/helpjp/helpview.aspx?si=6643/file05744/math211.png)
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| (2.1.3) |
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![Matrix(4, 4, proc (i, j) options operator, arrow; LieDerivative(XR[i], OmegaL[j]) end proc)](/support/helpjp/helpview.aspx?si=6643/file05744/math218.png)
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| (2.1.4) |
Let us check explicitly the left invariance of the vector field XL[4].
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![X := XL[4]](/support/helpjp/helpview.aspx?si=6643/file05744/math231.png)
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![_DG([["vector", G, []], [[[2], y], [[4], w]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math234.png)
| (2.1.5) |
Define points a and b and compute c = a*b.
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![a := [x1, y1, z1, w1]](/support/helpjp/helpview.aspx?si=6643/file05744/math248.png)
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![[x1, y1, z1, w1]](/support/helpjp/helpview.aspx?si=6643/file05744/math251.png)
| (2.1.6) |
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![b := [x = x2, y = y2, z = z2, w = w2]](/support/helpjp/helpview.aspx?si=6643/file05744/math255.png)
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![[x = x2, y = y2, z = z2, w = w2]](/support/helpjp/helpview.aspx?si=6643/file05744/math258.png)
| (2.1.7) |
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![_DG([["transformation", [[G, 0], [G, 0]], [], [Matrix(%id = 4583738882)]], [[x1*x, x], [x1*y+y1*w, y], [x1*z+z1, z], [w1*w, w]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math265.png)
| (2.1.8) |
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![[x = x1*x2, y = x1*y2+y1*w2, z = x1*z2+z1, w = w1*w2]](/support/helpjp/helpview.aspx?si=6643/file05744/math272.png)
| (2.1.9) |
Evaluate X at b and at c.
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![_DG([["vector", G, []], [[[2], y2], [[4], w2]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math289.png)
| (2.1.10) |
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![_DG([["vector", G, []], [[[2], x1*y2+y1*w2], [[4], w1*w2]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math296.png)
| (2.1.11) |
Pushforward X_b by muL . Since Y = X_c, the vector field X is invariant under left multiplication by muL.
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![_DG([["vector", G, []], [[[2], x1*y2+y1*w2], [[4], w1*w2]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math317.png)
| (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.
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![_DG([["vector", G, []], [[[2], y], [[4], w]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math334.png)
| (2.1.13) |
The left invariance of the form OmegaL[2] is similarly verified (by observing that theta = omega_c).
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![omega := OmegaL[2]](/support/helpjp/helpview.aspx?si=6643/file05744/math346.png)
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![_DG([["form", G, 1], [[[2], 1/x], [[4], -y/(x*w)]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math349.png)
| (2.1.14) |
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![_DG([["form", G, 1], [[[2], 1/(x1*x2)], [[4], -(x1*y2+y1*w2)/(x1*x2*w1*w2)]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math356.png)
| (2.1.15) |
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![_DG([["form", G, 1], [[[2], 1/x2], [[4], -y2/(x2*w2)]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math363.png)
| (2.1.16) |
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![_DG([["form", G, 1], [[[2], 1/x2], [[4], -y2/(x2*w2)]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math370.png)
| (2.1.17) |
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![_DG([["form", G, 1], [[[2], 1/x], [[4], -y/(x*w)]]])](/support/helpjp/helpview.aspx?si=6643/file05744/math377.png)
| (2.1.18) |
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