Nilpotent Elements of Vertex Algebras

TL;DR

This paper proves that the set of nilpotent elements in a vertex algebra forms an ideal and that the quotient by this ideal has no nonzero nilpotent elements, using commutative algebra techniques.

Contribution

It establishes a structural property of nilpotent elements in vertex algebras, showing they form an ideal and characterizing the quotient algebra.

Findings

Nilpotent elements form an ideal in vertex algebras.

The quotient algebra by nilpotent elements has no nonzero nilpotent elements.

Abstract

Using the method of commutative algebra, we show that the set $\mathfrak{R}$ of nilpotent elements of a vertex algebra $V$ forms an ideal, and $V/\mathfrak{R}$ has no nonzero nilpotent elements.