# Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case

@article{Liu2013BihamiltonianCA, title={Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case}, author={Si‐Qi Liu and You-jin Zhang}, journal={Communications in Mathematical Physics}, year={2013}, volume={324}, pages={897-935} }

We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless KdV hierarchy starting from any of its infinitesimal deformations.

## 30 Citations

### Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations

- Mathematics
- 2021

This series of papers is devoted to the study of deformations of Virasoro symmetries of the principal hierarchies associated to semisimple Frobenius manifolds. The main tool we use is a…

### Bihamiltonian Cohomologies and Integrable Hierarchies II: The Tau Structures

- MathematicsCommunications in Mathematical Physics
- 2018

Starting from a so-called flat exact semisimple bihamiltonian structure of hydrodynamic type, we arrive at a Frobenius manifold structure and a tau structure for the associated principal hierarchy.…

### Bihamiltonian Cohomologies and Integrable Hierarchies II: The Tau Structures

- MathematicsCommunications in Mathematical Physics
- 2018

Starting from a so-called flat exact semisimple bihamiltonian structure of hydrodynamic type, we arrive at a Frobenius manifold structure and a tau structure for the associated principal hierarchy.…

### Variational Bihamiltonian Cohomologies and Integrable Hierarchies II: Virasoro Symmetries

- MathematicsCommunications in Mathematical Physics
- 2022

We prove that for any tau-symmetric bihamiltonian deformation of the tau-cover of the Principal Hierarchy associated with a semisimple Frobenius manifold, the deformed tau-cover admits an infinite…

### Deformations of semisimple poisson pencils of hydrodynamic type are unobstructed

- Mathematics
- 2015

We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation…

### Bihamiltonian Cohomology of KdV Brackets

- Mathematics
- 2014

Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular, this proves a conjecture of Liu…

### Bihamiltonian Cohomology of KdV Brackets

- MathematicsCommunications in Mathematical Physics
- 2016

Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular, this proves a conjecture of Liu…

### Lecture Notes on Bihamiltonian Structures and Their Central Invariants

- Mathematics
- 2018

In these lecture notes, we give an introduction to the classification theorem of semisimple bihamiltonian structures, with as much details as possible. The equivalence classes of this classification…

### Towards a bihamiltonian structure for the double ramification hierarchy

- MathematicsLetters in Mathematical Physics
- 2021

We propose a remarkably simple and explicit conjectural formula for a bihamiltonian structure of the double ramification hierarchy corresponding to an arbitrary homogeneous cohomological field…

## References

SHOWING 1-10 OF 35 REFERENCES

### On the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type

- Mathematics
- 2006

### Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation

- Mathematics
- 1998

Abstract:We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a…

### Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

- Mathematics
- 2001

We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into the…

### Simple singularities and integrable hierarchies

- Mathematics
- 2005

The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In [12], it is proved that the total descendent potential corresponding to K.…

### On Deformation of Poisson Manifolds of Hydrodynamic Type

- Mathematics
- 2005

We study a class of deformations of infinite-dimensional Poisson manifolds of hydrodynamic type which are of interest in the theory of Frobenius manifolds. We prove two results. First, we show that…

### Lie algebras and equations of Korteweg-de Vries type

- Mathematics
- 1985

The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats-Moody and systems of differential equations generalizing the Korteweg-de Vries and…

### Bihamiltonian Systems of Hydrodynamic Type and Reciprocal Transformations

- Mathematics
- 2006

We prove that under certain linear reciprocal transformation, an evolutionary PDE of hydrodynamic type that admits a bihamiltonian structure is transformed to a system of the same type which is still…