The simplest cohomological invariants for vertex algebras

TL;DR

This paper develops a new cohomological framework for grading-restricted vertex algebras, introducing a bi-graded differential algebra structure and identifying fundamental cohomology classes that are independent of specific mappings.

Contribution

It introduces a multiplication on double complex spaces, establishes their structure as bi-graded differential algebras, and defines simple cohomology classes independent of mapping choices.

Findings

Cohomology classes are independent of mapping choices.

Double complex spaces form bi-graded differential algebras.

Orthogonality and bi-grading yield generators and relations for a Lie algebra.

Abstract

For the double complex structure of grading-restricted vertex algebra cohomology defined in \cite{Huang}, we introduce a multiplication of elements of double complex spaces. We show that the orthogonality and bi-grading conditions applied on double complex spaces, provide in relation among mappings and actions of co-boundary operators. Thus, we endow the double complex spaces with structure of bi-graded differential algebra. We then introduce the simples cohomology classes for a grading-restricted vertex algebra, and show their independence on the choice of mappings from double complex spaces. We prove that its cohomology class does not depend on mappings representing of the double complex spaces. Finally, we show that the orthogonality relations together with the bi-grading condition bring about generators and commutation relations for a continual Lie algebra.