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 purpose of the function LieAlgebraData is to convert various different presentations of a Lie algebra, which are commonly used in differential geometry and Lie theory, into the standard Lie algebra data structure required by the DGsetup command.

The types of Lie algebra presentations which can currently be converted to a Lie algebra data structure are:

The command LieAlgebraData returns a Lie algebra data structure.  The structure equations defined by this data structure are displayed.                      

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(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$C≔\mathrm{Array}\left(1..3\,1..3\,1..3\,0\right)\:$

$C\left[1,3,1\right]≔1\:$$C\left[3,1,1\right]≔-1\:$$C\left[2,3,2\right]≔1\:$$C\left[3,2,2\right]≔-1\:$

$\mathrm{L1}≔\mathrm{LieAlgebraData}\left(C\,\mathrm{Ex1}\right)$

$\mathrm{L1}≔\left[\left[\mathrm{e1}\,\mathrm{e3}\right]=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]=\mathrm{e2}\right]$

$\mathrm{VectStrEq}≔\left[\left[\mathrm{x1}\,\mathrm{x3}\right]=\mathrm{x1}\,\left[\mathrm{x2}\,\mathrm{x3}\right]=\mathrm{x1}+\mathrm{x2}\right],\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\right]$

$\mathrm{VectStrEq}≔\left[\left[\mathrm{x1}\,\mathrm{x3}\right]=\mathrm{x1}\,\left[\mathrm{x2}\,\mathrm{x3}\right]=\mathrm{x1}+\mathrm{x2}\right],\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\right]$

$\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{VectStrEq}\,\mathrm{Ex2}\right)$

$\mathrm{L2}≔\left[\left[\mathrm{e1}\,\mathrm{e3}\right]=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]=\mathrm{e1}+\mathrm{e2}\right]$

$\mathrm{FormStrEq}≔\left[d\left(\mathrm{\θ1}\right)=-\mathrm{\θ1}\&w\mathrm{\θ2}\,d\left(\mathrm{\θ2}\right)=-\mathrm{\θ2}\&w\mathrm{\θ3}\,d\left(\mathrm{\θ3}\right)=0\right],\left[\mathrm{\θ1}\,\mathrm{\θ2}\,\mathrm{\θ3}\right]$

$\mathrm{FormStrEq}≔\left[d\left(\mathrm{\θ1}\right)=-\mathrm{\θ1}\&w\mathrm{\θ2}\,d\left(\mathrm{\θ2}\right)=-\mathrm{\θ2}\&w\mathrm{\θ3}\,d\left(\mathrm{\θ3}\right)=0\right],\left[\mathrm{\θ1}\,\mathrm{\θ2}\,\mathrm{\θ3}\right]$

$\mathrm{L3}≔\mathrm{LieAlgebraData}\left(\mathrm{FormStrEq}\,\mathrm{Ex3}\right)$

$\mathrm{L3}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]=\mathrm{e2}\right]$

$\mathrm{MatrixAlg}≔\left[\mathrm{Matrix}\left(\left[\left[0\,1\right]\,\left[0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[1\,0\right]\,\left[0\,-1\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,0\right]\,\left[1\,0\right]\right]\right)\right]$

$\mathrm{MatrixAlg}≔\left[\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\,\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]\,\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]\right]$

$\mathrm{L4}≔\mathrm{LieAlgebraData}\left(\mathrm{MatrixAlg}\,\mathrm{Ex4}\right)$

$\mathrm{L4}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]=-2\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e3}\right]=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]=-2\mathrm{e3}\right]$

$\mathrm{L5}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Ex5}\,\left[3\right]\right]\,\left[\left[\left[1\,2\,1\right]\,1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L5}\right)$

$\mathrm{Lie\; algebra:\; Ex5}$

$\mathrm{LieAlgebraData}\left(\mathrm{Ex5}\,\mathrm{copyEx5}\right)$

$\left[\left[\mathrm{e1}\,\mathrm{e2}\right]=\mathrm{e1}\right]$

$L≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg5}\,\left[4\right]\right]\,\left[\left[\left[1\,4\,1\right]\,2\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,4\,2\right]\,1\right]\,\left[\left[3\,4\,3\right]\,1\right]\right]\right]\right)\;$$\mathrm{DGsetup}\left(L\right)\:$

$L≔\left[\left[\mathrm{e1}\,\mathrm{e4}\right]=2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]=\mathrm{e3}\right]$

$\mathrm{S1}≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]\:$

$\mathrm{L6}≔\mathrm{LieAlgebraData}\left(\mathrm{S1}\,\mathrm{Ex6}\right)$

$\mathrm{L6}≔\left[\left[\mathrm{e2}\,\mathrm{e3}\right]=\mathrm{e1}\right]$