Abelian Lie algebra

Let $\mathfrak{g}$ be a Lie algebra over a field $\Bbbk$ with Lie bracket $[\cdot,\cdot]$.

Definition. $\mathfrak{g}$ is abelian if

Equivalently, the derived subalgebra satisfies $[\mathfrak{g},\mathfrak{g}]=0$, and the center satisfies $Z(\mathfrak{g})=\mathfrak{g}$. In representation-theoretic terms, the adjoint representation $\operatorname{ad}:\mathfrak{g}\to \mathfrak{gl}(\mathfrak{g})$ is the zero map.

Context. Abelian Lie algebras are the “linearized” version of commutative groups: if $G$ is an abelian Lie group , then its Lie algebra $\mathfrak{g}$ is abelian. Conversely, for connected $G$, abelianness of $\mathfrak{g}$ forces the commutator subgroup to be discrete, hence trivial, so $G$ is abelian.