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The purpose of this Ph.D. thesis is to study and classify primitive ideals ofthe enveloping algebras $U(\mathfrak{o}(\infty))$ and$U(\mathfrak{sp}(\infty))$. Let $\mathfrak{g}(\infty)$ denote any of the Liealgebras $\mathfrak{o}(\infty)$ 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 ofbounded ideals, which is an analog of a coherent local system offinite-dimensional modules, as introduced earlier by A. Zhilinskii. As aresult, primitive ideals of $U(\mathfrak{g}(\infty))$ are parametrized bytriples $(x,y,Z)$ where $x$ is a nonnegative integer, $y$ is a nonnegativeinteger or half-integer, and $Z$ is a Young diagram. In the case of$\mathfrak{o}(\infty)$, each primitive ideal is integrable, and ourclassification reduces to a classification of integrable ideals going back toA. 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 $y$ isa half-integer.

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