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 , the first cohomology of V with coefficients in W introduced by the author is linearly isomorphic to the space of derivations from V to W. In particular, for are equal (and can be denoted using the same notation H (1)(V, W)). We also show that the second cohomology 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 corresponds bijectively to the set of equivalence classes of first order deformations of V.Physics, MathematicalSCI(E)0ARTICLEyzhuang@math.rutgers.edu1261-27832