convert a Lie algebra of vector fields to a Lie algebra data structure - Maple Programming Help

LieAlgebraData[VectorFields] - convert a Lie algebra of vector fields to a Lie algebra data structure

Calling Sequence

Parameters

Gamma    - a list of vector fields on a manifold $M$

Algname  - a name or string, the name to be assigned to the Lie algebra

option   - initialpointlist = [pt1, pt2, ...], a list of lists of coordinates for points of $M$

Description

 • In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra, that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.  The first argument for DGsetup is a Lie algebra data structure which contains the structure constants in a standard format used by the LieAlgebras package.
 • An important class of Lie algebras are provided by the infinitesimal generators (vector fields) for any r-parameter transformation group. These vector fields form a Lie algebra with respect the LieBracket operation. The program LieAlgebraData(Gamma, AlgName) will create the Lie algebra data structure for the Lie algebra of vector fields Gamma.
 • This program uses the GetComponents command with the option method = "real" to calculate the Lie brackets of the vector fields in Gamma as constant, real, linear combinations of the vectors in Gamma. When the component functions of the vector fields in Gamma are transcendental functions, it is best to specify a small list of points at which these component functions and their derivatives are easily evaluated.
 • The command LieAlgebraData is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form LieAlgebraData(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-LieAlgebraData(...).

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

 > DGsetup([x, y], M1);
 ${\mathrm{frame name: M1}}$ (2.1)
 M > Gamma1 := evalDG([D_x, D_y, x*D_x, y*D_x]);
 ${\mathrm{Γ1}}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{x}{}{\mathrm{D_x}}{,}{y}{}{\mathrm{D_x}}\right]$ (2.2)

We calculate the structure equations for this Lie algebra of vector fields.

 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}\right]$ (2.3)

Example 2.

 M > DGsetup([x, y, z], M2);
 ${\mathrm{frame name: M2}}$ (2.4)
 M2 > Gamma2 := evalDG([D_y, cos(y)*D_x-cos(x)/sin(x)*sin(y)*D_y+sin(y)/sin(x)*D_z, -sin(y)*D_x-cos(x)/sin(x)*cos(y)*D_y+cos(y)/sin(x)*D_z]);
 ${\mathrm{Γ2}}{:=}\left[{\mathrm{D_y}}{,}{\mathrm{cos}}{}\left({y}\right){}{\mathrm{D_x}}{-}\frac{{\mathrm{cos}}{}\left({x}\right){}{\mathrm{sin}}{}\left({y}\right){}{\mathrm{D_y}}}{{\mathrm{sin}}{}\left({x}\right)}{+}\frac{{\mathrm{sin}}{}\left({y}\right){}{\mathrm{D_z}}}{{\mathrm{sin}}{}\left({x}\right)}{,}{-}{\mathrm{sin}}{}\left({y}\right){}{\mathrm{D_x}}{-}\frac{{\mathrm{cos}}{}\left({x}\right){}{\mathrm{cos}}{}\left({y}\right){}{\mathrm{D_y}}}{{\mathrm{sin}}{}\left({x}\right)}{+}\frac{{\mathrm{cos}}{}\left({y}\right){}{\mathrm{D_z}}}{{\mathrm{sin}}{}\left({x}\right)}\right]$ (2.5)
 ${\mathrm{Γ2}}{:=}\left[{\mathrm{D_y}}{,}{\mathrm{cos}}{}\left({y}\right){}{\mathrm{D_x}}{-}\frac{{\mathrm{cos}}{}\left({x}\right){}{\mathrm{sin}}{}\left({y}\right){}{\mathrm{D_y}}}{{\mathrm{sin}}{}\left({x}\right)}{+}\frac{{\mathrm{sin}}{}\left({y}\right){}{\mathrm{D_z}}}{{\mathrm{sin}}{}\left({x}\right)}{,}{-}{\mathrm{sin}}{}\left({y}\right){}{\mathrm{D_x}}{-}\frac{{\mathrm{cos}}{}\left({x}\right){}{\mathrm{cos}}{}\left({y}\right){}{\mathrm{D_y}}}{{\mathrm{sin}}{}\left({x}\right)}{+}\frac{{\mathrm{cos}}{}\left({y}\right){}{\mathrm{D_z}}}{{\mathrm{sin}}{}\left({x}\right)}\right]$ (2.6)

We calculate the structure equations for this Lie algebra of vector fields.

 M2 > LieAlgebraData(Gamma2, Alg2, initialpointlist = [[x = Pi/2, y = 0, z = 0]]);
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (2.7)