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(...).

with(DifferentialGeometry): with(LieAlgebras):

VectStrEq := [[u, v] = x, [y, z] = y, [x, v] = x];

$\mathrm{VectStrEq}:=\left[\left[u\,v\right]\=x\,\left[y\,z\right]\=y\,\left[x\,v\right]\=x\right]$

Basis := [x, y, z, u, v];

$\mathrm{Basis}:=\left[x\,y\,z\,u\,v\right]$

L := LieAlgebraData(VectStrEq, Basis, Ex1,"LieAlgebraData");

$L:=\left[\left[\mathrm{e1}\,\mathrm{e5}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e2}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=\mathrm{e1}\right]$

DGsetup(L, ['X', 'Y', 'Z', 'U', 'V'], ['theta']);

$\mathrm{Lie\; algebra:\; L1}$

MultiplicationTable("LieBracket");

$\left[\left[X\,V\right]\=X\,\left[Y\,Z\right]\=Y\,\left[U\,V\right]\=X\right]$