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 $\mathcal{W}$-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 algebras - and the $\mathcal{W}$-algebras associated to the Casimir invariants of a Lie algebra is recovered. After carefully discussing the relations between the embeddings of $A_1=sl(2,\mathbb{C})$ into a simple Lie algebra $g$ and the elements of the Heisenberg subalgebras of $g^{(1)}$, we identify the class of $\mathcal{W}$-algebras that can be defined in this way. For $A_n$, this class only includes those associated to the embeddings labelled by partitions of the form $n+1=k(m)+q(1)$ and $n+1=k(m+1)+k(m)+q(1)$.

Improved version with more specific results concerning the class of W-algebras which are defined in terms of soliton equations. 48 pages. Plain TeX

References in corpus (3)

• Aspects of Non-Abelian Toda Theories
• Finite and Infinite W Algebras and their Applications
• Generalized Drinfeld-Sokolov Hierarchies and W-Algebras

Cited by in corpus (14)

• Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras
• Classical W-algebras for gl_N and associated integrable Hamiltonian hierarchies
• Constrained KP Models as Integrable Matrix Hierarchies
• Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction
• Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type
• Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras
• A Note on Fractional KdV Hierarchies
• Dirac operators and the Very Strange Formula for Lie superalgebras
• On classical finite and affine W-algebras
• KdV type systems and W-algebras in the Drinfeld-Sokolov approach
• Solutions to WDVV from generalized Drinfeld-Sokolov hierarchies
• MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $\mathcal{W}$-algebras
• Dressing Technique for Intermediate Hierarchies
• A geometric construction of integrable Hamiltonian hierarchies associated with the classical affine W-algebras