no. 1
Poisson vertex cohomology and Tate Lie algebroids
Bouaziz, Emile1 orcid
1Academia Sinica, No. 1, Sec. 4, Roosevelt Road, Da-an, 6F, Astronomy-Mathematics Building, Taipei 106319, Taiwan
Annals of Representation Theory, Volume 3 (2026) no. 1, pp. 99-111

We study sheaves on holomorphic spaces of loops (also referred to as arcs) and apply this to the study of the complex, defined in [2], governing deformations of the Poisson vertex algebra structure on the space of holomorphic loops into a Poisson variety. We describe this complex in terms of the (continuous) de Rham–Lie cohomology of an associated Lie algebroid object in locally linearly compact topological (alias Tate) sheaves of modules on $\mathcal{L}^{+}M$. In particular this allows us to easily compute the cohomology of the above in the case where $\pi $ is symplectic, we obtain de Rham cohomology of $M$.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/art.35
Classification: 17B69, 17B63, 81T40
Keywords: vertex algebras, Lie algebroids, Poisson cohomology

Bouaziz, Emile  1

1 Academia Sinica, No. 1, Sec. 4, Roosevelt Road, Da-an, 6F, Astronomy-Mathematics Building, Taipei 106319, Taiwan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ART_2026__3_1_99_0,
     author = {Bouaziz, Emile},
     title = {Poisson vertex cohomology and {Tate} {Lie} algebroids},
     journal = {Annals of Representation Theory},
     pages = {99--111},
     year = {2026},
     publisher = {The Publishers of ART},
     volume = {3},
     number = {1},
     doi = {10.5802/art.35},
     language = {en},
     url = {https://art.centre-mersenne.org/articles/10.5802/art.35/}
}
TY  - JOUR
AU  - Bouaziz, Emile
TI  - Poisson vertex cohomology and Tate Lie algebroids
JO  - Annals of Representation Theory
PY  - 2026
SP  - 99
EP  - 111
VL  - 3
IS  - 1
PB  - The Publishers of ART
UR  - https://art.centre-mersenne.org/articles/10.5802/art.35/
DO  - 10.5802/art.35
LA  - en
ID  - ART_2026__3_1_99_0
ER  - 
%0 Journal Article
%A Bouaziz, Emile
%T Poisson vertex cohomology and Tate Lie algebroids
%J Annals of Representation Theory
%D 2026
%P 99-111
%V 3
%N 1
%I The Publishers of ART
%U https://art.centre-mersenne.org/articles/10.5802/art.35/
%R 10.5802/art.35
%G en
%F ART_2026__3_1_99_0
Bouaziz, Emile. Poisson vertex cohomology and Tate Lie algebroids. Annals of Representation Theory, Volume 3 (2026) no. 1, pp. 99-111. doi: 10.5802/art.35

[1] Arakawa, Tomoyuki A remark on the C 2 -cofiniteness condition on vertex algebras, Math. Z., Volume 270 (2012) no. 1-2, pp. 559-575 | DOI | Zbl | MR

[2] Bakalov, Bojko; De Sole, Alberto; Heluani, Reimundo; Kac, Victor G. An Operadic Approach to Vertex Algebra and Poisson Vertex Algebra Cohomology, Jpn. J. Math. (3), Volume 14 (2019) no. 2, pp. 249-342 | DOI | Zbl | MR

[3] Bakalov, Bojko; Kac, Victor G.; Voronov, Alexander A. Cohomology of Conformal Algebras, Commun. Math. Phys., Volume 200 (1999) no. 3, pp. 561-598 | DOI | Zbl | MR

[4] Beilinson, Alexander Remarks on topological algebras, Mosc. Math. J., Volume 8 (2008) no. 1, pp. 1-20 | Zbl | DOI | MR

[5] Beilinson, Alexander; Drinfeld, Vladimir G. Chiral Algebras, AMS Colloquium Publications, 51, American Mathematical Society, 2004 | Zbl | DOI | MR

[6] Crainic, Marius; Moerdijk, Ieke Deformations of Lie brackets: cohomological aspects, J. Eur. Math. Soc., Volume 10 (2008) no. 4, pp. 1037-1059 | DOI | Zbl | MR

[7] Denef, Jan; Loeser, François Geometry on Arc Spaces of Algebraic Varieties, 3rd European congress of mathematics (ECM), Barcelona, Spain, July 10–14, 2000. Volume I (Progress in Mathematics), Volume 201, Birkhäuser, 2001, pp. 327-348 | Zbl | DOI | MR

[8] Drinfeld, Vladimir G. Infinite Dimensional Vector Bundles in Algebraic Geometry: an introduction, The Unity of Mathematics (Progress in Mathematics), Volume 244, Birkhäuser, 2004, pp. 263-304 | Zbl | DOI

[9] Li, Haisheng Abelianizing vertex algebras, Commun. Math. Phys., Volume 259 (2005) no. 2, pp. 391-411 | DOI | Zbl | MR

[10] Lichnerowicz, Andre Les variétés de Poisson et leurs algèbres de Lie associées, J. Differ. Geom., Volume 12 (1977), pp. 253-300 | DOI | Zbl | MR

[11] De Sole, Alberto; Kac, Victor G. Essential Variational Poisson Cohomology, Commun. Math. Phys., Volume 313 (2012) no. 3, pp. 837-864 | DOI | Zbl | MR

[12] De Sole, Alberto; Liu, Jiefeng; Valeri, Daniele Cohomology of Lie conformal algebroids (2025) | arXiv | Zbl

[13] The Stacks Project Authors The Stacks Project (Section 20.25, accessed: 02/09/2026, https://stacks.math.columbia.edu/tag/01FP)

Cited by Sources: