Lie subalgebra

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

Definition

A Lie subalgebra of $\mathfrak g$ is a $\Bbbk$-linear subspace $\mathfrak h\subseteq \mathfrak g$ such that

i.e. $[X,Y]\in \mathfrak h$ for all $X,Y\in \mathfrak h$.

With the restricted bracket, $\mathfrak h$ is itself a Lie algebra, and the inclusion $\mathfrak h\hookrightarrow \mathfrak g$ is a Lie algebra homomorphism .

Context and examples

• If $H\subseteq G$ is a Lie subgroup of a Lie group, then $\operatorname{Lie}(H)\subseteq \operatorname{Lie}(G)$ is a Lie subalgebra (see the Lie algebra of a subgroup lemma ).
• An ideal is a Lie subalgebra $\mathfrak h$ satisfying $[\mathfrak g,\mathfrak h]\subseteq \mathfrak h$; ideals are exactly kernels of Lie algebra homomorphisms and allow formation of quotient Lie algebras .