@article {2014,
title = {Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras},
number = {arXiv:1401.2082},
year = {2014},
note = {45 pages},
publisher = {SISSA},
abstract = {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.},
url = {http://hdl.handle.net/1963/7242},
author = {Alberto De Sole and Victor G. Kac and Daniele Valeri}
}
@article {2013,
title = {Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents},
journal = {Communications in Mathematical Physics 331, nr. 2 (2014) 623-676},
number = {arXiv:1306.1684;},
year = {2014},
note = {46 pages},
publisher = {SISSA},
abstract = {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{\textquoteright}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.},
doi = {10.1007/s00220-014-2049-2},
url = {http://hdl.handle.net/1963/6979},
author = {Alberto De Sole and Victor G. Kac and Daniele Valeri}
}
@article {2013,
title = {Dirac reduction for Poisson vertex algebras},
journal = {Communications in Mathematical Physics 331, nr. 3 (2014) 1155-1190},
number = {arXiv:1306.6589;},
year = {2014},
note = {31 pages},
publisher = {SISSA},
abstract = {We construct an analogue of Dirac{\textquoteright}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{\textquoteright}s reduction of an arbitrary non-local Poisson
structure. We apply this construction to an example of a generalized
Drinfeld-Sokolov hierarchy.},
doi = {10.1007/s00220-014-2103-0},
url = {http://hdl.handle.net/1963/6980},
author = {Alberto De Sole and Victor G. Kac and Daniele Valeri}
}
@article {2014,
title = {Integrability of Dirac reduced bi-Hamiltonian equations},
number = {arXiv:1401.6006;},
year = {2014},
note = {15 pages},
institution = {SISSA},
abstract = {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{\textquoteright}s, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies.},
url = {http://hdl.handle.net/1963/7247},
author = {Alberto De Sole and Victor G. Kac and Daniele Valeri}
}
@article {2014,
title = {Structure of classical (finite and affine) W-algebras},
number = {arXiv:1404.0715;},
year = {2014},
note = {40 pages},
institution = {SISSA},
abstract = {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.},
url = {http://hdl.handle.net/1963/7314},
author = {Alberto De Sole and Victor G. Kac and Daniele Valeri}
}
@article {2012,
title = {Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras},
journal = {Communications in Mathematical Physics 323, nr. 2 (2013) 663-711},
number = {arXiv:1207.6286;},
year = {2013},
note = {43 pages. Second version with minor editing and corrections},
publisher = {Springer},
abstract = {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.},
doi = {10.1007/s00220-013-1785-z},
url = {http://hdl.handle.net/1963/6978},
author = {Alberto De Sole and Victor G. Kac and Daniele Valeri}
}