Nilpotent Lie algebra

A Lie algebra whose lower central series reaches zero after finitely many steps.

Nilpotent Lie algebra

Let $\mathfrak g$ be a finite-dimensional Lie algebra .

Definition

The lower central series $\gamma_\bullet(\mathfrak g)$ is defined by $\gamma_1(\mathfrak g)=\mathfrak g$ and $\gamma_{k+1}(\mathfrak g)=[\mathfrak g,\gamma_k(\mathfrak g)]$ (see lower central series ).

The Lie algebra $\mathfrak g$ is nilpotent if there exists $s\ge 1$ such that

\gamma_{s+1}(\mathfrak g)=0.

The smallest such $s$ is the nilpotency class (or step).

Immediate consequences

$\gamma_2(\mathfrak g)=[\mathfrak g,\mathfrak g]$ is the derived subalgebra ; repeated commutators strictly decrease in size until they vanish.
In any nonzero nilpotent Lie algebra, the center is nontrivial (a standard structural feature used in many inductive arguments).
Nilpotent implies solvable ; see nilpotent implies solvable .

Examples

The Heisenberg Lie algebra is nilpotent of step $2$ ; see the Heisenberg algebra .
Strictly upper triangular matrices form a nilpotent Lie algebra under commutator; see strictly upper triangular matrices .

Context

Nilpotent Lie algebras are the infinitesimal counterparts of simply connected nilpotent Lie groups, where the exponential map has especially strong global behavior and the BCH formula truncates after finitely many terms.