find the derived series, lower central series, or upper central series of a Lie algebra or a Lie subalgebra - Maple Programming Help

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LieAlgebras[Series] - find the derived series, lower central series, or upper central series of a Lie algebra or a Lie subalgebra

Calling Sequences

Series(AlgName, keyword)

Series(S, keyword)

Parameters

AlgName    - (optional) the name of a Lie algebra $\mathrm{𝔤}$

keyword    - a string, one of "Derived", "Lower", "Upper"

S          - a list of vectors defining a basis for a Lie subalgebra of a Lie algebra $\mathrm{𝔤}$

Description

 • The derived series of a Lie algebra $\mathrm{𝔤}$ is the sequence of ideals defined inductively by and . See BracketOfSubspaces for the definition of the Lie bracket of 2 subspaces  Note thatThe derived series terminates whenor . The Lie algebra  is solvable if  .
 • The lower central series of a Lie algebra is a sequence of ideals defined inductively by and . Note that The lower central series terminates when or. The Lie algebra $\mathrm{𝔤}$ is nilpotent if   .
 • If  is an ideal, then the generalized center is for all The upper central series of a Lie algebra $\mathrm{𝔤}$is a sequence of ideals ${C}^{k}\left(\mathrm{𝔤}\right)$ defined inductively by and Note that . The upper central series terminates whenor .
 • Series(AlgName, keyword) calculates the series defined by the keyword for the Lie algebra AlgName. If the first argument AlgName is omitted, then the appropriate series of the current Lie algebra is found.
 • Series(S, keyword) calculates the series defined by the keyword for the Lie subalgebra S (viewed as a Lie algebra in its own right).
 • Series returns a list of list of vectors where is a basis for the term in the appropriate series. The list ends with if [i] ; or [ii] in case of the derived and lower series if; or [iii] in the case of the upper series .
 • The dimensions of the subalgebras in these series can be easily computed with the Maple map and nops commands.
 • The command Series is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Series(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Series(...).

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

First we initialize a Lie algebra and display the multiplication table.

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

The derived series:

 Alg1 > DS := Series("Derived"); map(nops, DS);
 ${\mathrm{DS}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}\right]{,}\left[{}\right]\right]$ $\left[{5}{,}{3}{,}{0}\right]$ (2.2)

The lower central series:

 Alg1 > LS := Series("Lower"); map(nops, LS);
 ${\mathrm{LS}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}\right]{,}\left[{-}{\mathrm{e1}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e4}}\right]{,}\left[{\mathrm{e4}}\right]\right]$ $\left[{5}{,}{3}{,}{2}{,}{1}{,}{1}\right]$ (2.3)

The upper central series:

 Alg1 > US := Series("Upper"); map(nops, US);
 ${\mathrm{US}}{:=}\left[\left[{\mathrm{e1}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e1}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e2}}{,}{\mathrm{e1}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}{,}{\mathrm{e2}}\right]\right]$ $\left[{1}{,}{2}{,}{3}{,}{3}\right]$ (2.4)

Example 2.

We compute the different series for the subalgebra .

 Alg1 > S1 := [e1, e2, e3, e4]:

The derived series:

 Alg1 > DS := Series(S1, "Derived"); map(nops, DS);
 ${\mathrm{DS}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e1}}\right]{,}\left[{}\right]\right]$ $\left[{4}{,}{1}{,}{0}\right]$ (2.5)

The lower central series:

 Alg1 > LS := Series(S1, "Lower"); map(nops, LS);
 ${\mathrm{LS}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e1}}\right]{,}\left[{}\right]\right]$ $\left[{4}{,}{1}{,}{0}\right]$ (2.6)

The upper central series:

 Alg1 > US := Series(S1, "Upper"); map(nops, US);
 ${\mathrm{US}}{:=}\left[\left[{\mathrm{e4}}{,}{\mathrm{e1}}\right]{,}\left[{\mathrm{e2}}{,}{\mathrm{e1}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}\right]\right]$ $\left[{2}{,}{4}\right]$ (2.7)