First and second cohomologies of grading-restricted vertex algebras

Details First and second cohomologies of grading-restricted vertex algebras First and second cohomologies of grading-restricted vertex algebras

2010-08-13

arxiv.org/abs/1008.2351v1

Mathematics

Quantum Algebra

24 pages

Scientific paper

Let $V$ be a grading-restricted vertex algebra and $W$ a $V$-module. We show that for any m in 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 N are equal (and can be denoted using the same notation H^{1}(V, W)). We also show that the second cohomology H^{2}_{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}_{1/2}(V, V) corresponds bijectively to the set of equivalence classes of first order deformations of V.

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