# Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types

## TL;DR

This paper characterizes and classifies centrally generated primitive ideals of the universal enveloping algebra of the nilradical in complex semisimple Lie algebras, especially focusing on exceptional types, using the Dixmier map and Kostant cascade.

## Contribution

It provides an explicit description of centrally generated primitive ideals for exceptional Lie algebras, extending previous results from classical types to exceptional cases.

## Key findings

- Explicit characterization of primitive ideals in exceptional types
- Classification of centrally generated primitive ideals for any semisimple algebra
- Extension of classical results to exceptional Lie algebras

## Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$, $\mathfrak{n}$ be the nilradical of $\mathfrak{b}$, and $U(\mathfrak{n})$ be the universal enveloping algebra of $\mathfrak{n}$. We study primitive ideals of $U(\mathfrak{n})$. Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center $Z(\mathfrak{n})$ of $U(\mathfrak{n})$. We present an explicit characterization of the centrally generated primitive ideals of $U(\mathfrak{n})$ in terms of the Dixmier map and the Kostant cascade in the case when $\mathfrak{g}$ is a simple algebra of exceptional type. (For classical simple Lie algebras, a similar characterization was obtained by Ivan Penkov and the first author.) As a corollary, we establish a classification of centrally generated primitive ideals of $U(\mathfrak{n})$ for an arbitrary semisimple algebra $\mathfrak{g}$.