TY - JOUR
T1 - Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
Y1 - 2014
A1 - Alberto De Sole
A1 - Victor G. Kac
A1 - Daniele Valeri
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/7242
N1 - 45 pages
ER -
TY - JOUR
T1 - Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents
JF - Communications in Mathematical Physics 331, nr. 2 (2014) 623-676
Y1 - 2014
A1 - Alberto De Sole
A1 - Victor G. Kac
A1 - Daniele Valeri
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/6979
N1 - 46 pages
U1 - 6967
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - JOUR
T1 - Dirac reduction for Poisson vertex algebras
JF - Communications in Mathematical Physics 331, nr. 3 (2014) 1155-1190
Y1 - 2014
A1 - Alberto De Sole
A1 - Victor G. Kac
A1 - Daniele Valeri
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/6980
N1 - 31 pages
U1 - 6968
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - RPRT
T1 - Integrability of Dirac reduced bi-Hamiltonian equations
Y1 - 2014
A1 - Alberto De Sole
A1 - Victor G. Kac
A1 - Daniele Valeri
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/7247
N1 - 15 pages
U1 - 7286
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - RPRT
T1 - Structure of classical (finite and affine) W-algebras
Y1 - 2014
A1 - Alberto De Sole
A1 - Victor G. Kac
A1 - Daniele Valeri
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/7314
N1 - 40 pages
U1 - 7359
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - JOUR
T1 - Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras
JF - Communications in Mathematical Physics 323, nr. 2 (2013) 663-711
Y1 - 2013
A1 - Alberto De Sole
A1 - Victor G. Kac
A1 - Daniele Valeri
AB - 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.
PB - Springer
UR - http://hdl.handle.net/1963/6978
N1 - 43 pages. Second version with minor editing and corrections
U1 - 6966
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -