check if a Lie algebra is semisimple - Maple Programming Help

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Query[Semisimple] - check if a Lie algebra is semisimple

Calling Sequences

Query(Alg, "Semisimple")

Query(S, "Semisimple")

Parameters

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

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

Description

 • There are various equivalent definitions for a semisimple Lie algebra. One definition, called Cartan's criterion, states that a Lie algebra is semisimple if its Killing form is non-degenerate. This is the test applied by Query.
 • Query(Alg, "Semisimple") returns true if Alg is a semisimple Lie algebra and false otherwise.  If the algebra is unspecified, then Query is applied to the current algebra.
 • Query(S, "Semisimple") returns true if the subalgebra S is a semisimple 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.

First initialize three Lie algebras.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[3\right]\right],\left[\left[\left[1,3,1\right],1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,3,2\right],1\right]\right]\right]\right)$
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (2.1)
 > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg2},\left[3\right]\right],\left[\left[\left[1,2,1\right],1\right],\left[\left[1,3,2\right],-2\right],\left[\left[2,3,3\right],1\right]\right]\right]\right)$
 ${\mathrm{L2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}\right]$ (2.2)
 > $\mathrm{L3}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg3},\left[5\right]\right],\left[\left[\left[1,3,2\right],-1\right],\left[\left[1,4,1\right],-1\right],\left[\left[2,4,2\right],1\right],\left[\left[2,5,1\right],-1\right],\left[\left[3,4,3\right],2\right],\left[\left[3,5,4\right],-1\right],\left[\left[4,5,5\right],2\right]\right]\right]\right)$
 ${\mathrm{L3}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e5}}\right]$ (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):$

The first and third algebras are not semisimple while the second algebra is semisimple.

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

The subalgebra span is a semisimple Lie subalgebra of Alg3.

 Alg3 > $S≔\left[\mathrm{z3},\mathrm{z4},\mathrm{z5}\right]:$
 Alg3 > $\mathrm{Query}\left(S,"Semisimple"\right)$
 ${\mathrm{true}}$ (2.7)