SYM Paclet Symbol

DerivedSeries

DerivedSeries[L]
gives the derived series for the Lie algebra L.

DerivedSeries[L, n]
gives the derived series for the Lie algebra L up to order n.

DerivedSeries[L, {n}]
gives the n^th member of the derived series for the Lie algebra L.

To use DerivedSeries, you first need to load the SYM Package using Needs["SYM`"].

The Lie algebra L can be given either from its structure constants or from its infinitesimal generator representation.

DerivedSeries returns the sequence of ideals L⁽²⁾ L⁽ⁱ⁾ where, L⁽ⁱ⁺¹⁾=[L⁽ⁱ⁾, L⁽ⁱ⁾], i≥1 and L⁽¹⁾=L. The second argument specifies whether a portion or a specific member of the chain will be returned.

Let the Lie algebra defined by the structure constants:

The derived series is:

Obviously, the Lie algebra is solvable and terminates for i=2. To get the chain up to the first member other than the Lie algebra:

To get only the first member of the chain: