# [Plenary] Enveloping Algebras of Infinite-Dimensional Lie Algebras

The algebraic structure of a Lie algebra can be thought of as a linear version of a group structure, so a Lie algebra is a vector space that is a "first approximation to a group''. Lie algebras are not associative, but if $L$ is a Lie algebra, there is an associative ring $U(L)$, called the {\em universal enveloping algebra of $L$} which has the same representation theory as $L$.

For finite-dimensional $L$, the rings $U(L)$ are some of the most well-understood objects in noncommutative algebra, and are known to have many good properties: for example, they are left and right noetherian, as they are deformations of polynomial rings in finitely many variables. For infinite-dimensional $L$, on the other hand, the rings $U(L)$ are much more mysterious. We give a survey on what is known about universal enveloping algebras of infinite-dimensional Lie algebras, focussing on chain conditions such as noetherianity. We will assume minimal background in noncommutative ring theory or in Lie theory.