Nilpotent Lie algebra

A Lie algebra is nilpotent when its lower central series eventually becomes zero.

Every Adjoint representation operator raises the lower central series of a Lie algebra filtration. In a basis adapted to that finite filtration, all adjoint operators are strictly triangular. Their products have zero trace, so the Killing form vanishes. Consequently its radical of the Killing form is the whole algebra. For strictly upper triangular matrices the filtration can be seen directly by the gap between the column and row indices.

// Target: algebra.bigb

Each term is the preimage of the center of a Lie algebra after quotienting by the preceding term. Equivalently, $Z_{i+1}=\{x:[\mathfrak{g},x]\subseteq Z_i\}$. A Lie algebra is nilpotent exactly when this series reaches the whole algebra after finitely many steps. For the Heisenberg Lie algebra, the series is $0\subset \mathbb{C}c\subset \mathfrak{g}$.

Over an algebraically closed field, a finite-dimensional Lie algebra is nilpotent exactly when every two-dimensional Lie subalgebra is abelian. A nonnilpotent Adjoint representation matrix has a nonzero eigenvalue $\lambda$ with eigenvector $y$, so $[x,y]=\lambda y$ generates a nonabelian two-dimensional subalgebra. Conversely a subalgebra of a nilpotent Lie algebra is nilpotent, while a nonabelian two-dimensional algebra has a basis $u,v$ with $[u,v]=v$ and a nonnilpotent adjoint matrix. Algebraic closure is what guarantees the required eigenvector.

Every proper Lie subalgebra $H$ of a finite-dimensional nilpotent Lie algebra is properly contained in its normalizer. Choose the first term of the lower central series contained in $H$; the preceding term contains an element outside $H$ that normalizes it.

A finite-dimensional Lie algebra is nilpotent if and only if every adjoint map ${\mathrm{ad}}_x$ is nilpotent. A Lie algebra of nilpotent endomorphisms can be simultaneously represented by strictly upper triangular matrices.

First prove the Engel lemma by induction on the dimension of a Lie algebra of nilpotent endomorphisms. If $x^m=0$, then $\mathrm{ad}x=L_x-R_x$ on the endomorphism algebra has $(\mathrm{ad}x{)}^{2m-1}=0$, since left and right multiplication commute. Thus a proper Lie subalgebra $M$ acts nilpotently on $L/M$, and induction gives a nonzero class normalized by $M$. This proves the Engel normalizer lemma.

For a maximal proper $M$, its normalizer is all of $L$, so $M$ is an ideal. Maximality then forces $\dim L/M=1$, since every line in a Lie algebra is a subalgebra. Induction gives a nonzero common annihilator $W$ of $M$ in the representation space. Ideality makes $W$ stable under $L$. A generator $x$ of $L/M$ is nilpotent on $W$ and has nonzero kernel there, providing a vector killed by all of $L$. In the possibly nonfaithful quotient actions, apply induction to the image, whose dimension is at most $\dim M$.

Repeatedly apply this common-kernel statement to quotient representation spaces to obtain a complete flag with $x{V}_j\subseteq V_{j-1}$. Hence the operators are simultaneously strictly upper triangular. Applying this to the Adjoint representation shows that a finite-dimensional Lie algebra whose every adjoint map is nilpotent has terminating lower central series. Conversely, termination of that series immediately makes every adjoint map nilpotent. This proves Engel theorem.

// Target: semisimple-lie-algebra.bigb

A finite-dimensional Lie algebra representation in which every representing endomorphism is nilpotent has a nonzero vector annihilated by the entire Lie algebra. Applying this to the Adjoint representation is the key step in Engel theorem.

If $L$ is a finite-dimensional Lie algebra of nilpotent endomorphisms, every proper Lie subalgebra $M$ is strictly contained in its normalizer of a Lie subalgebra. The Adjoint representation of $M$ on $L/M$ consists of nilpotent maps by nilpotence of commutation by a nilpotent endomorphism. In the inductive proof of Engel theorem, the lower-dimensional Engel lemma gives a nonzero class $y+M$ with $[M,y]\subseteq M$. A maximal proper $M$ is consequently an ideal of a Lie algebra of codimension one.