670 citations · 3.3k across the 47 of their papers we have counts for
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Perturbative criteria for the ergodicity of interacting dissipative quantum lattice systems
Lorenzo Bertini, Alberto De Sole, Gustavo Posta +1
We introduce a class of quantum Markov semigroups describing the evolution of interacting quantum lattice systems, specified either as generic qudits or as fermions. The correspond…
Poisson vertex algebras and Hamiltonian PDE
Alberto De Sole, Victor G. Kac, Daniele Valeri
We discuss the theory of Poisson vertex algebras and their generalizations in relation to integrability of Hamiltonian PDE. In particular, we discuss the theory of affine classical…
Trace distance ergodicity for quantum Markov semigroups
Lorenzo Bertini, Alberto De Sole, Gustavo Posta
We discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral de…
Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras
Alberto De Sole, Victor G. Kac, Daniele Valeri
We prove that any classical affine W-algebra W(g,f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of…
Classical W-algebras for gl_N and associated integrable Hamiltonian hierarchies
Alberto De Sole, Victor G. Kac, Daniele Valeri
We apply the new method for constructing integrable Hamiltonian hierarchies of Lax type equations developed in our previous paper, to show that all W-algebras W(gl_N,f) carry such…
A new scheme of integrability for (bi)Hamiltonian PDE
Alberto De Sole, Victor G. Kac, Daniele Valeri
We develop a new method for constructing integrable Hamiltonian hierarchies of Lax type equations, which combines the fractional powers technique of Gelfand and Dickey, and the cla…
Double Poisson vertex algebras and non-commutative Hamiltonian equations
Alberto De Sole, Victor G. Kac, Daniele Valeri
We develop the formalism of double Poisson vertex algebras (local and non-local) aimed at the study of non-commutative Hamiltionan PDEs. This is a generalization of the theory of d…
On integrability of some bi-Hamiltonian two field systems of PDE
Alberto De Sole, Victor G. Kac, Refik Turhan
We continue the study of integrability of bi-Hamiltonian systems with a compatible pair of local Poisson structures (H_0,H_1), where H_0 is a strongly skew-adjoint operator. This i…
Structure of classical (finite and affine) W-algebras
Alberto De Sole, Victor G. Kac, Daniele Valeri
First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to…
Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
Alberto De Sole, Victor G. Kac, Daniele Valeri
We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, t…
Integrability of Dirac reduced bi-Hamiltonian equations
Alberto De Sole, Victor G. Kac, Daniele Valeri
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 rem…
On classical finite and affine W-algebras
Alberto De Sole
This paper is meant to be a short review and summary of recent results on the structure of finite and affine classical W-algebras, and the application of the latter to the theory o…
A new approach to the Lenard-Magri scheme of integrability
Alberto De Sole, Victor G. Kac, Refik Turhan
We develop a new approach to the Lenard-Magri scheme of integrability of bi-Hamiltonian PDE's, when one of the Poisson structures is a strongly skew-adjoint differential operator.
Non-local Hamiltonian structures and applications to the theory of integrable systems I
Alberto De Sole, Victor G. Kac
We develop a rigorous theory of non-local Hamiltonian structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee ap…
Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras
Alberto De Sole, Victor G. Kac, Daniele Valeri
We provide a description of the Drinfeld-Sokolov Hamiltonian reduction for the construction of classical W-algebras within the framework of Poisson vertex algebras. In this context…
Essential variational Poisson cohomology
Alberto De Sole, Victor G. Kac
In our recent paper [DSK11] we computed the dimension of the variational Poisson cohomology for any quasiconstant coefficient matrix differential operator K of arbitrary order with…
The variational Poisson cohomology
Alberto De Sole, Victor G. Kac
It is well known that the validity of the so called Lenard-Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corre…
Poisson vertex algebras in the theory of Hamiltonian equations
Aliaa Barakat, Alberto De Sole, Victor G. Kac
We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is c…
Non-linear Lie conformal algebras with three generators
Bojko Bakalov, Alberto De Sole
We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sl_2. In doing so we discov…
Finite vs. affine W-algebras
Alberto De Sole, Victor Kac
In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the lambda-bracket. In Se…
Subalgebras of $\gc_N$ and Jacobi polynomials
Alberto De Sole, Victor G. Kac
We classify the subalgebras of the general Lie conformal algebra $\gc_N$ that act irreducibly on $\C[\partial]^N$ and that are normalized by the --part of a Vi…