 MinimalIdeal - Maple Help

LieAlgebras[MinimalIdeal] - find the smallest ideal containing a given set of vectors

Calling Sequences

MinimalIdeal(S)

Parameters

S        - a list of vectors in a Lie algebra Description

 • MinimalIdeal(S) calculates the smallest ideal $J$ containing the list of vectors S in an Lie algebra $\mathrm{𝔤}$.
 • A list of vectors giving a basis for is returned.
 • The command MinimalIdeal is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form MinimalIdeal(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MinimalIdeal(...). Examples

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

Example 1.

First initialize a Lie algebra and display the multiplication table.

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

Find the minimal ideal containing the vectors

 Alg1 > $\mathrm{S1}≔\left[\mathrm{e1},\mathrm{e3}\right]:$
 Alg1 > $\mathrm{I1}≔\mathrm{MinimalIdeal}\left(\mathrm{S1}\right)$
 ${\mathrm{I1}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]$ (2.2)

Find the minimal ideal containing the vectors

 Alg1   > $\mathrm{S2}≔\left[\mathrm{e2},\mathrm{e4}\right]:$
 Alg1 > $\mathrm{I2}≔\mathrm{MinimalIdeal}\left(\mathrm{S2}\right)$
 ${\mathrm{I2}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}\right]$ (2.3)
 Alg1 > $\mathrm{Query}\left(\mathrm{S2},"Ideal"\right)$
 ${\mathrm{false}}$ (2.4)
 Alg1 > $\mathrm{Query}\left(\mathrm{I2},"Ideal"\right)$
 ${\mathrm{true}}$ (2.5)