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

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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 $\mathrm{𝔤}$

M         - a Maple name or string, the name of an initialized manifold with the same dimension as that of $\mathrm{𝔤}$

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 $\mathrm{Γ}$ on an $n$-dimensional manifold with structure constants $C$.
 • The command LiesThirdTheorem(Alg, M) produces a globally defined Lie algebra of vector fields $\mathrm{Γ}$ in the special case that 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

 > with(DifferentialGeometry): with(LieAlgebras): with(GroupActions): with(Library):

Example 1.

We obtain a Lie algebra from the DifferentialGeometry library using the Retrieve command and initialize it.

 > L := Retrieve("Winternitz", 1, [4, 4], Alg1);
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{+}{\mathrm{e2}}\right]$ (2.1)
 > DGsetup(L):

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

 Alg1 > DGsetup([x, y, z, w], M1);
 ${\mathrm{frame name: M1}}$ (2.2)
 M1 > Gamma1 := LiesThirdTheorem(Alg1, M1);
 ${\mathrm{Γ1}}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}{,}\left({x}{+}{y}\right){}{\mathrm{D_x}}{+}\left({y}{+}{z}\right){}{\mathrm{D_y}}{+}{z}{}{\mathrm{D_z}}{+}{\mathrm{D_w}}\right]$ (2.3)
 M1 > Omega1 := LiesThirdTheorem(Alg1, M1, output = "forms");
 ${\mathrm{Ω1}}{:=}\left[{\mathrm{dx}}{-}\left({x}{+}{y}\right){}{\mathrm{dw}}{,}{\mathrm{dy}}{-}\left({y}{+}{z}\right){}{\mathrm{dw}}{,}{-}{\mathrm{dw}}{}{z}{+}{\mathrm{dz}}{,}{\mathrm{dw}}\right]$ (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.

 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{+}{\mathrm{e2}}\right]$ (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 > L2 := LieAlgebraData([e4, e2 - e4, e3, e1 + e3], Alg2);
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e4}}{+}{\mathrm{e3}}{-}{\mathrm{e2}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{-}{\mathrm{e2}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{-}{\mathrm{e2}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{+}{\mathrm{e2}}{+}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{+}{\mathrm{e2}}{+}{\mathrm{e1}}\right]$ (2.6)
 Alg2 > DGsetup(L2);
 ${\mathrm{Lie algebra: Alg2}}$ (2.7)
 Alg2 > Gamma2 := LiesThirdTheorem(Alg2, M1);
 ${\mathrm{Γ2}}{:=}\left[\left({x}{-}{y}\right){}{\mathrm{D_x}}{+}\left({y}{+}{z}\right){}{\mathrm{D_y}}{+}{z}{}{\mathrm{D_z}}{+}{\mathrm{D_w}}{,}{-}\left({x}{-}{y}\right){}{\mathrm{D_x}}{-}\left({-}{1}{+}{y}{+}{z}\right){}{\mathrm{D_y}}{-}{z}{}{\mathrm{D_z}}{-}{\mathrm{D_w}}{,}{\mathrm{D_z}}{,}{-}{\mathrm{D_x}}{+}{\mathrm{D_z}}\right]$ (2.8)
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e4}}{+}{\mathrm{e3}}{-}{\mathrm{e2}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{-}{\mathrm{e2}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{-}{\mathrm{e2}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{+}{\mathrm{e2}}{+}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{+}{\mathrm{e2}}{+}{\mathrm{e1}}\right]$ (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 > L3 := Retrieve("Winternitz", 1, [5, 25], Alg3);
 ${\mathrm{L3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{_p}}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{_p}}{}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{_p}}{}{\mathrm{e3}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{_b}}{}{\mathrm{e4}}\right]$ (2.10)
 M1 > DGsetup(L3):
 $\left[\begin{array}{ccccc}{-}{2}{}{\mathrm{_p}}& {0}& {0}& {0}& {0}\\ {0}& {-}{\mathrm{_p}}& {1}& {0}& {0}\\ {0}& {-}{1}& {-}{\mathrm{_p}}& {0}& {0}\\ {0}& {0}& {0}& {-}{\mathrm{_b}}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]$ (2.11)
 Alg3 > DGsetup([x, y, z, u ,v], M3);
 ${\mathrm{frame name: M3}}$ (2.12)
 M3 > Gamma3 := LiesThirdTheorem(Alg3, M3);
 ${\mathrm{Γ3}}{:=}\left[{\mathrm{D_x}}{,}{-}{\mathrm{D_x}}{}{z}{+}{\mathrm{D_y}}{,}{\mathrm{D_z}}{,}{\mathrm{D_u}}{,}\left(\frac{{1}}{{2}}{}{{z}}^{{2}}{-}\frac{{1}}{{2}}{}{{y}}^{{2}}{+}{2}{}{\mathrm{_p}}{}{x}\right){}{\mathrm{D_x}}{+}\left({\mathrm{_p}}{}{y}{-}{z}\right){}{\mathrm{D_y}}{+}\left({\mathrm{_p}}{}{z}{+}{y}\right){}{\mathrm{D_z}}{+}{\mathrm{_b}}{}{u}{}{\mathrm{D_u}}{+}{\mathrm{D_v}}\right]$ (2.13)
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{_p}}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{_p}}{}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{_p}}{}{\mathrm{e3}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{_b}}{}{\mathrm{e4}}\right]$ (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.

