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

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 ${\mathfrak{sl}}_n(\mathbb{C})$, ${\mathfrak{sp}}_{2n}(\mathbb{C})$.

20 pages

References in corpus (1)

• Cartan subalgebras of root-reductive Lie algebras