Abstract
We prove that, for a Poisson vertex algebra \({\cal V}\), the canonical injective homomorphism of the variational cohomology of \({\cal V}\) to its classical cohomology is an isomorphism, provided that \({\cal V}\), viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.
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Acknowledgements
This research was partially conducted during the authors’ visits to the University of Rome La Sapienza, to MIT, and to IHES. The first author was supported in part by a Simons Foundation grant 584741. The second author was partially supported by the national PRIN fund n. 2015ZWST2C_001 and the University funds n. RM116154CB35DFD3 and RM11715C7FB74D63. The third author was partially supported by CPNq grant 409582/2016-0. The fourth author was partially supported by the Bert and Ann Kostant fund and by a Simons Fellowship. We would like to thank Pavel Etingof for providing a proof of the differential HKR theorem for the algebra of differential polynomials, included in Appendix A. We are grateful to the referee for carefully reading the paper and suggesting improvements of the exposition.
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Communicated by: Yasuyuki Kawahigashi
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Bakalov, B., De Sole, A., Heluani, R. et al. Classical and variational Poisson cohomology. Jpn. J. Math. 16, 203–246 (2021). https://doi.org/10.1007/s11537-021-2109-2
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DOI: https://doi.org/10.1007/s11537-021-2109-2
Keywords and phrases
- Poisson vertex algebra (PVA)
- classical operad
- classical PVA cohomology
- variational PVA cohomology
- sesquilinear Hochschild and Harrison cohomology