2,129 research outputs found
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.Comment: 47 pages. In version 2 we fixed the proof of Corollary 4.15 (which is
now Theorem 4.14), and we added some reference
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.Comment: 31 pages. Corrected some typos and added missing equations in Section
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.Comment: 48 pages. Minor revisions and a correction to formulas (7.25) and
(7.48
Finite W-algebras for glN
We study the quantum finite W -algebras W (glN, f ), associ-ted to the Lie algebra glN, and its arbitrary nilpotent element f . We construct for such an algebra an r1× r1 matrix L(z) of Yangian type, where r1 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 (glN, 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 element
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.Comment: 46 pages. Corrected an error in Section 6.2 which has led to
additional equations in the case of g=sl_n and its minimal nilpotent element
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