IsotropySubalgebra - Maple Help
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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

     p         - a list of equations  specifying the coordinates of point

     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

Examples

Description

• 

Let be a Lie algebra of vector fields on a manifold and letThe isotropy subalgebra  of the Lie algebra of vector fields at the point is defined by . The Lie bracket of vector fields defines a natural representation of  on the tangent space  by  for  , and  any vector field on  such that . The representation  is called the linear isotropy representation.

• 

IsotropySubalgebra(Gamma, p) returns a list of vectors whose span defines the isotropy subalgebra as a subalgebra of  .

• 

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

• 

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

• 

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  and  

M > 

(2.1)
M > 

(2.2)
M > 

(2.3)
Alg1 > 

 

We illustrate some different possible outputs from the IsotropySubalgebra program.

Alg1 > 

(2.4)
M > 

(2.5)
Alg1 > 

Alg1 > 

(2.6)
Alg1 > 

Note that the vectors in Iso2 all vanish at  

 

It is apparent from the multiplication table that the pair Alg1, S1 is a symmetric pair with respect to the complementary subspace. We can check this with the command Query/"SymmetricPair".

Alg1 > 

(2.7)

 

The linear isotropy representation can be converted to a representation.

Alg1 > 

(2.8)
Alg1 > 

(2.9)
iso1 > 

iso1 > 

(2.10)

See Also

DifferentialGeometry

GroupActions

Library

LieAlgebras

LieAlgebraData

MultiplicationTable

Query

Representation

Retrieve

 


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