Nilpotent Lie algebras #
Like groups, Lie algebras admit a natural concept of nilpotency. More generally, any Lie module carries a natural concept of nilpotency. We define these here via the lower central series.
Main definitions #
lie algebra, lower central series, nilpotent
The lower central series of Lie submodules of a Lie module.
A Lie module is nilpotent if its lower central series reaches 0 (in a finite number of steps).
For a nilpotent Lie module, the weight space of the 0 weight is the whole module.
This result will be used downstream to show that weight spaces are Lie submodules, at which time it will be possible to state it in the language of weight spaces.