 DifferentialGeometry/LieAlgebras/Query/RegularElement - Maple Help

Query[RegularElement] - check if an element of a Lie algebra is regular

Calling Sequences

Query()

Parameters

X        - a vector in a Lie algebra

options  - the keywords arguments rank = $m,$ algebratype = "semisimple" Description

 • Let g be a Lie algebra. For each , let This is the generalized null space of ). The rank of g is defined as rank{dimfor }. An element is called regular if dimrankIf is a regular element, then the centralizer is a Cartan subalgebra.  Conversely, if $Z\left(x\right)$ is a Cartan subalgebra, then is a regular element.
 • Alternatively, for each , set Then the rank of g is the smallest integer (as varies) such that the coefficient of in ${p}_{x}\left({\mathrm{\lambda }}_{}\right)$ is nonzero.
 • With the calling sequence Query(), the centralizer $Z\left(X\right)$ of $X$ is computed. It is then determined if this centralizer is a Cartan subalgebra.
 • With the calling sequence Query(), the calculations are simplified using the fact that the Cartan subalgebra must be Abelian (in general, it need only be nilpotent).
 • With the calling sequence Query( rank = ${\mathrm{m}}{,}$"), the regularity of   is determined by calculating the generalized null space of This is the fastest method for checking if an element is regular, assuming that the rank of is known. Examples

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

Example 1.

We check for regular elements in the Lie algebra First use the command SimpleLieAlgebraData to obtain the structure equations for $\mathrm{so}\left(5\right)$.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("so\left(5\right)",\mathrm{so5}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e8}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so5}}$ (2.2)

The vector is not regular.

 > $\mathrm{Query}\left(\mathrm{e1},"RegularElement"\right)$
 ${\mathrm{false}}$ (2.3)
 > $\mathrm{Query}\left(\mathrm{e1},\mathrm{rank}=2,"RegularElement"\right)$
 ${\mathrm{false}}$ (2.4)

The element  is regular.

 > $\mathrm{Query}\left(\mathrm{evalDG}\left(\mathrm{e1}+\mathrm{e5}-2\mathrm{e8}\right),"RegularElement"\right)$
 ${\mathrm{true}}$ (2.5)
 so5 > $\mathrm{Query}\left(\mathrm{evalDG}\left(\mathrm{e1}+\mathrm{e5}-2\mathrm{e8}\right),\mathrm{rank}=2,"RegularElement"\right)$
 ${\mathrm{true}}$ (2.6)
 so5 > $\mathrm{Query}\left(\mathrm{evalDG}\left(\mathrm{e1}+\mathrm{e5}-2\mathrm{e8}\right),\mathrm{algebratype}="semisimple","RegularElement"\right)$
 ${\mathrm{true}}$ (2.7)