The orbit method for locally nilpotent infinite-dimensional Lie algebras

Let n be a locally nilpotent infinite-dimensional Lie algebra over C. Let U(n) and S(n) be its respective universal enveloping algebra and symmetric algebra. Consider the Jacobson topology on the primitive spectrum of U(n), and the Poisson topology on the primitive Poisson spectrum of S(n). We provide a homeomorphism between the corresponding topological spaces (at the level of points, it gives a bijection between the primitive ideals of U(n) and S(n)). We also show that all primitive ideals of S(n) from an open set in a properly chosen topology are generated by their intersections with the Poisson center. Under the assumption that n is a nil-Dynkin Lie algebra, we give two criteria for primitive ideals I(lambda) subset of S(n) and J(lambda) subset of U(n), lambda is an element of n*, to be nonzero. Most of these results generalize known facts about primitive and Poisson spectrum for finite-dimensional nilpotent Lie algebras (but note that for a finite-dimensional nilpotent Lie algebra all primitive ideals I(lambda), J(lambda) are nonzero). (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The paper discusses the universal enveloping algebra and symmetric algebra of a locally nilpotent infinite-dimensional Lie algebra over C, with a focus on their primitive and Poisson spectra. It provides a homeomorphism between the corresponding topological spaces and shows that primitive ideals of S(n) are mostly generated by intersections with the Poisson center. Additionally, the paper presents two criteria for determining nonzero primitive ideals in the context of nil-Dynkin Lie algebras.