Mathieu-Zhao Subspaces of Vertex Algebras

TL;DR

This paper introduces Mathieu-Zhao subspaces within vertex algebras, classifies many such subspaces via group actions, and discusses the LNED conjecture, extending algebraic concepts to vertex algebra structures.

Contribution

It formally defines Mathieu-Zhao subspaces in vertex algebras, classifies numerous examples, and proposes the LNED conjecture for these structures.

Findings

Classified infinite non-trivial Mathieu-Zhao subspaces in vertex algebras.

Established connections between Mathieu-Zhao subspaces and associative algebra concepts.

Proposed the LNED conjecture for vertex algebras.

Abstract

A Mathieu-Zhao subspace is a generalization of an ideal of an associative algebra $\mathcal A$ over a unital ring $R$ first formalized in 2010. A vertex algebra is an algebraic structure first developed in conjunction with string theory in the 1960s and later axiomatized by mathematicians in the 1980s. We formally introduce the definition of a Mathieu-Zhao subspace $M$ of a vertex algebra $V$. Building on natural connections to associative algebras, we classify an infinite set of non-trivial, non-ideal Mathieu-Zhao subspaces for simple and general vertex algebras by group action eigenspace decomposition. Finally, we state the locally nilpotent $\varepsilon$-derivation (LNED) conjecture for vertex algebras.