First and second cohomologies of grading-restricted vertex algebras

TL;DR

This paper studies the first and second cohomologies of grading-restricted vertex algebras, establishing their connections to derivations, extensions, and deformations, thus advancing the understanding of their algebraic structure.

Contribution

It introduces and characterizes the first and second cohomology groups of grading-restricted vertex algebras, linking them to derivations, extensions, and deformations.

Findings

First cohomology is isomorphic to the space of derivations.

Second cohomology classifies square-zero extensions.

Second cohomology corresponds to first order deformations.

Abstract

Let $V$ be a grading-restricted vertex algebra and $W$ a $V$-module. We show that for any $m\in \mathbb{Z}_{+}$, the first cohomology $H^{1}_{m}(V, W)$ of $V$ with coefficients in $W$ introduced by the author is linearly isomorphic to the space of derivations from $V$ to $W$. In particular, $H^{1}_{m}(V, W)$ for $m\in \mathbb{N}$ are equal (and can be denoted using the same notation $H^{1}(V, W)$). We also show that the second cohomology $H^{2}_{\frac{1}{2}}(V, W)$ of $V$ with coefficients in $W$ introduced by the author corresponds bijectively to the set of equivalence classes of square-zero extensions of $V$ by $W$. In the case that $W=V$, we show that the second cohomology $H^{2}_{\frac{1}{2}}(V, V)$ corresponds bijectively to the set of equivalence classes of first order deformations of $V$.