FormStructureEquations - Maple Help

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

Calling Sequence

Parameters

FormStructureEquations  - a list of equations of the form   (sum on with

Basis                   - a list of unassigned names  which defines a basis for the dual 1-forms of the Lie algebra

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

Description

 • 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.
 • One commonly used format for the structure equations of a Lie algebra $\mathrm{𝔤}$ is the set of exterior derivative equations for the dual 1-forms of the Lie algebra.  For a 1-form $\mathrm{θ}$ in the dual of a Lie algebra, the exterior derivative is the 2-form defined by , for . The function LieAlgebraData enables one to create a Lie algebra in Maple from a list of exterior derivative 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 structure equations for the exterior derivatives of the dual 1-forms. The structure equations contain arbitrary constants  and we determine for which values of these parameters the Jacobi identities actually hold. First, we create the list of structure equations.  The variables t1, t2, and t3 must be unassigned names.  They simply serve as placeholders for the purpose of entering in the structure equations.

 >
 > FormStrEq := [d(t1) = -t2 &w t3, d(t2) = - t1 &w t3, d(t3) = a*t1 &w t2 + b*t2 &w t3 + c*t1 &w t3];
 ${\mathrm{FormStrEq}}{:=}\left[{d}{}\left({\mathrm{t1}}\right){=}{-}{\mathrm{t2}}{&w}{\mathrm{t3}}{,}{d}{}\left({\mathrm{t2}}\right){=}{-}{\mathrm{t1}}{&w}{\mathrm{t3}}{,}{d}{}\left({\mathrm{t3}}\right){=}{a}{}{\mathrm{t1}}{&w}{\mathrm{t2}}{+}{b}{}{\mathrm{t2}}{&w}{\mathrm{t3}}{+}{c}{}{\mathrm{t1}}{&w}{\mathrm{t3}}\right]$ (2.1)
 > Basis := [t1, t2, t3];
 ${\mathrm{Basis}}{:=}\left[{\mathrm{t1}}{,}{\mathrm{t2}}{,}{\mathrm{t3}}\right]$ (2.2)
 > L := LieAlgebraData(FormStrEq, Basis, Ex1);
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{a}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{-}{c}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{-}{b}{}{\mathrm{e3}}\right]$ (2.3)
 > DGsetup(L);
 ${\mathrm{Lie algebra: Ex1}}$ (2.4)
 Ex1 > MultiplicationTable("ExteriorDerivative");
 ${d}{}\left({\mathrm{θ1}}\right){=}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$
 ${d}{}\left({\mathrm{θ2}}\right){=}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}$
 ${d}{}\left({\mathrm{θ3}}\right){=}{a}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{+}{c}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{+}{b}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$ (2.5)
 Ex1 > TF, EQ, SOLN, LIEALG := Query({a, b, c}, "Jacobi");
 ${\mathrm{TF}}{,}{\mathrm{EQ}}{,}{\mathrm{SOLN}}{,}{\mathrm{LIEALG}}{:=}{\mathrm{true}}{,}\left\{{0}{,}{b}{,}{-}{c}\right\}{,}\left[\left\{{a}{=}{a}{,}{b}{=}{0}{,}{c}{=}{0}\right\}\right]{,}\left[\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{a}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]\right]$ (2.6)

We conclude that the structure equations define a Lie Algebra for arbitrary with