arXiv subjects

Daniele Valeri

Publications and source records attributed to Daniele Valeri.

35 records · Page 2Linked to original sources

arXiv2018

Local and non-local multiplicative Poisson vertex algebras and differential-difference equations

We develop the notions of multiplicative Lie conformal and Poisson vertex algebras, local and non-local, and their connections to the theory of integrable differential-difference Hamiltonian equations. We establish relations of these notions to $q$-deformed $W$-algebras and lattice Poisson algebras. We introduce the notion of Adler type pseudodifference operators and apply them to integrability of differential-difference Hamiltonian equations.

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arXiv2018

Poisson $λ$-brackets for differential-difference equations

We introduce the notion of a multiplicative Poisson $λ$-bracket, which plays the same role in the theory of Hamiltonian differential-difference equations as the usual Poisson $λ$-bracket plays in the theory of Hamiltonian PDE. We classify multiplicative Poisson $λ$-brackets in one difference variable up to order 5. Applying the Lenard-Magri scheme to a compatible pair of multiplicative Poisson $λ$-brackets of order 1 and 2, we establish integrability of some differential-difference equations, generalizing the Volterra chain.

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arXiv2018

Generators of the quantum finite W-algebras in type A

We prove a conjecture proposed in [DSKV16] describing the Lax type operator L(z) for the quantum finite W-algebras of gl_N in terms of a PBW generating system for the W-algebra. In doing so, we extend this result to an arbitrary good grading and an arbitrary isotropic subspace of g[1/2].

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arXiv2018

Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras

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 Lax type equations. This is based on the theories of generalized Adler type operators and of generalized quasideterminants, which we develop in the paper. Moreover, we show that under certain conditions, the product of two generalized Adler type operators is a Lax type operator. We use this fact to construct a large number of integrable Hamiltonian systems, recovering, as a special case, all KdV type hierarchies constructed by Drinfeld and Sokolov.

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arXiv2017

MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $\mathcal{W}$-algebras

We give an introduction to the Mathematica packages "MasterPVA" and "MasterPVAmulti used to compute lambda-brackets in Poisson vertex algebras, which play an important role in the theory of infinite-dimensional Hamiltonian systems. As an application, we give an introduction to the Mathematica package "WAlg" aimed to compute the lambda-brackets among the generators of classical affine W-algebras. The use of these packages is shown by providing some explicit examples.

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arXiv2016

Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case

We study the ODE/IM correspondence for ODE associated to $\hat{\mathfrak g}$-valued connections, for a simply-laced Lie algebra $\mathfrak g$. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called $Ψ$-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.

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arXiv2016

Bethe Ansatz and the Spectral Theory of affine Lie algebra--valued connections II. The non simply--laced case

We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $Ψ$-system among subdominant solutions. We then use the $Ψ$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.

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arXiv2016

Classical W-algebras for gl_N and associated integrable Hamiltonian hierarchies

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 hierarchy. As an application, we show that all vector constrained KP hierarchies and their matrix generalizations are obtained from these hierarchies by Dirac reduction, which provides the former with a bi-Poisson structure.

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arXiv2016

Finite W-algebras for gl_N

We study the quantum finite W-algebras W(gl_N,f), associated to the Lie algebra gl_N, and its arbitrary nilpotent element f. We construct for such an algebra an r_1 x r_1 matrix L(z) of Yangian type, where r_1 is the number of maximal parts of the partition corresponding to f. The matrix L(z) is the quantum finite analogue of the operator of Adler type which we introduced in the classical affine setup. As in the latter case, the matrix L(z) is obtained as a generalized quasideterminant. It should encode the whole structure of W(gl_N,f), including explicit formulas for generators and the commutation relations among them. We describe in all detail the examples of principal, rectangular and minimal nilpotent elements.

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arXiv2016

Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation

We study the one dimensional periodic derivative nonlinear Schrödinger (DNLS) equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion $\int h_k$, $k\in \mathbb{Z}_{+}$. In each $\int h_{2k}$ the term with the highest regularity involves the Sobolev norm $\dot H^{k}(\mathbb{T})$ of the solution of the DNLS equation. We show that a functional measure on $L^2(\mathbb{T})$, absolutely continuous w.r.t. the Gaussian measure with covariance $(\mathbb{I}+(-Δ)^{k})^{-1}$, is associated to each integral of motion $\int h_{2k}$, $k\geq1$.

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arXiv2015

Double Poisson vertex algebras and non-commutative Hamiltonian equations

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 double Poisson algebras, developed by Van den Bergh, which is used in the study of Hamiltonian ODEs. We apply our theory of double Poisson vertex algebras to non-commutative KP and Gelfand-Dickey hierarchies. We also construct the related non-commutative de Rham and variational complexes.

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arXiv2014

Integrability of Dirac reduced bi-Hamiltonian equations

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.

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arXiv2014

Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents

We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter number of g. In the case when g is sl_2 both these equations coincide with the KdV equation. In the case when g is not of type C_n, we associate to the minimal nilpotent element of g yet another generalized Drinfeld-Sokolov hierarchy.

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arXiv2014

Dirac reduction for Poisson vertex algebras

We construct an analogue of Dirac's reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac's reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld-Sokolov hierarchy.

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arXiv2014

Structure of classical (finite and affine) W-algebras

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 a simple Lie algebra g and its nilpotent element f. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine W-algebra W(g,f). As an immediate consequence, we obtain a Poisson algebra isomorphism between W^{fin}(g,f) and the Zhu algebra of W(g,f). We also study the generalized Miura map for classical W-algebras.

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arXiv2014

Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras

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, together with its generalizations and reduction to the N-th KdV hierarchy, using the formal distribution calculus and the lambda-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (non-local) bi-Poisson structures of the matrix KP and the matrix N-th KdV hierarchies, and we prove integrability of the N-th matrix KdV hierarchy.

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arXiv2013

Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras

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, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.

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