Universal enveloping algebras, thanks to their so-called universal properties, are one of the main tools in the ring-theoretical approach to the representation theory of Lie algebras. Indeed, having a representation of a Lie algebra on a vector space is equivalent to having a module over its universal enveloping algebra (categorically speaking, one says that we have an isomorphism of categories). Furthermore, from a mathematical perspective, universal enveloping algebras enjoy a very rich structure, which affects also their representation theory: that of a Hopf algebra.
In this talk, starting from the familiar setting of Lie algebras over a field, we will recall the universal properties of universal enveloping algebras for Lie-Rinehart algebras and some of the main theorems associated with them (the Poincaré-Birkhoff-Witt and the Milnor-Moore Theorems) and we will discuss some recent developments regarding anchored Lie algebras and their universal enveloping algebras as well.
The novel part of the presentation will be based on arXiv:2009.14656 and arXiv:2102.01553