Solvable - Maple Help
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Query[Solvable] - check if a Lie algebra is solvable

Calling Sequences

Query(Alg, "Solvable")

Query(S, "Solvable")

Parameters

Alg     - (optional) name or string, the name of an initialized Lie algebra

S       - a list of vectors defining a basis for a subalgebra

Description

 • A Lie algebra is solvable if the $k$-th ideal in the derived series for  is 0 for some $k\ge 0$. Every nilpotent Lie algebra is solvable.
 • Query(Alg, "Solvable") returns true if Alg is a solvable Lie algebra and false otherwise. If the algebra is unspecified, then Query is applied to the current algebra.
 • Query(S, "Solvable") returns true if the subalgebra S is a solvable Lie algebra and false otherwise.
 • 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.

We initialize three different Lie algebras.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[3\right]\right],\left[\left[\left[2,3,1\right],1\right]\right]\right]\right)$
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (2.1)
 > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg2},\left[3\right]\right],\left[\left[\left[1,3,1\right],1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,3,2\right],1\right]\right]\right]\right)$
 ${\mathrm{L2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (2.2)
 > $\mathrm{L3}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg3},\left[3\right]\right],\left[\left[\left[1,2,1\right],1\right],\left[\left[1,3,2\right],-2\right],\left[\left[2,3,3\right],1\right]\right]\right]\right)$
 ${\mathrm{L3}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}\right]$ (2.3)
 > $\mathrm{DGsetup}\left(\mathrm{L1},\left[x\right],\left[a\right]\right):$$\mathrm{DGsetup}\left(\mathrm{L2},\left[y\right],\left[b\right]\right):$$\mathrm{DGsetup}\left(\mathrm{L3},\left[z\right],\left[c\right]\right):$

Alg1 and Alg2 are solvable but Alg3 is not. (Alg1 is actually nilpotent while Alg3 is semisimple.)

 Alg3 > $\mathrm{Query}\left(\mathrm{Alg1},"Solvable"\right)$
 ${\mathrm{true}}$ (2.4)
 Alg1 > $\mathrm{Query}\left(\mathrm{Alg2},"Solvable"\right)$
 ${\mathrm{true}}$ (2.5)
 Alg2 > $\mathrm{Query}\left(\mathrm{Alg3},"Solvable"\right)$
 ${\mathrm{false}}$ (2.6)

The subalgebra spanis a solvable subalgebra of Alg3. (The algebra Alg3 is  and is a Borel subalgebra.)

 Alg3 > $\mathrm{Query}\left(\left[\mathrm{z1},\mathrm{z2}\right],"Solvable"\right)$
 ${\mathrm{true}}$ (2.7)

 See Also