arXiv subjects

Daniele Valeri

Publications and source records attributed to Daniele Valeri.

At least 19 recordsLinked to original sources

arXiv2025

Cohomology of Lie conformal algebroids

We study Lie conformal algebroids (LCAd) and their representations using the language of lambda-brackets and Lie conformal algebras. We describe several general constructions, such as the LCAd of conformal derivations CDer(A) of a differential algebra A, the gauge LCAd G(A,M) associated to a differential algebra A and its module M, the current LCAd F^ of a Lie algebroid F, and the LCAd structure of the space Omega(V) of Kahler differentials over a Poisson vertex algebra (PVA) V. We develop the cohomology theories of LCAd and we relate them to the corresponding cohomology theories of PVA. In particular, we find an isomorphism between the cohomology of a PVA V with coefficients in a module M and the corresponding cohomology of the LCAd Omega(V) with coefficients in the same module.

math.RT↗

arXiv2025

Double Poisson (vertex) algebra cohomology

A noncommutative (NC) version of Poisson geometry was initiated by Van den Bergh by introducing at the level of associative algebras the formalism of double Poisson brackets. Their key property is to induce (standard) Poisson brackets under each representation functor. Then, Pichereau and Van de Weyer developed and studied the corresponding cohomology theory under the assumption that there exists a NC bivector defining the double Poisson bracket. Our first main result is that one can remove this assumption by constructing a completed double Poisson cohomology valid in any situation, hence generalizing the approach of Pichereau-Van de Weyer. As an application, we show that the double Poisson cohomology complex associated to the path algebra of a quiver is acyclic. Furthermore, we show that this new double Poisson cohomology theory can be adapted to weaker forms of double Poisson brackets (called quasi-Poisson and gauged Poisson), and that it is compatible with representation functors. A second focus of this memoir concerns the formalism of double Poisson vertex algebras. These were introduced by De Sole, Kac and the second author, as NC versions of Poisson vertex algebras, which induce the latter structures under each representation functor. Our second main result is the development of cohomology theories for double Poisson vertex algebras. These are NC analogues of the basic, reduced and variational Poisson vertex algebra cohomologies. More importantly, we prove that under each representation functor these cohomology theories are compatible with their commutative counterparts. As an application, we compute the double Poisson vertex algebra cohomology of the generalized NC de Rham complex and of the generalized NC variational complex. Finally, we describe the relation between the double Poisson algebra and double Poisson vertex algebra cohomologies using jet and quotient functors.

math.RT↗

arXiv2024

Multi-component Hamiltonian difference operators

In this paper we study local Hamiltonian operators for multi-component evolutionary differential-difference equations. We address two main problems: the first one is the classification of low order operators for the two-component case. On the one hand, this extends the previously known results in the scalar case; on the other hand, our results include the degenerate cases, going beyond the foundational investigation conducted by Dubrovin. The second problem is the study and the computation of the Poisson cohomology for a two-component (-1,1)-order Hamiltonian operator with degenerate leading term appearing in many integrable differential-difference systems, notably the Toda lattice. The study of its Poisson cohomology sheds light on its deformation theory and the structure of the bi-Hamiltonian pairs where it is included in, as we demonstrate in a series of examples.

math-ph↗

arXiv2024

Remarks on intersection numbers and integrable hierarchies. II. Tau-structure

For systems of evolutionary partial differential equations the tau-structure is an important notion which originated from the deep relation between integrable systems and quantum field theories. We show that, under a certain non-degeneracy condition, existence of a tau-structure implies integrability. As an example, we apply this principle to provide a new proof of the integrability of the Drinfeld--Sokolov hierarchy associated to an arbitrary Kac--Moody algebra and a choice of a vertex of its Dynkin diagram.

math-ph↗

arXiv2023

Flat coordinates of algebraic Frobenius manifolds in small dimensions

Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide polynomial Frobenius manifolds. Flat coordinates of the Frobenius metric $η$ are Saito polynomials which are distinguished basic invariants of the Coxeter group. Algebraic Frobenius manifolds are typically related to quasi-Coxeter conjugacy classes in finite Coxeter groups. We find explicit relations between flat coordinates of the Frobenius metric $η$ and flat coordinates of the intersection form $g$ for most known examples of algebraic Frobenius manifolds up to dimension 4. In all the cases, flat coordinates of the metric $η$ appear to be algebraic functions on the orbit space of the Coxeter group.

math.DG↗

arXiv2023

Poisson vertex algebras and Hamiltonian PDE

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 W-algebras and apply it to construct a large class of integrable Hamiltonian PDE. We also discuss non commutative Hamiltonian PDE in the framework of double PVA, and differential-difference Hamiltonian equations in the framework of multiplicative PVA.

math-ph↗

arXiv2022

Affine Kac-Moody Algebras and Tau-Functions for the Drinfeld-Sokolov Hierarchies: the Matrix-Resolvent Method

For each affine Kac-Moody algebra $X_n^{(r)}$ of rank $\ell$, $r=1,2$, or $3$, and for every choice of a vertex $c_m$, $m=0,\dots,\ell$, of the corresponding Dynkin diagram, by using the matrix-resolvent method we define a gauge-invariant tau-structure for the associated Drinfeld-Sokolov hierarchy and give explicit formulas for generating series of logarithmic derivatives of the tau-function in terms of matrix resolvents, extending the results of [Mosc. Math. J. 21 (2021), 233-270, arXiv:1610.07534] with $r=1$ and $m=0$. For the case $r=1$ and $m=0$, we verify that the above-defined tau-structure agrees with the axioms of Hamiltonian tau-symmetry in the sense of [Adv. Math. 293 (2016), 382-435, arXiv:1409.4616] and [arXiv:math.DG/0108160].

