First and Second Cohomologies of Grading-Restricted Vertex Algebras

Let V be a grading-restricted vertex algebra and W a V-module. We show that for any ${min mathbb{Z}_{+}}$ , 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 ${min mathbb{N}}$ 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.