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GroupActions[IsotropyFiltration] - find the infinitesimal isotropy filtration for a Lie algebra of vector fields
Calling Sequences
IsotropyFiltration(Gamma, pt, option)
Parameters
Gamma - a list of vector fields on a manifold M
pt - a list of coordinate values [x1 = p1, x2 = p2, ...] specifying a point p in M
option - the optional argument output = O, where O is a list containing the keywords "Vector" and/or the name of an initialized abstract algebra for the Lie algebra of vector fields Gamma.
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Description
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The isotropy filtration of a Lie algebra of vector fields Gamma is the decreasing nested sequence of subalgebras Gamma^k_p = {X in Gamma | the coefficients of X and all their derivatives to order k vanish}. If X in Gamma^k_p and Y in Gamma^l_p, then [X, Y] in Gamma^(k + l)_p.
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The command IsotropyFiltration is part of the DifferentialGeometry:-GroupActions package. It can be used in the form IsotropyFiltration(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-IsotropyFiltration(...).
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Examples
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Example 1.
First we obtain a Lie algebra of vector fields from the paper by Gonzalez-Lopez, Kamran, Olver in the DifferentialGeometry Library using the Retrieve command and then we compute the isotropy filtration.
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![DGsetup([x, y], M)](/support/helpjp/helpview.aspx?si=6636/file05745/math90.png)
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| (2.1) |
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M >
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![GAMMA := Retrieve("Gonzalez-Lopez", 1, [27, 4], manifold = M)](/support/helpjp/helpview.aspx?si=6636/file05745/math97.png)
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![Gamma := [D_x, 2*x*D_x+4*y*D_y, x^2*D_x+4*x*y*D_y, D_y, x*D_y, x^2*D_y, x^3*D_y, x^4*D_y]](/support/helpjp/helpview.aspx?si=6636/file05745/math100.png)
| (2.2) |
We calculate the isotropy filtration as a subalgebra of Gamma.
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M >
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![F1 := IsotropyFiltration(GAMMA, [x = 0, y = 0])](/support/helpjp/helpview.aspx?si=6636/file05745/math110.png)
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![F1 := [[2*x*D_x+4*y*D_y, x^2*D_x+4*x*y*D_y, x*D_y, x^2*D_y, x^3*D_y, x^4*D_y], [x^2*D_x+4*x*y*D_y, x^2*D_y, x^3*D_y, x^4*D_y], [x^3*D_y, x^4*D_y], [x^4*D_y], []]](/support/helpjp/helpview.aspx?si=6636/file05745/math113.png)
| (2.3) |
We calculate the isotropy filtration as a subalgebra of the abstract Lie algebra defined by Gamma. To this end, we first calculate the structure constants for Gamma and initialize the result as Alg1.
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M >
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![L := [[e1, e2] = 2*e1, [e1, e3] = e2, [e1, e5] = e4, [e1, e6] = 2*e5, [e1, e7] = 3*e6, [e1, e8] = 4*e7, [e2, e3] = 2*e3, [e2, e4] = -4*e4, [e2, e5] = -2*e5, [e2, e7] = 2*e7, [e2, e8] = 4*e8, [e3, e4] = -4*e5, [e3, e5] = -3*e6, [e3, e6] = -2*e7, [e3, e7] = -e8]](/support/helpjp/helpview.aspx?si=6636/file05745/math130.png)
| (2.4) |
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M >
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Re-run the IsotropyFiltration command with the 3rd argument output = [Alg1].
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Alg1 >
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![F := IsotropyFiltration(GAMMA, [x = 0, y = 0], output = [Alg1])](/support/helpjp/helpview.aspx?si=6636/file05745/math146.png)
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![F := [[e2, e3, e5, e6, e7, e8], [e3, e6, e7, e8], [e7, e8], [e8], []]](/support/helpjp/helpview.aspx?si=6636/file05745/math149.png)
| (2.5) |
We check that F does indeed define a filtration (note that there is an index shift Gamma^k_p = F[k + 1]).
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Alg1 >
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![BracketOfSubspaces(F[1], F[1])](/support/helpjp/helpview.aspx?si=6636/file05745/math161.png)
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![[2*e3, -2*e5, 2*e7, 4*e8, -3*e6]](/support/helpjp/helpview.aspx?si=6636/file05745/math164.png)
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Alg1 >
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![BracketOfSubspaces(F[1], F[2])](/support/helpjp/helpview.aspx?si=6636/file05745/math168.png)
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![[2*e3, 2*e7, 4*e8, 3*e6]](/support/helpjp/helpview.aspx?si=6636/file05745/math171.png)
| (2.7) |
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Alg1 >
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![BracketOfSubspaces(F[1], F[3])](/support/helpjp/helpview.aspx?si=6636/file05745/math175.png)
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![[2*e7, 4*e8]](/support/helpjp/helpview.aspx?si=6636/file05745/math178.png)
| (2.8) |
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Alg1 >
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![BracketOfSubspaces(F[1], F[4])](/support/helpjp/helpview.aspx?si=6636/file05745/math182.png)
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![[4*e8]](/support/helpjp/helpview.aspx?si=6636/file05745/math185.png)
| (2.9) |
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Alg1 >
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![BracketOfSubspaces(F[2], F[3])](/support/helpjp/helpview.aspx?si=6636/file05745/math189.png)
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![[-e8]](/support/helpjp/helpview.aspx?si=6636/file05745/math192.png)
| (2.10) |
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Alg1 >
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![BracketOfSubspaces(F[2], F[4])](/support/helpjp/helpview.aspx?si=6636/file05745/math196.png)
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![[]](/support/helpjp/helpview.aspx?si=6636/file05745/math199.png)
| (2.11) |
All these brackets can be checked at once with Query/"filtration".
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Alg1 >
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| (2.12) |
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