ChangeLieAlgebraTo - Maple Help

LieAlgebras[ChangeLieAlgebraTo] - change the current frame to the frame for a Lie algebra

Calling Sequences

ChangeLieAlgebraTo(AlgName)

Parameters

AlgName     - name or string, the name of an initialized Lie algebra

Description

 • With the DifferentialGeometry package, the user can create many manifolds (coordinate systems) and Lie algebras within a single Maple session.  The structure equations and other attributes of these manifolds and Lie algebras are all stored in memory and are accessible at any time.  However, at any given moment one manifold or Lie algebra has the preferred status of the current frame.  The default value of the current frame is the frame in which the most recent calculation was performed or the most recent object displayed.
 • The DifferentialGeometry:-DGinfo("CurrentFrame") returns the value of the current frame.
 • The command ChangeLieAlgebraTo(AlgName) simply changes the current frame to the frame for the Lie algebra AlgName. It returns the name of the previous frame.
 • The command ChangeLieAlgebraTo is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form ChangeLieAlgebraTo(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-ChangeLieAlgebraTo(...).

Examples

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

Example 1.

First initialize a pair of Lie algebras.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[3\right]\right],\left[\left[\left[2,3,1\right],1\right]\right]\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{Alg1}}{,}\left[{3}\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]\right]\right]\right)$ (2.1)
 Alg1   > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$
 Alg1   > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg2},\left[2\right]\right],\left[\right]\right]\right)$
 > $\mathrm{DGsetup}\left(\mathrm{L2}\right):$
 ${\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{Alg2}}{,}\left[{2}\right]\right]{,}\left[{}\right]\right]\right)$ (2.2)
 Alg2 > $\mathrm{Tools}:-\mathrm{DGinfo}\left("CurrentFrame"\right)$
 ${\mathrm{Alg2}}$ (2.3)
 Alg2 > $\mathrm{ChangeLieAlgebraTo}\left(\mathrm{Alg1}\right)$
 ${\mathrm{Alg2}}$ (2.4)
 Alg1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left("CurrentFrame"\right)$
 ${\mathrm{Alg1}}$ (2.5)

We used the same names e1 and e2 for vectors in the two algebras Alg1 and Alg2. It is important then to remember that e1 will refer to the first vector in the Lie algebra of the current frame. We can illustrate this by using lprint to display the internal representation of the vector e1 or by using the command Tools:-DGinfo with the keyword "ObjectFrame".

 Alg1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left("CurrentFrame"\right)$
 ${\mathrm{Alg1}}$ (2.6)
 Alg1 > $\mathrm{lprint}\left(\mathrm{e1}\right)$
 _DG([["vector", Alg1, []], [[[1], 1]]])
 Alg1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{e1},"ObjectFrame"\right)$
 ${\mathrm{Alg1}}$ (2.7)

Now change to the Lie algebra Alg2 and run the lprint and Tools:-DGinfo commands with the keyword "ObjectFrame".

 Alg1 > $\mathrm{ChangeLieAlgebraTo}\left(\mathrm{Alg2}\right)$
 ${\mathrm{Alg1}}$ (2.8)
 Alg2 > $\mathrm{lprint}\left(\mathrm{e1}\right)$
 _DG([["vector", Alg2, []], [[[1], 1]]])
 Alg2 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{e1},"ObjectFrame"\right)$
 ${\mathrm{Alg2}}$ (2.9)