GeneralizedCenter - Maple Help

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LieAlgebras[GeneralizedCenter] - find the generalized center of an ideal

Calling Sequences

GeneralizedCenter(S1, S2)

Parameters

S1     - a list of vectors defining a basis for an idealin a Lie algebra $\mathrm{𝔤}$

S2     - (optional) list of vectors defining a basis for a subalgebra with

.Description

 • Let be a Lie algebra, a subalgebra of $\mathrm{𝔤}$, and an ideal with . Then the generalized center of with respect to is the ideal  for all  In particular, the generalized center of in $\mathrm{𝔤}$ is the inverse image of the center of the quotient algebra $\mathrm{𝔤}\mathit{/}\mathrm{𝔥}$ with respect to the canonical projection map $\mathrm{𝔤}\mathit{/}\mathrm{𝔥}$.
 • A list of vectors defining a basis for the generalized center of in is returned. If the optional argument S2 is omitted, then the default is If the generalized center of $\mathrm{𝔥}$ in is trivial, then an empty list is returned.
 • The command GeneralizedCenter is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form GeneralizedCenter(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-GeneralizedCenter(...).

Examples

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

Example 1.

First initialize a Lie algebra.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[5\right]\right],\left[\left[\left[2,5,1\right],1\right],\left[\left[3,5,2\right],1\right],\left[\left[4,5,4\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

Calculate the generalized center of [e1, e2] in the Lie algebra Alg1.

 Alg1 > $\mathrm{S1}≔\left[\mathrm{e1},\mathrm{e2}\right]:$
 Alg1 > $\mathrm{GeneralizedCenter}\left(\left[\mathrm{e1},\mathrm{e2}\right]\right)$
 $\left[{\mathrm{e3}}{,}{\mathrm{e2}}{,}{\mathrm{e1}}\right]$ (2.1)

Calculate the generalized center of [e1, e4] in [e1, e2, e4, e5].

 Alg1 > $\mathrm{S2}≔\left[\mathrm{e1},\mathrm{e4}\right]:$$\mathrm{S3}≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e4},\mathrm{e5}\right]:$
 Alg1 > $\mathrm{GeneralizedCenter}\left(\mathrm{S2},\mathrm{S3}\right)$
 $\left[{\mathrm{e5}}{,}{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e4}}\right]$ (2.2)