W-Algebras from Soliton Equations and Heisenberg Subalgebras

Abstract

We derive sufficient conditions under which the "second" Hamiltonian structure of a class of generalized KdV-hierarchies defines one of the classical W-algebras obtained through Drinfel'd-Sokolov Hamiltonian reduction. These integrable hierarchies are associated to the Heisenberg subalgebras of an untwisted affine Kac-Moody algebra. When the principal Heisenberg subalgebra is chosen, the well known connection between the Hamiltonian structure of the generalized Drinfel'd-Sokolov hierarchies -the Gel'fand-Dickey algebrasand the W-algebras associated to the Casimir invariants of a Lie algebra is recovered. After carefully discussing the relations between the embeddings of A 1 = sl(2, C) into a simple Lie algebra g and the elements of the Heisenberg subalgebras of g(1) , we identify the class of W-algebras that can be defined in this way. For A n , this class only includes those 1 pousa@gaes.usc.es 2 gallas@gaes.usc.es 3 miramontes@gaes.usc.es 4 joaquin@gaes.usc.es associated to the embeddings labelled by partitions of the form n + 1 = k (m) + q (1) and n + 1 = k (m + 1) + k (m) + q (1). August 1994 1.Introduction.

A note on fractional KdV hierarchies

“…As a side remark, we note that the MR reduction process yields a more direct construction of the W (2) 3 algebra, since it allows to dispose from the beginning of the Casimir function φ appearing in [4] (see also [14]). To define the hierarchy it is most useful to consider the associated linear problem [11,5] − |ψ > x +(S + λA)|ψ >= k|ψ > .…”

𝒲-gauge structures and their anomalies: An algebraic approach

“…For Σ(z,z) transforming as 9) in some representation if the Lie group, the covariant derivative is DΣ = dΣ + AΣ, (2.10) ‡ in the general case, the SL(2) decomposition makes it necessary to use this triple index notation, which we hope not to be too confusing, in particular for the structure constants…”

“…In general [4], [5], for a given Lie algebra G there are several possibilities to identify such a SL(2) subalgebra [6], [7], [8], [9]. Different embeddings lead to different W-structures.…”

Classical $${\mathcal{W}}$$ W -Algebras and Generalized Drinfeld-Sokolov Bi-Hamiltonian Systems Within the Theory of Poisson Vertex Algebras

“…In [dGHM92,BdGHM93,FGMS95,FGMS96] they constructed the corresponding generalized KdV hierarchies, starting with a Heisenberg subalgebra H ⊂ g((z −1 )). In this approach, they cover all classical W-algebras associated to nilpotent elements f ∈ g, for which there exists a graded semisimple element of the form f + zs ∈ H (the existence of such a graded semisimple element is also studied, in the regular, or "type I", case, in [FHM92,DF95], using results in [KP85], and, for g of type A n , in [FGMS95,FGMS96]). …”

“…The main result here is Theorem 4.18, where we construct an integrable hierarchy of bi-Hamiltonian equations associated to a nilpotent element f ∈ g, a homogeneous element s ∈ g such that [s, n] = 0 and f +zs ∈ g((z −1 )) is semisimple, and an element a(z) ∈ Z(Ker ad(f + zs))\Z(g((z −1 )). In Section 4.10 we discuss, in the case of gl n , for which nilpotent elements f Theorem 4.18 can be applied, obtaining the same restrictions as in [FGMS95]. third author were visiting the Mathematics Department.…”