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LieAlgebras[InfinitesimalCoadjointAction] - find the vector fields defining the infinitesimal co-adjoint action of a Lie group on its Lie algebra
Calling Sequences
InfinitesimalCoadjointAction(Algebra, Manifold)
Parameters
Algebra - name or string, the name of an initialized Lie algebra
Manifold - name or string, the name of an initialized manifold
Description
Let be an -dimensional Lie group with Lie algebra and let be the structure equations for . If are coordinates for the dual vector space , then the infinitesimal generators for the co-adjoint action of onare the vector fields .
The command InfinitesimalCoadjointAction(Algebra, Manifold) calculates the vector fields for the Lie algebra Algebra using the coordinates for the dual space provide by Manifold.
The command InfinitesimalCoadjointAction is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form InfinitesimalCoadjointAction(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:- LieAlgebras:- InfinitesimalCoadjointAction(...).
Examples
Example 1.
First we initialize a Lie algebra.
Now define coordinates for the dual of the Lie algebra.
Calculate the infinitesimal generators for the co-adjoint action.
The center of the Lie algebra is trivial and therefore the structure equations for the Lie algebra are the same as those for .
The vector fields may be calculated directly using the Adjoint and convert/DGvector commands. For example, we obtain the last vector in as follows.
Example 2.
First we initialize a 4-dimensional Lie algebra.
In this example, the Lie algebra has a non-trivial center and now the structure equations for are those for the quotient of by its center.
Example 3.
The invariants for the co-adjoint action are called generalized Casimir operators (See J. Patera, R. T. Sharp , P. Winternitz and H. Zassenhaus, Invariants of real low dimensional Lie algebras, J. Math. Phys. vol 17, No 6, June 1976, 966--994).
We calculate the generalized Casimir operators for the Lie algebra [5,12] from this article. First use the Retrieve command to obtain the structure equations for this algebra and initialize the Lie algebra.
We use the InvariantGeometricObjectFields command to calculate the functions which invariant under the group generated by .
Functional combinations of these invariants give the formulas for the generalized Casimir operators in the Patera, Sharp, et al. paper.
See Also
DifferentialGeometry, GroupActions, Library, LieAlgebras, convert/DGvector, LieAlgebraData, Adjoint, Retrieve, InvariantGeometricObjectFields
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