Centrally generated primitive ideals of $U(\mathfrak{n})$ in types $B$ and $D$

TL;DR

This paper characterizes centrally generated primitive ideals of the universal enveloping algebra of the nilpotent radical in certain Lie algebras of types B and D, using the Dixmier map and Kostant cascade, including infinite-dimensional cases.

Contribution

It provides a detailed description of these primitive ideals in types B and D, extending to infinite-dimensional cases with large centers, using advanced algebraic tools.

Findings

Classification of primitive ideals in finite-dimensional cases.

Extension of results to infinite-dimensional nilpotent radicals.

Use of the Dixmier map and Kostant cascade for characterization.

Abstract

We study the centrally generated primitive ideals of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the (locally) nilpotent radical of a (splitting) Borel subalgebra of a simple complex Lie algebra $\mathfrak{g}=\mathfrak{o}_{2n+1}(\mathbb{C})$, $\mathfrak{o}_{2n}(\mathbb{C})$, $\mathfrak{o}_{\infty}(\mathbb{C})$. In the infinite-dimensional setting, there are infinitely many isomorphism classes of Lie algebras $\mathfrak{n}$, and we fix $\mathfrak{n}$ with "largest possible" center of $U(\mathfrak{n})$. We characterize the centrally generated primitive ideals of $U(\mathfrak{n})$ in terms of the Dixmier map and the Kostant cascade.