Two-sided ideals of enveloping algebras of $sl_\infty$, $so_\infty$, $sp_\infty$

Fix $n$. Let $U(sl_n)$ be the universal enveloping algebra of a Lie algebra $sl_n$. We say that an ideal $I$ of $U(sl_n)$ is integrable if it is an intersection of ideals of finite codimension in $U(sl_n)$. Such ideals allows (yet unknown, at least to me!) classification based on the representation theory of finite-dimensional modules of $sl_n$. First result which I will discuss is that, for big enough $N$, a radical of intersection of $I$ with $U(sl_n)$ is integrable for any two-sided ideal $I$ of $U(sl_N)$. As I will show this result implies that a "radical" of any ideal of $U(sl_\infty)$ (here $sl_\infty$ is the direct limit of $sl_n$) is integrable. Note that all integrable ideals of $U(sl_\infty)$ are described 20 years ago (and I recall this description).

The talk will be based on the joint work with I. Penkov --- arXiv:1210.0466.