find a Lie algebra of pointwise independent vector fields with prescribed structure equations (solvable algebras only)

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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.

(2.1)

We define a manifold M of dimension 4 (the same dimension as the Lie algebra).

Alg1 >

(2.2)

M1 >

(2.3)

M1 >

(2.4)

We calculate the structure equations for the Lie algebra of vector fields Gamma1 and check that these structure equations coincide with those for Alg1.

M1 >

(2.5)

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.)

M1 >

(2.6)

Alg2 >

(2.7)

Alg2 >

(2.8)

M1 >

(2.9)

Example 3.

Here is an example where one of the adjoint matrices has complex eigenvalues. The Lie algebra contains parameters p and b.

M1 >

(2.10)

M1 >

Alg3 >

Matrix(5, 5, {(1, 1) = -2*p, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (1, 5) = 0, (2, 1) = 0, (2, 2) = -p, (2, 3) = 1, (2, 4) = 0, (2, 5) = 0, (3, 1) = 0, (3, 2) = -1, (3, 3) = -p, (3, 4) = 0, (3, 5) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = -b, (4, 5) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = 0})

(2.11)

Alg3 >

(2.12)

M3 >

[_DG([["vector", M3, []], [[[1], 1]]]), _DG([["vector", M3, []], [[[1], -z], [[2], 1]]]), _DG([["vector", M3, []], [[[3], 1]]]), _DG([["vector", M3, []], [[[4], 1]]]), _DG([["vector", M3, []], [[[1], (1/2)*z^2-(1/2)*y^2+2*p*x], [[2], y*p-z], [[3], z*p+y], [[4], b*u], [[5], 1]]])]

(2.13)

M3 >

(2.14)

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.

(2.15)

MaurerCartan := Matrix(4, 4, {(1, 1) = -dw, (1, 2) = -dw, (1, 3) = `0`*dx, (1, 4) = dx+dy+(-2*y-z-x)*dw, (2, 1) = `0`*dx, (2, 2) = -dw, (2, 3) = -dw, (2, 4) = dy+dz+(-2*z-y)*dw, (3, 1) = `0`*dx, (3, 2) = `0`*dx, (3, 3) = -dw, (3, 4) = dz-z*dw, (4, 1) = `0`*dx, (4, 2) = `0`*dx, (4, 3) = `0`*dx, (4, 4) = `0`*dx})

(2.16)

Note that the elements of this matrix coincide with the appropriate linear combinations of the forms in the list from Example 1.

Alg1 >

(2.17)

Alg1 >

(2.18)

Alg1 >

(2.19)

Alg1 >

(2.20)

	See Also
	DifferentialGeometry, GroupActions, Library, LieAlgebras, Representation, Adjoint, LieAlgebraData, Representation, Retrieve, SolvableRepresentation