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LieAlgebras

 LieAlgebraData
 convert different realizations of a Lie algebra to a Lie algebra data structure

 Calling Sequence LieAlgebraData(LieAlgebraPresentation)

Parameters

 LieAlgebraPresentation - one of several different formats for defining a 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 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(...).

Examples

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

Example 1.  LieAlgebraData(StructureConstants)

In this example we create a 3 dimensional Array, C, of structure constants and use this Array to create a Lie algebra data structure for a Lie algebra called Ex1.

 > $C≔\mathrm{Array}\left(1..3,1..3,1..3,0\right):$
 > $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:$
 > $\mathrm{L1}≔\mathrm{LieAlgebraData}\left(C,\mathrm{Ex1}\right)$
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}\right]$ (1)

Example 2.  LieAlgebraData(VectorStructureEquations)

In this example we create a Lie algebra data structure for a Lie algebra called Ex2 from a list of structure equations for the Lie brackets.

 > $\mathrm{VectStrEq}≔\left[\left[\mathrm{x1},\mathrm{x3}\right]=\mathrm{x1},\left[\mathrm{x2},\mathrm{x3}\right]=\mathrm{x1}+\mathrm{x2}\right],\left[\mathrm{x1},\mathrm{x2},\mathrm{x3}\right]$
 ${\mathrm{VectStrEq}}{≔}\left[\left[{\mathrm{x1}}{,}{\mathrm{x3}}\right]{=}{\mathrm{x1}}{,}\left[{\mathrm{x2}}{,}{\mathrm{x3}}\right]{=}{\mathrm{x1}}{+}{\mathrm{x2}}\right]{,}\left[{\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}\right]$ (2)
 > $\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{VectStrEq},\mathrm{Ex2}\right)$
 ${\mathrm{L2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (3)

Example 3.  LieAlgebraData(FormStructureEquations)

In this example we create a Lie algebra data structure for a Lie algebra called Ex3 from a list of structure equations for the exterior derivatives of the dual 1-forms.

 > $\mathrm{FormStrEq}≔\left[d\left(\mathrm{θ1}\right)=-\mathrm{θ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ2},d\left(\mathrm{θ2}\right)=-\mathrm{θ2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ3},d\left(\mathrm{θ3}\right)=0\right],\left[\mathrm{θ1},\mathrm{θ2},\mathrm{θ3}\right]$
 ${\mathrm{FormStrEq}}{≔}\left[{d}{}\left({\mathrm{θ1}}\right){=}{-}{\mathrm{θ1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{θ2}}{,}{d}{}\left({\mathrm{θ2}}\right){=}{-}{\mathrm{θ2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&w}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{θ3}}{,}{d}{}\left({\mathrm{θ3}}\right){=}{0}\right]{,}\left[{\mathrm{θ1}}{,}{\mathrm{θ2}}{,}{\mathrm{θ3}}\right]$ (4)
 > $\mathrm{L3}≔\mathrm{LieAlgebraData}\left(\mathrm{FormStrEq},\mathrm{Ex3}\right)$
 ${\mathrm{L3}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}\right]$ (5)

Example 4.  LieAlgebraData(MatrixAlgebra)

In this example we create a Lie algebra data structure for a Lie algebra called Ex4 from a list of matrices.

 > $\mathrm{MatrixAlg}≔\left[\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[1,0\right]\right]\right)\right]$
 ${\mathrm{MatrixAlg}}{≔}\left[\left[\begin{array}{cc}{0}& {1}\\ {0}& {0}\end{array}\right]{,}\left[\begin{array}{cc}{1}& {0}\\ {0}& {-1}\end{array}\right]{,}\left[\begin{array}{cc}{0}& {0}\\ {1}& {0}\end{array}\right]\right]$ (6)
 > $\mathrm{L4}≔\mathrm{LieAlgebraData}\left(\mathrm{MatrixAlg},\mathrm{Ex4}\right)$
 ${\mathrm{L4}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}\right]$ (7)

Example 5.  LieAlgebraData(LieAlgebraName)

In this example we create a Lie algebra data structure for a Lie algebra called Ex5 from a previously initialized Lie algebra.

First we initialize a Lie algebra Ex5.

 > $\mathrm{L5}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Ex5},\left[3\right]\right],\left[\left[\left[1,2,1\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{L5}\right)$
 ${\mathrm{Lie algebra: Ex5}}$ (8)

Applying LieAlgebraData to Ex5 gives back the Lie algebra data structure we started from.

 > $\mathrm{LieAlgebraData}\left(\mathrm{Ex5},\mathrm{copyEx5}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}\right]$ (9)

Example 6.  LieAlgebraData(Subalgebra, "LieAlgebraData")

In this example we create a Lie algebra data structure from a subalgebra of a Lie algebra.

First we initialize a Lie algebra Alg5.

 > $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):$
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (10)

The vectors [e1, e2, e3] define a Lie subalgebra which we wish to initialize as a Lie algebra in its own right.

 > $\mathrm{S1}≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right]:$
 > $\mathrm{L6}≔\mathrm{LieAlgebraData}\left(\mathrm{S1},\mathrm{Ex6}\right)$
 ${\mathrm{L6}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (11)

 See Also