Ideals in Enveloping Algebras of Loop Algebras

Abstract

Let $\mathfrak g$ be a finite-dimensional simple complex Lie algebra, and consider the loop algebra $L\mathfrak g = \mathfrak g[s, s^{-1}]$ and the affine Lie algebra $\widehat{\mathfrak g}$, which is an extension of $L\mathfrak g$ by a central element $c$. It is known that the rings $\operatorname{U}(\widehat{\mathfrak g})/(c-λ)$ are simple for any nonzero scalar $λ$. On the other hand, the universal enveloping algebra $\mathrm{U}(L\mathfrak g) \cong \mathrm{U}(\widehat{\mathfrak g})/(c)$ has many two-sided ideals, and in this paper we investigate their structure. We show that the two-sided structure of $\mathrm{U}(L\mathfrak g)$ is both intricate and well-behaved. Specifically, $\mathrm{U}(L\mathfrak g)$ does not satisfy the ascending chain condition on two-sided ideals, but the two-sided ideals still have a nice structure: there is a canonical nested collection of ideals $I_n$, parameterised by positive integers, so that any nonzero two-sided ideal of $\mathrm{U}(L\mathfrak g)$ contains all $I_n$ for $n \gg 0$. The ideals $I_n$ can be thought of as universal annihilators of natural classes of finite-dimensional representations of $L\mathfrak g$. We also study the Poisson structure of the symmetric algebra $\mathrm{S}(L\mathfrak g)$ and give applications to the representation theory of $L\mathfrak g$.