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 $\hat{\mathfrak{g}}$, which is an extension of $L\mathfrak{g}$ by a central element $c$. It is known that the rings $U(\hat{\mathfrak{g}})/(c-\lambda )$ are simple for any nonzero scalar $\lambda$. On the other hand, the universal enveloping algebra $\mathrm{U}(L\mathfrak{g})\cong \mathrm{U}(\hat{\mathfrak{g}})/(c)$ 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 $L\mathfrak{g}$.