math-ph↗

arXiv2022

Double Multiplicative Poisson Vertex Algebras

We develop the theory of double multiplicative Poisson vertex algebras. These structures, defined at the level of associative algebras, are shown to be such that they induce a classical structure of multiplicative Poisson vertex algebra on the corresponding representation spaces. Moreover, we prove that they are in one-to-one correspondence with local lattice double Poisson algebras, a new important class among Van den Bergh's double Poisson algebras. We derive several classification results, and we exhibit their relation to non-abelian integrable differential-difference equations. A rigorous definition of double multiplicative Poisson vertex algebras in the non-local and rational cases is also provided.

math.RT↗

arXiv2022

Adler-Oevel-Ragnisco type operators and Poisson vertex algebras

The theory of triples of Poisson brackets and related integrable systems, based on a classical R-matrix R in End_F(g), where g is a finite dimensional associative algebra over a field F viewed as a Lie algebra, was developed by Oevel-Ragnisco and Li-Parmentier [OR89,LP89]. In the present paper we develop an "affine" analogue of this theory by introducing the notion of a continuous Poisson vertex algebra and constructing triples of Poisson lambda-brackets. We introduce the corresponding Adler type identities and apply them to integrability of hierarchies of Hamiltonian PDEs.

nlin.SI↗

arXiv2022

Weakly nonlocal Poisson brackets: tools, examples, computations

We implement an algorithm for the computation of Schouten bracket of weakly nonlocal Hamiltonian operators in three different computer algebra systems: Maple, Reduce and Mathematica. This class of Hamiltonian operators encompass almost all the examples coming from the theory of (1+1)-integrable evolutionary PDEs

math-ph↗

arXiv2021

On Lax operators

We define a Lax operator as a monic pseudodifferential operator $L(\partial)$ of order $N\geq 1$, such that the Lax equations $\dfrac{\partial L(\partial)}{\partial t_k}=[(L^{\frac kN}(\partial))_+,L(\partial)]$ are consistent and non-zero for infinitely many positive integers $k$. Consistency of an equation means that its flow is defined by an evolutionary vector field. In the present paper we demonstrate that the traditional theory of the KP and the $N$-th KdV hierarchies holds for arbitrary scalar Lax operators.

nlin.SI↗

arXiv2021

Integrability of classical affine W-algebras

We prove that all classical affine W-algebras W(g,f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G_2, one in F_4, and five in E_8.

math-ph↗

arXiv2021

Integrable triples in semisimple Lie algebras

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple $(f,0,e)$ in $\mathfrak{sl}_2$ corresponds to the KdV hierarchy, and the triple $(f,0,e_θ)$, where $f$ is the sum of negative simple root vectors and $e_θ$ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld-Sokolov hierarchy.

nlin.SI↗

arXiv2020

p-reduced multicomponent KP hierarchy and classical W-algebras W(gl_N,p)

For each partition p of an integer N \geq 2, consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the p-reduced r-component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction of the generators of the corresponding classical W-algebra W(gl_N,p), and write down explicit formulas for evolution of these generators along the Hamiltonian flows.

math-ph↗

arXiv2020

$W$-algebras via Lax type operators

$W$-algebras are certain algebraic structures associated to a finite dimensional Lie algebra $\mathfrak g$ and a nilpotent element $f$ via Hamiltonian reduction. In this note we give a review of a recent approach to the study of (classical affine and quantum finite) $W$-algebras based on the notion of Lax type operators. For a finite dimensional representation of $\mathfrak g$ a Lax type operator for $W$-algebras is constructed using the theory of generalized quasideterminants. This operator carries several pieces of information about the structure and properties of the $W$-algebras and shows the deep connection of the theory of $W$-algebras with Yangians and integrable Hamiltonian hierarchies of Lax type equations.

math-ph↗

arXiv2018

A Lax type operator for quantum finite W-algebras

For a reductive Lie algebra g, its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra W(g,f). We show that for the classical linear Lie algebras gl_N, sl_N, so_N and sp_N, the operator L(z) satisfies a generalized Yangian identity. The operator L(z) is a quantum finite analogue of the operator of generalized Adler type which we recently introduced in the classical affine setup. As in the latter case, L(z) is obtained as a generalized quasideterminant.

math.RT↗

arXiv2018

Invariant measures for the periodic derivative nonlinear Schrödinger equation

We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation (DNLS) for small data in $L^2$ and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of Gaussian measures on $L^2$.

math-ph↗

arXiv2018

A new scheme of integrability for (bi)Hamiltonian PDE

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 classical Hamiltonian reduction technique of Drinfeld and Sokolov. The method is based on the notion of an Adler type matrix pseudodifferential operator and the notion of a generalized quasideterminant. We also introduce the notion of a dispersionless Adler type series, which is applied to the study of dispersionless Hamiltonian equations. Non-commutative Hamiltonian equations are discussed in this framework as well.

math-ph↗