Centralizer - Maple Help
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LieAlgebras[Centralizer] - find the centralizer of a list of vectors

Calling Sequences

Centralizer(S, h)

Parameters

S     - a list of vectors in a Lie algebra $\mathrm{𝔤}$ or a general algebra $\mathrm{𝔸}$

h     - (optional) a subalgebra of or $\mathrm{𝔸}$

Description

 • The centralizer of a set of vectors relative to a subalgebra is the subalgebra of vectors in which commute with all the vectors in $S.$
 • Centralizer(S, h) calculates the centralizer of the list Sin the subalgebra $h$. If the second argument h is not specified then the centralizer of S in the entire algebra is calculated.
 • A list of vectors defining a basis for the centralizer of S is returned. If the centralizer of S is trivial, then an empty list is returned.
 • The command Centralizer is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Centralizer(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Centralizer(...).

Examples

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

Example 1.

First initialize a Lie algebra.

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

Calculate the centralizer of  in the Lie algebra Alg1.

 Alg1 > $S≔\left[\mathrm{e3}\right]$
 ${S}{:=}\left[{\mathrm{e3}}\right]$ (2.2)
 Alg1 > $\mathrm{Centralizer}\left(S\right)$
 $\left[{\mathrm{e3}}{,}{\mathrm{e2}}{,}{\mathrm{e1}}\right]$ (2.3)

Calculate the centralizer of  relative to the subalgebras spanned by and  .

 Alg1 > $S≔\left[\mathrm{e4},\mathrm{e5}\right]$
 ${S}{:=}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (2.4)
 Alg1 > $h≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e4},\mathrm{e5}\right]$
 ${h}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (2.5)
 Alg1 > $\mathrm{Centralizer}\left(S,h\right)$
 $\left[{\mathrm{e5}}{,}{\mathrm{e4}}{,}{\mathrm{e1}}\right]$ (2.6)
 Alg1 > $h≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right]$
 ${h}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]$ (2.7)
 Alg1 > $\mathrm{Centralizer}\left(S,h\right)$
 $\left[{\mathrm{e1}}\right]$ (2.8)

Example 2.

Calculate the centralizer of a set of vectors in the algebra $\mathrm{𝕆}$ of octonions.

 Alg1 > $\mathrm{L2}≔\mathrm{AlgebraLibraryData}\left("Octonions",\mathrm{Oct}\right)$
 ${\mathrm{L2}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e1}}{.}{\mathrm{e5}}{=}{\mathrm{e5}}{,}{\mathrm{e1}}{.}{\mathrm{e6}}{=}{\mathrm{e6}}{,}{\mathrm{e1}}{.}{\mathrm{e7}}{=}{\mathrm{e7}}{,}{\mathrm{e1}}{.}{\mathrm{e8}}{=}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e2}}{.}{\mathrm{e5}}{=}{\mathrm{e6}}{,}{\mathrm{e2}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e2}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e8}}{=}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e3}}{.}{\mathrm{e5}}{=}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e6}}{=}{\mathrm{e8}}{,}{\mathrm{e3}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e3}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e4}}{.}{\mathrm{e5}}{=}{\mathrm{e8}}{,}{\mathrm{e4}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e4}}{.}{\mathrm{e7}}{=}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e1}}{=}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e5}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e5}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e8}}{,}{{\mathrm{e5}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e5}}{.}{\mathrm{e6}}{=}{\mathrm{e2}}{,}{\mathrm{e5}}{.}{\mathrm{e7}}{=}{\mathrm{e3}}{,}{\mathrm{e5}}{.}{\mathrm{e8}}{=}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e1}}{=}{\mathrm{e6}}{,}{\mathrm{e6}}{.}{\mathrm{e2}}{=}{\mathrm{e5}}{,}{\mathrm{e6}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e6}}{.}{\mathrm{e4}}{=}{\mathrm{e7}}{,}{\mathrm{e6}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e6}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e6}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e8}}{=}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e1}}{=}{\mathrm{e7}}{,}{\mathrm{e7}}{.}{\mathrm{e2}}{=}{\mathrm{e8}}{,}{\mathrm{e7}}{.}{\mathrm{e3}}{=}{\mathrm{e5}}{,}{\mathrm{e7}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e7}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e6}}{=}{\mathrm{e4}}{,}{{\mathrm{e7}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e7}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e2}}{,}{\mathrm{e8}}{.}{\mathrm{e1}}{=}{\mathrm{e8}}{,}{\mathrm{e8}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e8}}{.}{\mathrm{e3}}{=}{\mathrm{e6}}{,}{\mathrm{e8}}{.}{\mathrm{e4}}{=}{\mathrm{e5}}{,}{\mathrm{e8}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e4}}{,}{\mathrm{e8}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e8}}{.}{\mathrm{e7}}{=}{\mathrm{e2}}{,}{{\mathrm{e8}}}^{{2}}{=}{-}{\mathrm{e1}}\right]$ (2.9)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2}\right)$
 ${\mathrm{algebra name: Oct}}$ (2.10)
 Alg1 > $S≔\left[\mathrm{e3}\right]$
 ${S}{:=}\left[{\mathrm{e3}}\right]$ (2.11)
 Alg1 > $\mathrm{Centralizer}\left(S\right)$
 $\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]$ (2.12)