check if a Lie algebra is decomposable as a direct sum of Lie algebras over the real numbers - Maple Programming Help

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Query[Indecomposable] - check if a Lie algebra is decomposable as a direct sum of Lie algebras over the real numbers

Query[AbsolutelyIndecomposable] - check if a Lie algebra is decomposable as a direct sum of Lie algebras over the complex numbers

Calling Sequences

Query(Alg, "Indecomposable")

Query(Alg, "AbsolutelyIndecomposable")

Parameters

Alg     - (optional) the name of an initialized Lie algebra or a Lie algebra data structure

Description

 • A collection of subalgebras  of a Lie algebra $\mathrm{𝔤}$ defines a direct sum decomposition of  if  (vector space direct sum)  and  for
 • Query(Alg, "Indecomposable") returns false if the Lie algebra Alg is decomposable as a direct sum of Lie algebras over the real numbers, otherwise true is returned.
 • Query(Alg, "AbsolutelyIndecomposable") returns false if the Lie algebra Alg is decomposable as a direct sum of Lie algebras over the complex numbers, otherwise true is returned.
 • The command Query is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).

Examples

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

Example 1.

In this example we illustrate the fact that the result of Inquiry("Indecomposable") does not depend upon the choice of basis for the Lie algebra. First we initialize a Lie algebra.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[1,2,1\right],1\right],\left[\left[3,4,3\right],1\right]\right]\right]\right)$
 ${\mathrm{L1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

Now we make a change of basis in the Lie algebra.  In this basis it is not possible to see that the Lie algebra is decomposable by examining the multiplication table.

 Alg1 > $\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\left[\mathrm{e1}+\mathrm{e4},\mathrm{e2}-\mathrm{e3},\mathrm{e2}+\mathrm{e4},\mathrm{e1}\right],\mathrm{Alg2}\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e2}}{+}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{-}{\mathrm{e3}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}\right]$ (2.2)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2}\right):$

Both Alg1 and Alg2 are seen to be decomposable.

 Alg2 > $\mathrm{Query}\left(\mathrm{Alg1},"Indecomposable"\right)$
 ${\mathrm{false}}$ (2.3)
 Alg1 > $\mathrm{Query}\left(\mathrm{Alg2},"Indecomposable"\right)$
 ${\mathrm{false}}$ (2.4)

Example 2

Here is the simplest example of a solvable Lie algebra which is absolutely decomposable but not decomposable. First we initialize the Lie algebra and display the multiplication table.

 Alg2 > $L≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg3},\left[4\right]\right],\left[\left[\left[1,3,1\right],1\right],\left[\left[2,4,1\right],1\right],\left[\left[1,4,2\right],-1\right],\left[\left[2,3,2\right],1\right]\right]\right]\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}\right]$ (2.5)
 Alg2 > $\mathrm{DGsetup}\left(L\right):$

The algebra is indecomposable over the real numbers.

 Alg3 > $\mathrm{Query}\left(L,"Indecomposable"\right)$
 ${\mathrm{true}}$ (2.6)

The algebra is decomposable over the complex numbers.

 Alg3 > $\mathrm{Query}\left(L,"AbsolutelyIndecomposable"\right)$
 ${\mathrm{false}}$ (2.7)

The explicit decomposition of this Lie algebra is given in the help page for the command Decompose.

 See Also