# Integrability of Dirac reduced bi-Hamiltonian equations

@article{Sole2014IntegrabilityOD, title={Integrability of Dirac reduced bi-Hamiltonian equations}, author={Alberto De Sole and Victor G. Kac and Daniele Valeri}, journal={arXiv: Mathematical Physics}, year={2014}, pages={13-32} }

First, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three hierarchies of bi-Hamiltonian PDE’s, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies.

## 6 Citations

### Introduction to vertex algebras, Poisson vertex algebras, and integrable Hamiltonian PDE

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These lectures were given in Session 1: “Vertex algebras, W-algebras, and applications” of INdAM Intensive research period “Perspectives in Lie Theory” at the Centro di Ricerca Matematica Ennio De…

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. Logarithmic vertex algebras were introduced in our previous paper, motivated by logarithmic conformal ﬁeld theory [BV]. Non-local Poisson vertex algebras were introduced by De Sole and Kac,…

### Erratum to: Classical W-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents (Commun. Math. Phys., (2014), 331, (623-676), 10.1007/s00220-014-2049-2)

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- 2014

There is an error in Sect. 6.2 of the original article in the computation of equations of the generalized Drinfeld–Sokolov hierarchy associated to the minimal nilpotent element f of the Lie algebra g…

### Classical and Quantum $${\mathcal {W}}$$-Algebras and Applications to Hamiltonian Equations

- MathematicsSpringer INdAM Series
- 2019

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### Singular Degree of a Rational Matrix Pseudodifferential Operator

- Mathematics
- 2015

In our previous work we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H=A/B, where A and B are matrix differential operators, and B is non-degenerate of…

### Classical $${\mathcal{W}}$$W-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

- Mathematics
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We derive explicit formulas for λ-brackets of the affine classical $${\mathcal{W}}$$W -algebras attached to the minimal and short nilpotent elements of any simple Lie algebra $${\mathfrak{g}}$$g .…

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