### Abstract

Let V be a grading-restricted vertex algebra and W a V-module. We show that for any m ∈ 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 ∈ 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.

Original language | English (US) |
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Pages (from-to) | 261-278 |

Number of pages | 18 |

Journal | Communications In Mathematical Physics |

Volume | 327 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2014 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics