Advertisement

Communications in Mathematical Physics

, Volume 327, Issue 1, pp 261–278 | Cite as

First and Second Cohomologies of Grading-Restricted Vertex Algebras

  • Yi-Zhi HuangEmail author
Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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
  2. H.
    Huang, Y.-Z.: A cohomology theory of grading-restricted vertex algebras. Commun. Math. Phys. doi: 10.1007/s00220-014-1940-1 (2014)
  3. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchPeking UniversityBeijingChina
  2. 2.Kavali Institute For Theoretical Physics ChinaCASBeijingChina
  3. 3.Department of MathematicsRutgers UniversityPiscatawayUSA

Personalised recommendations