# What is the point of (Lie) derivations?

Probably some very naive questions, but ...

Definition
Let $$A$$ be a vector space together with a bilinear map $$\mu: A \times A \rightarrow A$$. We call a map $$D: A \rightarrow A$$ a derivation if it satisfies the Leibniz rule $$D(\mu (a,b))=\mu(D(a), b) + \mu(a, D(b))$$ for all $$a,b \in A$$. There are many examples of derivations, differential operators on a suitable associative algebra being one of them.

Lie derivations
One can look at derivations of Lie algebras (where $$\mu$$ is the Lie bracket). Given a Lie algebra $$\mathfrak g$$ (in fact any vector space with bilinear product) one can show that the derivations on $$\mathfrak g$$ form a Lie subalgebra of $$End(\mathfrak g)$$ with respect to the commutator. Furthermore, the Jacobi identity is precisely the statement that $$ad_x$$ is a derivation with respect to the Lie bracket for any $$x \in\mathfrak g$$.

Question(s)
Why should we care? What is the relevance of Lie derivations? Why study them? What do they tell us about a Lie algebra? Are they somehow a generalization of differential operators? Are they related to the Lie algebra-Lie group correspondence? Is there any category theoretic perspective on them?

• Lie algebra derivations are as important to Lie algebras as homomorphisms or automorphisms to groups. They tell us a lot about the Lie algebra, but not everything (have a look at the posts here). Commented Jan 31, 2022 at 13:44
• For the link between Lie algebra derivations and automorphisms see for example this post, and others. Commented Jan 31, 2022 at 13:54
• @DietrichBurde: The automorphism group of a Lie group $G$ is again a Lie group (how?). Its associated Lie algebra is the Lie algebra of derivations on the Lie algebra associated to $G$. Correct? Is that what you mean? Commented Feb 1, 2022 at 11:39
• Yes. Furthermore derivations are $1$-cocycles for Lie algebra cohomology, and we can construct extensions with it. This also plays a role in geometry (your question is so broad, that it is difficult to mention all aspects). Commented Feb 1, 2022 at 11:43