Lie algebra homomorphism

Let $\mathfrak g,\mathfrak h$ be Lie algebras over a field $\Bbbk$ (typically $\Bbb R$ or $\Bbb C$), with Lie brackets $[\ ,\ ]_{\mathfrak g}$ and $[\ ,\ ]_{\mathfrak h}$.

Definition

A Lie algebra homomorphism is a $\Bbbk$-linear map $\varphi:\mathfrak g\to\mathfrak h$ such that for all $X,Y\in\mathfrak g$,

Equivalently, $\varphi$ is a morphism in the category of Lie algebras: it intertwines the brackets.

Basic consequences

• The kernel $\ker(\varphi)$ is an ideal in $\mathfrak g$.
• The image $\operatorname{im}(\varphi)$ is a Lie subalgebra of $\mathfrak h$.
• There is an induced injective homomorphism $\overline\varphi:\mathfrak g/\ker(\varphi)\to \operatorname{im}(\varphi)$, giving an isomorphism of Lie algebras $\mathfrak g/\ker(\varphi)\cong \operatorname{im}(\varphi)$ (an instance of the first isomorphism theorem, formulated using quotients of Lie algebras ).

Context

If $f:G\to H$ is a Lie group homomorphism , then its differential at the identity, $df_e:\operatorname{Lie}(G)\to \operatorname{Lie}(H)$, is a Lie algebra homomorphism (see the differential–bracket compatibility ). This is the bridge between global group structure and infinitesimal algebra structure.