# First and Second Cohomologies of Grading-Restricted Vertex Algebras

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## 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*.

## Keywords

Equivalence Class Vertex Operator Identity Property Vertex Operator Algebra Absolute Convergence## Preview

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## References

- FHL.Frenkel, I., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc. 104, 1–64 (1993)MathSciNetGoogle Scholar
- H.Huang, Y.-Z.: A cohomology theory of grading-restricted vertex algebras. Commun. Math. Phys. doi: 10.1007/s00220-014-1940-1 (2014)
- W.Weibel, C.:
*An introduction to homological algebras*. In: Cambridge Studies in Advanced Mathematics, Vol.**38**, pp. 1–450, Cambridge: Cambridge University Press (1994)Google Scholar