find the infinitesimal isotropy subalgebra of a Lie algebra of vector fields and the representation of the isotropy subalgebra on the tangent space

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GroupActions[IsotropySubalgebra] - find the infinitesimal isotropy subalgebra of a Lie algebra of vector fields and the representation of the isotropy subalgebra on the tangent space

Calling Sequences

IsotropySubalgebra(Gamma, p, option)

Parameters

Gamma - a list of vector fields on a manifold M

p - 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", "Representation", and/or the name of an initialized abstract algebra for the Lie algebra of vector fields Gamma.

Description

•	The isotropy subalgebra Gamma_p of a Lie algebra of vector fields Gamma at the point p is defined by Gamma_p = {X in Gamma \| X_p = 0}. The Lie bracket defines a natural representation rho of Gamma_p on the tangent space T_pM by rho(X)(Y) = [X, Y], where X in Gamma_p and Y in T_pM.

•	IsotropySubalgebra returns a list of vectors giving the isotropy subalgebra Gamma_p as a subalgebra of Gamma.

•	With output = ["Vector", "Representation"], two lists are returned. The first is a list of vectors giving the isotropy subalgebra Gamma_p as a subalgebra of Gamma and the second is the list of matrices defining the linear isotropy representation with respect to the standard basis for T_pM.

•	Let algname be the name of the abstract Lie algebra g created from Gamma. With output = ["Vector", algname], the second list returned gives the isotropy subalgebra as a subalgebra of the abstract Lie algebra g.

•

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

Examples

Example 1.

We use the Retrieve command to obtain a Lie algebra of vector fields in the paper by Gonzalez-Lopez, Kamran, and Olver from the DifferentialGeometry Library. We compute the isotropy subalgebra and isotropy representation at the points [x = 0, y = 0] and [x = 1, y = 1].

iso1 >

M >

(2.1)

M >

(2.2)

M >

(2.3)

Alg1 >

[[[,`|`,e1,e2,e3,e4,e5],[,-`--`,-`---`,-`---`,-`---`,-`---`,-`---`],[e1,`|`,0,0,e1,0,e2],[e2,`|`,0,0,-e2,e1,0],[e3,`|`,-e1,e2,0,-2 e4,2 e5],[e4,`|`,0,-e1,2 e4,0,-e3],[e5,`|`,-e2,0,-2 e5,e3,0]]]

(2.4)

We illustrate some different possible outputs from the IsotropySubalgebra program.

Alg1 >

(2.5)

M >

(2.6)

Alg1 >

$Iso1, A1, S1 := [[x*D_x-y*D_y, y*D_x, x*D_y]], [[e3, e4, e5]], [[Matrix(2, 2, {(1, 1) = -1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = -1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = -1, (2, 2) = 0})]]$

(2.7)

Alg1 >

(2.8)

Alg1 >

$Iso2, A2, S2 := [[(-y+1)*D_x, (-x+1)*D_y, (-y+x)*D_x+(-y+x)*D_y]], [[e1-e4, e2-e5, e3-e4+e5]], [[Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = -1, (1, 2) = 1, (2, 1) = -1, (2, 2) = 1})]]$

(2.9)

Note that the vectors in Iso2 all vanish at [x = 1, y =1]

It is apparent from the multiplication table that the pair [Alg1, S1] is a symmetric pair with respect to the complementary subspace T = [e1, e2]. Of course, we can check this with the command Query/"SymmetricPair".

Alg1 >

(2.10)

The isotropy representation can be converted to a representation.

Alg1 >

(2.11)

Alg1 >

(2.12)

iso1 >

rho := [[[[e1, Matrix(2, 2, {(1, 1) = -1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1})]], [[e2, Matrix(2, 2, {(1, 1) = 0, (1, 2) = -1, (2, 1) = 0, (2, 2) = 0})]], [[e3, Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = -1, (2, 2) = 0})]]]]