Infinite Kostant cascades and centrally generated primitive ideals of $U(\mathfrak{n})$ in types $A_{\infty}$, $C_{\infty}$

TL;DR

This paper investigates the center of the universal enveloping algebra of the nilpotent radical in infinite-dimensional Lie algebras of types A and C, providing explicit generators and characterizations of primitive ideals.

Contribution

It explicitly determines generators of the center of $U( )$ for infinite-dimensional Lie algebras and characterizes centrally generated primitive ideals in these cases.

Findings

Explicit generators of the center of $U( )$ for types $A_{}$ and $C_{}$.

Characterization of centrally generated primitive ideals in infinite-dimensional cases.

Extension of primitive ideal characterization to finite-dimensional classical Lie algebras.

Abstract

We study the center of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the locally nilpotent radical of a splitting Borel subalgebra of a simple complex Lie algebra $\mathfrak{g}=\mathfrak{sl}_{\infty}(\mathbb{C})$, $\mathfrak{so}_{\infty}(\mathbb{C})$, $\mathfrak{sp}_{\infty}(\mathbb{C})$. There are infinitely many isomorphism classes of Lie algebras $\mathfrak{n}$, and we provide explicit generators of the center of $U(\mathfrak{n})$ in all cases. We then fix $\mathfrak{n}$ with "largest possible" center of $U(\mathfrak{n})$ and characterize the centrally generated primitive ideals of $U(\mathfrak{n})$ for $\mathfrak{g}=\mathfrak{sl}_{\infty}(\mathbb{C})$, $\mathfrak{sp}_{\infty}(\mathbb{C})$ in terms of the above generators. As a preliminary result, we provide a characterization of the centrally generated primitive ideals in the enveloping algebra of the nilradical of a Borel subalgebra of $\mathfrak{sl}_n(\mathbb{C})$, $\mathfrak{sp}_{2n}(\mathbb{C})$.