 Radical - Maple Help

Calling Sequences

Parameters

LieAlgName - (optional) name or string, the name of a Lie algebra Description

 • The radical of a Lie algebra is the largest solvable ideal contained in $\mathrm{𝔤}$. The radical of can be calculated as the orthogonal complement of the derived algebra $\mathrm{𝔤}\mathit{'}$ of with respect to the Killing form $B$, that is, rad$\left(\mathrm{𝔤}\right)$ = for all . See, for example, Fulton and Harris Representation Theory, Graduate Texts in Mathematics 129, Springer 1991, Proposition C.22 page 484.
 • Radical(LieAlgName) calculates the radical of the Lie algebra $\mathrm{𝔤}$ defined by LieAlgName. If no argument is given, then the radical of the current Lie algebra is found.
 • A list of vectors defining a basis for the rad($\mathrm{𝔤})$is returned. If rad($\mathrm{𝔤})$ is trivial, then an empty list is returned.
 • The command Radical is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Radical(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Radical(...). Examples

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

Example 1.

First we initialize a Lie algebra.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[7\right]\right],\left[\left[\left[1,2,2\right],2\right],\left[\left[1,3,3\right],-2\right],\left[\left[2,3,1\right],1\right],\left[\left[1,4,4\right],1\right],\left[\left[1,5,5\right],-1\right],\left[\left[2,5,4\right],1\right],\left[\left[3,4,5\right],1\right],\left[\left[4,5,6\right],1\right],\left[\left[4,7,4\right],1\right],\left[\left[5,7,5\right],1\right],\left[\left[6,7,6\right],2\right]\right]\right]\right)$
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e6}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

We calculate the radical of Alg1 to be the 4-dimensional ideal with basis $\left\{{e}_{4},{e}_{5},{e}_{6},{e}_{7}\right\}$and check that the result is indeed a solvable ideal.

 Alg1 > $\mathrm{rad}≔\mathrm{Radical}\left(\right)$
 ${\mathrm{rad}}{≔}\left[{\mathrm{e7}}{,}{\mathrm{e6}}{,}{\mathrm{e5}}{,}{\mathrm{e4}}\right]$ (2.2)
 Alg1 > $\mathrm{Query}\left(\mathrm{rad},"Solvable"\right)$
 ${\mathrm{true}}$ (2.3)
 Alg1 > $\mathrm{Query}\left(\mathrm{rad},"Ideal"\right)$
 ${\mathrm{true}}$ (2.4)

We remark that the span of the vectors $\left\{{e}_{1},{e}_{4},{e}_{6},{e}_{7}\right\}$is a 4-dimensional solvable subalgebra but it is not an ideal.

 Alg1 > $A≔\left[\mathrm{e1},\mathrm{e4},\mathrm{e5},\mathrm{e6},\mathrm{e7}\right]$
 ${A}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}\right]$ (2.5)
 Alg1 > $\mathrm{Query}\left(A,"Solvable"\right)$
 ${\mathrm{true}}$ (2.6)
 Alg1 > $\mathrm{Query}\left(A,"Ideal"\right)$
 ${\mathrm{false}}$ (2.7)