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GroupActions[LiesThirdTheorem] - find a Lie algebra of pointwise independent vector fields with prescribed structure equations (solvable algebras only)
Calling Sequences
LiesThirdTheorem(Alg, M, option)
LiesThirdTheorem(A, M)
Parameters
Alg - a Maple name or string, the name of an initialized Lie algebra g
M - a Maple name or string, the name of an initialized manifold with the same dimension as that of g
option - with output = "forms" the dual 1-forms (Maurer-Cartan forms) are returned
A - a list of square matrices, defining a matrix Lie algebra
Description
Let g be an n-dimensional Lie algebra with structure constants C. Then Lie's Third Theorem (see, for example, Flanders, page 108) asserts that there is, at least locally, a Lie algebra of n pointwise independent vector fields Gamma on an n-dimensional manifold M with structure constants C.
The command LiesThirdTheorem(Alg, M) produces a globally defined Lie algebra of vector fields Gamma in the special case that g is solvable. More general cases will be handled in subsequent versions of the DifferentialGeometry package.
The command LiesThirdTheorem(A, M) produces a globally defined matrix of 1-forms (Maurer-Cartan forms) in the special case that the list of matrices A defines a solvable Lie algebra.
The command LiesThirdTheorem is part of the DifferentialGeometry:-GroupActions package. It can be used in the form LiesThirdTheorem(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-LiesThirdTheorem(...).
Examples
Example 1.
We obtain a Lie algebra from the DifferentialGeometry library using the Retrieve command and initialize it.
We define a manifold M of dimension 4 (the same dimension as the Lie algebra).
We calculate the structure equations for the Lie algebra of vector fields Gamma1 and check that these structure equations coincide with those for Alg1.
Example 2.
We re-work the previous example in a more complicated basis. In this basis the adjoint representation is not upper triangular, in which case LiesThirdTheorem first calls the program SolvableRepresentation to find a basis for the algebra in which the adjoint representation is upper triangular. (Remark: It is almost always useful, when working with solvable algebras, to transform to a basis where the adjoint representation is upper triangular.)
Example 3.
Here is an example where one of the adjoint matrices has complex eigenvalues. The Lie algebra contains parameters p and b.
Example 4.
We calculate the Maurer-Cartan matrix of 1-forms for a solvable matrix algebra, namely the matrices defining the adjoint representation for Alg1 from Example 1.
Note that the elements of this matrix coincide with the appropriate linear combinations of the forms in the list from Example 1.
See Also
DifferentialGeometry, GroupActions, Library, LieAlgebras, Representation, Adjoint, LieAlgebraData, Representation, Retrieve, SolvableRepresentation
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