 Series - Maple Help

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

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

Example 1.

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

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

The derived series:

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

The lower central series:

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

The upper central series:

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

Example 2.

We compute the different series for the subalgebra .

 Alg1 > $\mathrm{S1}≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3},\mathrm{e4}\right]:$

The derived series:

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

The lower central series:

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

The upper central series:

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