Cohomology theory of Novikov algebras and applications

Abstract

In this paper, first we give a new characterization of the cohomology of pre-Lie algebras using the Chevalley-Eilenberg cohomology associated to a morphism from the operad of Lie algebras to Hadamard product of the operad of pre-Lie algebras and its Koszul dual operad. Then we apply the same approach to study the cohomology of Novikov algebras, and give the cochain complex explicitly. The cochain complex of the underlying pre-Lie algebra is shown to be isomorphic to the quotient of the cochain complex of a Novikov algebra. Consequently, there is a long exact sequence connecting the cohomologies of a Novikov algebra and the underlying pre-Lie algebra. The cohomology of a Novikov algebra with coefficients in a representation is introduced using pseudo-tensor categories. As applications, we show that infinitesimal deformations and abelian extensions are classified by the second cohomology groups with different coefficients. Various examples are given to illustrate the difference between the cohomology of a Novikov algebra and that of the underlying pre-Lie algebra.

Classification of Novikov-Poisson Algebras and Their Applications
Ziyi Zhang, Zeyu Hao, Yining Sun +1
arXiv26 Aug 2026PreprintCC0Full textAlgebra & number theoryarXiv
δ-Mock-Novikov Algebras: Structural Theory, Operad and Poisson-Type Constructions
Jie Ruan, Zafar Normatov
arXiv7 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
A Cohomology Theory of Supercommutative Algebras and Grading-Restricted Vertex Superalgebras
Paul Johnson, Fei Qi
Symmetry, Integrability and Geometry: Methods and Applications29 Oct 2026Peer-reviewedCC BY-SA 4.0Mathematical physicsCrossref
A closed formula for the Poisson cohomology of scalar Dubrovin-Novikov brackets in arbitrary dimension
Matteo Casati
arXiv1 Nov 2026PreprintarXiv licenceFull textGeometry & topology Mathematical physicsarXiv
Classification of Finite-Dimensional Simple Novikov Algebras and Their Irreducible Bimodules in Positive Characteristic
Dong Liu, Yufeng Pei, Limeng Xia
arXiv26 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theory Mathematical physicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{kolesnikov2026cohomology,
  title = {{Cohomology theory of Novikov algebras and applications}},
  author = {Pavel Kolesnikov and Yue Li and Yunhe Sheng and Nanyan Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06304},
  eprint = {2609.06304},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06304},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Cohomology theory of Novikov algebras and applications
AU  - Pavel Kolesnikov
AU  - Yue Li
AU  - Yunhe Sheng
AU  - Nanyan Xu
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06304
UR  - https://arxiv.org/abs/2609.06304
AB  - In this paper, first we give a new characterization of the cohomology of pre-Lie algebras using the Chevalley-Eilenberg cohomology associated to a morphism from the operad of Lie algebras to Hadamard product of the operad of pre-Lie algebras and its Koszul dual operad. Then we apply the same approach to study the cohomology of Novikov algebras, and give the cochain complex explicitly. The cochain complex of the underlying pre-Lie algebra is shown to be isomorphic to the quotient of the cochain complex of a Novikov algebra. Consequently, there is a long exact sequence connecting the cohomologies of a Novikov algebra and the underlying pre-Lie algebra. The cohomology of a Novikov algebra with coefficients in a representation is introduced using pseudo-tensor categories. As applications, we show that infinitesimal deformations and abelian extensions are classified by the second cohomology groups with different coefficients. Various examples are given to illustrate the difference between the cohomology of a Novikov algebra and that of the underlying pre-Lie algebra.
ER  -

Abstract

Related

Cite