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Nilpotency class of a Lie algebra
Definition
If is nilpotent, its nilpotency class is
where is the lower central series from Lower central series and nilpotent Lie algebras. Nilpotence makes the displayed subset of nonempty, so its least element exists.
This convention gives . A nonzero Lie algebra has class exactly when it is abelian. The class is left undefined for a nonnilpotent Lie algebra.
Depends on
Used by
- Abelian Lie algebras are nilpotent of class one Example
- Series of a standard filiform Lie algebra Example
- Strictly upper triangular matrices form a nilpotent Lie algebra Example
- The Heisenberg Lie algebra is two-step nilpotent Example
- Derivations preserve the nilradical in characteristic zero Proposition
- Subalgebras, quotients, and finite products of nilpotent Lie algebras Proposition
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Milne, Lie Algebras, §2.1 (standard reference, not scraped)