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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.
Description
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.
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(...).
Examples
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.
We calculate the isotropy filtration as a subalgebra of Gamma.
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.
Re-run the IsotropyFiltration command with the 3rd argument output = [Alg1].
We check that F does indeed define a filtration (note that there is an index shift Gamma^k_p = F[k + 1]).
All these brackets can be checked at once with Query/"filtration".
See Also
DifferentialGeometry, GroupActions, Library, LieAlgebras, BracketOfSubspaces, IsotropySubalgebra, LieAlgebraData, Query
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