Ideals of enveloping algebras of loop algebras

Abstract: Let g be a finite-dimensional simple Lie algebra, and consider the loop algebra $L_g = g[t, t^{-1}]$ and the affine Lie algebra $\hat{g}$, which is an extension of $L_g$ by a central element $c$. We investigate two-sided ideals in the universal enveloping algebra $U(L_g)$. It is known that the rings $U(\hat{g})/(c-\lambda)$ are simple for any nonzero scalar $\lambda$, but the two-sided structure of $U(L_g) = U(\hat{g}/(c))$ is more complicated. We show that $U(L_g)$ does not satisfy the ascending chain condition on two-sided ideals, but that the two-sided ideals still have a nice structure: there is a canonical collection of ideals $I_n$, parameterised by positive integers, so that any two-sided ideal of $U(L_g)$ contains some $I_n$. The ideals $I_n$ can be thought of as universal annihilators of classes of finite-dimensional representations of $L_g$. This is a preliminary report on joint work with Alexey Petukhov.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar