Related papers: On the Classical $W_{4}^{(2)}$ Algebra

In analogy with the Liouville case we study the $sl_3$ Toda theory on the lattice and define the relevant quadratic algebra and out of it we recover the discrete $W_3$ algebra. We define an integrable system with respect to the latter and…

We present a systematic construction of classical extended superconformal algebras from the hamiltonian reduction of a class of affine Lie superalgebras, which include an even subalgebra $sl(2)$. In particular, we obtain the doubly extended…

This is an introductory survey written in Japanese on classical integrability from the viewpoint of symmetry reduction of 4-dimensional anti-self-dual Yang-Mills equations. Twistor construction methods are discussed in detail in…

A method to construct both classical and quantum completely integrable systems from (Jordan-Lie) comodule algebras is introduced. Several integrable models based on a so(2,1) comodule algebra, two non-standard Schrodinger comodule algebras,…

By generalizing the Miura transformation for $\Ww_N$ to other classical $\Ww$ algebras obtained by hamiltonian reduction, we find realisations of these algebras in terms of relatively simple non-abelian current algebras, e.g.…

The original Miura transformation, considered as a nonlinear potential transformation, is applicable to a continual class of evolution equations, not only to discrete integrable equations and their hierarchies. The same continual class of…

Starting from known solutions of the functional Yang-Baxter equations, we exhibit Miura type of transformations leading to various known integrable quad equations. We then construct, from the same list of Yang-Baxter maps, a series of…

Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete…

It has been known for some time that $W$ algebras can be realised in terms of an energy-momentum tensor together with additional free scalar fields. Some recent results have shown that more general realisations are also possible. In this…

We study the quantum $N=2$ super-$W_{3}$ algebra using the free field realization, which is obtained from the supersymmetric Miura transformation associated with the Lie superalgebra $A(2|1)$. We compute the full operator product expansions…

We consider the algebraic setting of classical defects in discrete and continuous integrable theories. We derive the "equations of motion" on the defect point via the space-like and time-like description. We then exploit the structural…

We point out that the Miura transformation is related to a holomorphic foliation in a relative flag manifold over a Riemann Surface. Certain differential operators corresponding to a free field description of $W$--algebras are thus…

The Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions of the Miura transformation, which have been used to relate different kinds of integrable equations and to classify the…

Let R be a ring. A construction method for flexible quadratic algebras with scalar involution over R is presented which unifies various classical constructions in the literature, in particular those to construct composition algebras.

There is a constrained-WZNW--Toda theory for any simple Lie algebra equipped with an integral gradation. It is explained how the different approaches to these dynamical systems are related by gauge transformations. Combining Gauss…

A class of classical affine W-algebras are shown to be isomorphic as differential algebras to the coordinate rings of double coset spaces of certain prounipotent proalgebraic groups. As an application, integrable Hamiltonian hierarchies…

A framework analogous to path integrals in quantum physics is set up for abstract dynamical systems in a W*-algebraic setting. We consider spaces of evolutions, defined in a specific way, of a W*-algebra A as an analogue of spaces of…

A unified description of the relationship between the Hamiltonian structure of a large class of integrable hierarchies of equations and W-algebras is discussed. The main result is an explicit formula showing that the former can be…

We construct a quadratic basis of generators of matrix-extended $\mathcal{W}_{1+\infty}$ using a generalization of the Miura transformation. This makes it possible to conjecture a closed-form formula for the operator product expansions…

We construct the classical W-algebras for some non-abelian Toda systems associated with the Lie groups GL(2n,R) and Sp(n,R). We start with the set of characteristic integrals and find the Poisson brackets for the corresponding Hamiltonian…