Remarks on Mathieu-Zhao Subspaces of Commutative Associative Algebras and Vertex Algebras

TL;DR

This paper introduces Mathieu-Zhao subspaces in vertex algebras, linking their properties to commutative algebra and the Jacobian conjecture, and classifies certain subspaces in specific vertex algebras.

Contribution

It defines Mathieu-Zhao subspaces for vertex algebras and establishes their equivalence with subspaces in commutative associative algebras, connecting to the Jacobian conjecture.

Findings

Characterization of Mathieu-Zhao subspaces in vertex algebras

Equivalence with subspaces in commutative associative algebras

Classification of Mathieu-Zhao subspaces in C_2-cofinite vertex operator algebras

Abstract

We introduce a notion of Mathieu-Zhao subspaces of vertex algebras. Among other things, we show that for a vertex algebra $V$ and its subspace $M$ that contains $C_2(V)$, $M$ is a Mathieu-Zhao subspace of $V$ if and only if the quotient space $M/C_2(V)$ is a Mathieu-Zhao subspace of a commutative associative algebra $V/C_2(V)$. As a result, one can study the famous Jacobian conjecture in terms of Mathieu-Zhao subspaces of vertex algebras. In addition, for a $CFT$-type vertex operator algebra $V$ that satisfies the $C_2$-cofiniteness condition, we classify all Mathieu-Zhao subspaces $M$ that contain $C_2(V)$.