The derived series of a Lie algebra $\mathrm{\𝔤}$ is the sequence of ideals ${\mathrm{D}}^k\left(\mathrm{\𝔤}\right)\subset \mathrm{\𝔤}\mathit{ }$defined inductively by ${\mathrm{D}}^0\left(\mathrm{\𝔤}\right) \= \mathrm{\𝔤}\mathit{ }$and ${\mathrm{D}}^{k\+1}\left(\mathrm{\𝔤}\right) \= \left[{\mathrm{D}}^k\left(\mathrm{\𝔤} \right)\, {\mathrm{D}}^k\left(\mathrm{\𝔤}\right)\right]$. See BracketOfSubspaces for the definition of the Lie bracket $\left[A\, B\right]$of 2 subspaces $A\, B \subset \mathrm{\𝔤}\mathit{\.}$ Note that${\mathrm{D}}^{k\+1}\left(\mathrm{\𝔤}\right) \subset {\mathrm{D}}^k\left(\mathrm{\𝔤} \right)\.$The derived series terminates when${\mathrm{D}}^{k\+1}\left(𝔤\right) \= 0$or ${\mathrm{D}}^{k\+1}\left(𝔤\right) \= {\mathrm{D}}^k\left(𝔤\right)$. The Lie algebra $\mathrm{\𝔤}\mathit{ }$ is solvable if  ${\mathrm{D}}^{k\+1}\left(𝔤\right) \= 0$.

The lower central series of a Lie algebra $\mathrm{\𝔤}\mathit{ }$is a sequence of ideals defined inductively by $L^0\left(\mathrm{\𝔤}\right) \= \mathrm{\𝔤}\mathit{ }$and $L^{k\+1}\left(\mathrm{\𝔤}\right) \= \left[\mathrm{\𝔤} \, L^k\left(\mathrm{\𝔤}\right)\right]$. Note that $L^{k\+1}\left(\mathrm{\𝔤}\right) \subset L^k\left(\mathrm{\𝔤} \right)\.$The lower central series terminates when$L^{k\+1}\left(𝔤\right) \= 0$ or$L^{k\+1}\left(𝔤\right) \= L^k\left(𝔤\right)$. The Lie algebra $\mathrm{\𝔤}$ is nilpotent if  $L^{k\+1}\left(𝔤\right) \= 0$ .

If $h\subset \mathrm{\𝔤}$ is an ideal, then the generalized center is $\mathrm{GC}\left(h\right) \= \{x \in \mathrm{\𝔤}\mathit{ }\mathit{\|}\mathit{ }\left[x\, y\right] \in h$for all $y \in \mathrm{\𝔤}\}\.$The upper central series of a Lie algebra $\mathrm{\𝔤}$$$is a sequence of ideals $C^k\left(\mathrm{\𝔤}\right)$ defined inductively by $C^0\left(\mathrm{\𝔤}\right) \= \mathrm{GC}\left(0\right)$and $C^{k\+1}\left(\mathrm{\𝔤}\right) \=\mathrm{GC}\left(C^k\left(\mathrm{\𝔤}\right)\right)\.$Note that $C^k\left(𝔤\right) \subset C^{k\+1}\left(𝔤 \right)$. The upper central series terminates when$C^{k\+1}\left(𝔤\right) \= \mathrm{\𝔤}$or $C^{k\+1}\left(𝔤\right) \= C^k\left(𝔤\right)$.

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 $L \= \left[A_1\, A_2\, ..\.\right]$where $A_k$is a basis for the $\left(k-1\right)$term in the appropriate series. The list $L$ends with $A_m$if [i] $A_{m-1} \= A_m$; or [ii] in case of the derived and lower series if$A_m \=\left[\right]$; or [iii] in the case of the upper series $A_m\= \mathrm{\𝔤}\mathit{ }$.

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(...).

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

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

$\mathrm{L1}≔\left[\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e3}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=\mathrm{e4}\right]$

$\mathrm{DGsetup}\left(\mathrm{L1}\right)\:$

$\mathrm{DS}≔\mathrm{Series}\left(''Derived''\right)\;$$\mathrm{map}\left(\mathrm{nops}\,\mathrm{DS}\right)$

$\mathrm{DS}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\,\mathrm{e5}\right]\,\left[\mathrm{e1}\,\mathrm{e3}\,\mathrm{e4}\right]\,\left[\right]\right]$

$\left[5\,3\,0\right]$

$\mathrm{LS}≔\mathrm{Series}\left(''Lower''\right)\;$$\mathrm{map}\left(\mathrm{nops}\,\mathrm{LS}\right)$

$\mathrm{LS}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\,\mathrm{e5}\right]\,\left[\mathrm{e1}\,\mathrm{e3}\,\mathrm{e4}\right]\,\left[-\mathrm{e1}\,\mathrm{e4}\right]\,\left[\mathrm{e4}\right]\,\left[\mathrm{e4}\right]\right]$

$\left[5\,3\,2\,1\,1\right]$

$\mathrm{US}≔\mathrm{Series}\left(''Upper''\right)\;$$\mathrm{map}\left(\mathrm{nops}\,\mathrm{US}\right)$

$\mathrm{US}≔\left[\left[\mathrm{e1}\right]\,\left[\mathrm{e3}\,\mathrm{e1}\right]\,\left[\mathrm{e3}\,\mathrm{e2}\,\mathrm{e1}\right]\,\left[\mathrm{e1}\,\mathrm{e3}\,\mathrm{e2}\right]\right]$

$\left[1\,2\,3\,3\right]$

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

$\mathrm{DS}≔\mathrm{Series}\left(\mathrm{S1}\,''Derived''\right)\;$$\mathrm{map}\left(\mathrm{nops}\,\mathrm{DS}\right)$

$\mathrm{DS}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\right]\,\left[\mathrm{e1}\right]\,\left[\right]\right]$

$\left[4\,1\,0\right]$

$\mathrm{LS}≔\mathrm{Series}\left(\mathrm{S1}\,''Lower''\right)\;$$\mathrm{map}\left(\mathrm{nops}\,\mathrm{LS}\right)$

$\mathrm{LS}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\right]\,\left[\mathrm{e1}\right]\,\left[\right]\right]$

$\left[4\,1\,0\right]$

$\mathrm{US}≔\mathrm{Series}\left(\mathrm{S1}\,''Upper''\right)\;$$\mathrm{map}\left(\mathrm{nops}\,\mathrm{US}\right)$

$\mathrm{US}≔\left[\left[\mathrm{e4}\,\mathrm{e1}\right]\,\left[\mathrm{e2}\,\mathrm{e1}\,\mathrm{e3}\,\mathrm{e4}\right]\right]$

$\left[2\,4\right]$