Uniqueness of Joseph ideal
Let \(\mathfrak g\) be a simple complex Lie algebra. For any ideal \(\mathcal J\) of the universal enveloping algebra \(U({\mathfrak g})\) of \(\mathfrak g\) one associates an ideal gr\(({\mathcal J})\) in the symmetric algebra \(U({\mathfrak g})\) of \(\mathfrak g\). If \(\mathcal J\) is primitive, then the associated variety Ass\(({\mathcal J})\) (i.e. the zero set of gr\(({\mathcal J})\) in \(\mathfrak g\)) is contained in the nilpotent cone of \(\mathfrak g\). In 1976 Joseph constructed a completely prime two-sided primitive ideal of \(U({\mathfrak g})\) whose associated variety is the closure of the minimal orbit \({\mathbf O}_{min}\) of \(\mathfrak g\). If \(\mathfrak g\) is not of type A, then the Joseph ideal is unique with this property. In the paper under review the authors provide a new simple proof of the uniqueness of the Joseph ideal which is different from the original proof of Joseph.
scientific article; zbMATH DE number 2152362