Lie algebra isomorphism

Let $\mathfrak g,\mathfrak h$ be Lie algebras .

Definition

A Lie algebra isomorphism is a map $\varphi:\mathfrak g\to\mathfrak h$ that is

1. a Lie algebra homomorphism , and
2. a bijection (equivalently, a linear isomorphism).

In this case the inverse map $\varphi^{-1}:\mathfrak h\to\mathfrak g$ is automatically a Lie algebra homomorphism as well, so $\mathfrak g$ and $\mathfrak h$ are “the same” as Lie algebras.

Automorphisms

An isomorphism $\varphi:\mathfrak g\to\mathfrak g$ is a Lie algebra automorphism ; the set of all such maps forms the group $\operatorname{Aut}(\mathfrak g)$ under composition.

Context

Lie algebra isomorphism is the correct equivalence relation for “infinitesimal symmetry.” In particular, by Lie’s third theorem , isomorphism classes of finite-dimensional Lie algebras correspond to connected, simply connected Lie groups up to isomorphism (see also simply connected groups are determined by their Lie algebra ).