How much does the universal enveloping algebra of a Lie algebra remember about the Lie algebra itself? This is the statement of a classical problem in algebra called ``the isomorphism problem for enveloping algebras". In this talk, I'll explain how modern tools from algebraic homotopy theory and deformation theory allow us to make new progress on this problem. Along the way, we'll see that this is closely related to the following intriguing question: in the realm of homotopy theory, is being a commutative algebra simply a property, or is it an additional structure?  We will make sense of this second question and see that this turns out to be interesting for algebraic topologists as well.

(Joint with D. Petersen, D. Robert-Nicoud, F. Wierstra.)