Classification of primitive ideals of $U(\mathfrak{o}(\infty))$ and $U(\mathfrak{sp}(\infty))$

Abstract

The purpose of this Ph.D. thesis is to study and classify primitive ideals of the enveloping algebras $U(\mathfrak{o}(\infty))$ and $U(\mathfrak{sp}(\infty))$. Let $\mathfrak{g}(\infty)$ denote any of the Lie algebras $\mathfrak{o}(\infty)$ or $\mathfrak{sp}(\infty)$. Then\break $\mathfrak{g}(\infty)=\bigcup_{n\geq 2} \mathfrak{g}(2n)$ for $\mathfrak{g}(2n)=\mathfrak{o}(2n)$ or $\mathfrak{g}(2n)=\mathfrak{sp}(2n)$, respectively. We show that each primitive ideal $I$ of $U(\mathfrak{g}(\infty))$ is weakly bounded, i.e., $I\cap U(\mathfrak{g}(2n))$ equals the intersection of annihilators of bounded weight $\mathfrak{g}(2n)$-modules. To every primitive ideal $I$ of $\mathfrak{g}(\infty)$ we attach a unique irreducible coherent local system of bounded ideals, which is an analog of a coherent local system of finite-dimensional modules, as introduced earlier by A. Zhilinskii. As a result, primitive ideals of $U(\mathfrak{g}(\infty))$ are parametrized by triples $(x,y,Z)$ where $x$ is a nonnegative integer, $y$ is a nonnegative integer or half-integer, and $Z$ is a Young diagram. In the case of $\mathfrak{o}(\infty)$, each primitive ideal is integrable, and our classification reduces to a classification of integrable ideals going back to A. Zhilinskii, A. Penkov and I. Petukhov. In the case of $\mathfrak{sp}(\infty)$, only 'half' of the primitive ideals are integrable, and nonintegrable primitive ideals correspond to triples $(x,y,Z)$ where $y$ is a half-integer.