 ${A}{:=}\left[\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{-}{1}& {-}{1}& {0}& {0}\\ {0}& {-}{1}& {-}{1}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]$ (2.15)
 > MaurerCartan := LiesThirdTheorem(A, M1);
 ${\mathrm{MaurerCartan}}{:=}\left[\begin{array}{cccc}{-}{\mathrm{dw}}& {-}{\mathrm{dw}}& {0}{}{\mathrm{dx}}& {\mathrm{dx}}{+}{\mathrm{dy}}{-}\left({2}{}{y}{+}{z}{+}{x}\right){}{\mathrm{dw}}\\ {0}{}{\mathrm{dx}}& {-}{\mathrm{dw}}& {-}{\mathrm{dw}}& {\mathrm{dy}}{+}{\mathrm{dz}}{-}\left({2}{}{z}{+}{y}\right){}{\mathrm{dw}}\\ {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {-}{\mathrm{dw}}& {-}{\mathrm{dw}}{}{z}{+}{\mathrm{dz}}\\ {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}\end{array}\right]$ (2.16)

Note that the elements of this matrix   coincide with the appropriate linear combinations of the forms in the list ${\mathrm{Ω}}_{1}$ from Example 1.

 Alg1 > MaurerCartan[1,4], Omega1[1] &plus Omega1[2];
 ${\mathrm{dx}}{+}{\mathrm{dy}}{-}\left({2}{}{y}{+}{z}{+}{x}\right){}{\mathrm{dw}}{,}{\mathrm{dx}}{+}{\mathrm{dy}}{-}\left({2}{}{y}{+}{z}{+}{x}\right){}{\mathrm{dw}}$ (2.17)
 Alg1 > MaurerCartan[2,4], Omega1[2] &plus Omega1[3];
 ${\mathrm{dy}}{+}{\mathrm{dz}}{-}\left({2}{}{z}{+}{y}\right){}{\mathrm{dw}}{,}{\mathrm{dy}}{+}{\mathrm{dz}}{-}\left({2}{}{z}{+}{y}\right){}{\mathrm{dw}}$ (2.18)
 Alg1 > MaurerCartan[3,4], Omega1[3];
 ${-}{\mathrm{dw}}{}{z}{+}{\mathrm{dz}}{,}{-}{\mathrm{dw}}{}{z}{+}{\mathrm{dz}}$ (2.19)
 Alg1 > MaurerCartan[1,1], (-1) &mult Omega1[4];
 ${-}{\mathrm{dw}}{,}{-}{\mathrm{dw}}$ (2.20)