Maps from the enveloping algebra of the positive Witt algebra to regular algebras

TL;DR

This paper constructs algebra homomorphisms from the enveloping algebra of the positive Witt algebra to regular algebras, revealing non-noetherian properties of related infinite-dimensional algebras.

Contribution

It introduces explicit homomorphisms to Artin-Schelter regular algebras and characterizes their kernels and images, providing new proofs of non-noetherianity.

Findings

Universal enveloping algebras are not noetherian

Constructed explicit algebra homomorphisms

Determined kernels and images of these maps

Abstract

We construct homomorphisms from the universal enveloping algebra of the positive (part of the) Witt algebra to several different Artin-Schelter regular algebras, and determine their kernels and images. As a result, we produce elementary proofs that the universal enveloping algebras of the Virasoro algebra, the Witt algebra, and the positive Witt algebra are neither left nor right noetherian.