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Classical W-algebras within the theory of Poisson vertex algebras
No full textWe review the Poisson vertex algebra theory approach to classical W-algebras. First, we provide a description of the Drinfeld-Sokolov Hamiltonian reduction for the construction of classical W-algebras within the framework of Poisson vertex algebras and we 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. Then we provide a Poisson vertex algebra analogue of the Gelfand-Dickey construction of classical W-algebras and we show the relations with the Drinfeld-Sokolov Hamiltonian reduction. It will be also shown that classical W-algebras are the Poisson vertex algebras which are of interest from the conformal field theory point of view
Bethe Ansatz and the spectral theory of affine Lie algebra–valued connections II: the non simply–laced case
Full text linkWe assess the ODE/IM correspondence for the quantum g-KdV model, for a non-simply laced Lie algebra g. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra 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 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
Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation
Full text linkWe study the one-dimensional periodic derivative nonlinear Schrödinger equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion ∫hk , k∈Z+. In each ∫h2k the term with the highest regularity involves the Sobolev norm H˙k(T) of the solution of the DNLS equation. We show that a functional measure on L2(T) , absolutely continuous w.r.t. the Gaussian measure with covariance (I+(−Δ)k)−1, is associated to each integral of motion ∫h2k , k≥1
Invariant measures for the periodic derivative nonlinear Schrödinger equation
Full text linkWe construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation (DNLS) for small data in L2 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 the Gaussian measure on L2 with covariance (I+(−Δ)k)−1 for any k⩾2
MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine -algebras
Full text linkWe give an introduction to the Mathematica packages MasterPVA and MasterPVAmulti used to compute λ-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 λ-brackets among the generators of classical affine W-algebras. The use of these packages is shown by providing some explicit examples
Flat coordinates of algebraic Frobenius manifolds in small dimensions
Full text linkOrbit spaces of the reflection representation of finite irreducible Coxeter groups provide
polynomial Frobenius manifolds. Flat coordinates of the Frobenius manifold 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 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
Classical affine W -Algebras for glN and associated integrable Hamiltonian hierarchies
Full text linkWe 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(,) 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
Structure of classical (finite and affine) W-algebras
Full text linkFirst, we derive an explicit formula for the Poisson bracket of the classical finite W- Algebra Wfin(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 Wfin(g, f) and the Zhu algebra of W(g, f).We also study the generalized Miura map for classicalW- Algebras
A new scheme of integrability for (bi)Hamiltonian PDE
Full text linkWe 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
Adler–Oevel-Ragnisco type operators and Poisson vertex algebras
Full text linkThe theory of triples of Poisson brackets and related integrable systems, based on a classical R-matrix R ∈ EndF (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 λ-brackets. We introduce the corresponding Adler type identities and apply them to integrability of hierarchies of Hamiltonian PDEs
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