 VectorStructureEquations - Maple Help

LieAlgebraData[VectorStructureEquations] - convert a list of exterior derivative equations to a Lie algebra data structure

Calling Sequence

Parameters

VectorStructureEquations  - a list of equations of the form

Basis                     - a list of unassigned names which defines a basis for the Lie algebra

AlgName                   - a name or string, the name to be assigned to the Lie algebra 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.
 • The most common format for describing the structure equations of a Lie algebra is to list the non-zero Lie brackets. The function LieAlgebraData enables one to create a Lie algebra in Maple from a list of Lie bracket equations.
 • 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.

In this example we create a Lie algebra data structure for a Lie algebra called Ex1 from a list of Lie bracket equations. First we create the list of structure equations.  The variables x, y, z$,$u$,$v must be unassigned names. They simply serve as placeholders for the purpose of entering in the structure equations.

 > VectStrEq := [[u, v] = x, [y, z] = y, [x, v] = x];
 ${\mathrm{VectStrEq}}{:=}\left[\left[{u}{,}{v}\right]{=}{x}{,}\left[{y}{,}{z}\right]{=}{y}{,}\left[{x}{,}{v}\right]{=}{x}\right]$ (2.1)
 > Basis := [x, y, z, u, v];
 ${\mathrm{Basis}}{:=}\left[{x}{,}{y}{,}{z}{,}{u}{,}{v}\right]$ (2.2)
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}\right]$ (2.3)

To assign the names [X, Y, Z, U, V] to the basis vectors for this Lie algebra, use the optional argument form of DGsetup.

 > DGsetup(L, ['X', 'Y', 'Z', 'U', 'V'], ['theta']);
 ${\mathrm{Lie algebra: L1}}$ (2.4)
 L1 > MultiplicationTable("LieBracket");
 $\left[\left[{X}{,}{V}\right]{=}{X}{,}\left[{Y}{,}{Z}\right]{=}{Y}{,}\left[{U}{,}{V}\right]{=}{X}\right]$ (2.5)