Poisson-Nijenhuis Structures and Sato Hierarchy

Abstract

We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie-Poisson brackets, which are pairwise compatible and define a multi-Hamiltonian structure; to this structure one can associate a recursion operator and a Kac-Moody algebra of Hamiltonian vector fields. If the initial Lie algebra is taken to be an associative algebra of differential operators, a suitable family of Hamiltonian vector fields reproduce either the n-th Gel'fand-Dikii hierarchy (for n finite) or Sato's hierarchy (for n = ∞). Within the same framework, it is also possible to recover a class of integro-differential hierarchies involving a finite number of fields, which generalize the Gel'fand-Dikii equations and are equivalent to Sato's hierarchy.

New integrable hierarchies from vertex operator representations of polynomial Lie algebras

“…The aim of this first section is to construct in a few different ways a class of finite dimensional Lie algebras (called in what follows polynomial Lie algebras to keep the name usually used in the literature, see for example [17] and [21]) which are going to play a crucial role in the whole paper. The reader will surely recognise that actually our constructions can be straightforward generalised to the realm of the infinite dimensional Lie algebras; but, although we shall need for our porpoises this generalisation, let us here for sake of concreteness restrict ourself to the maybe simpler finite dimensional case.…”

“…We should point out here that the Lie algebras g (n) , which is obviously equivalent to g (n) (λ), was already defined and considered in [17] (in particular definition 2.4 coincides with the definition of the polynomial Lie algebras with bracket [·, ·] 0 (in their notation) contained in their proposition 4.4). From this definition one would natural define on g (n) the symmetric bilinear form…”

“…The paper is organised as follows: in the second section we shall describe a class of finite dimensional Lie algebras known in the literature as polynomial Lie algebras [17], [21] which, roughly speaking, can be regarded as direct sum of semisimple Lie algebras endowed with a non canonical Lie bracket. We shall show that these Lie algebras can be constructed in completely different ways: namely as particular finite dimensional quotients of an infinite dimensional algebra and as a Wigner contraction of a direct sum of finite dimensional semisimple Lie algebras, or finally as tensor product between a finite dimensional Lie algebra g and a nilpotent commutative ring.…”

The method of Poisson pairs in the theory of nonlinear PDEs

“…The motivations can be found for instance in [20] (see also [25]). Later on we shall see how to extend this definition to the case of an arbitrary number of copies.…”

Hamiltonian Structure of PI Hierarchy

“…The Poisson brackets of the matrix elements of the L-matrix take the form of 'generalized linear brackets' [20]. (Actually, this system has a multi-Hamiltonian structure [21,22], but this is beyond the scope of this paper.) The Lax equations can be thereby expressed in the Hamiltonian form ∂ t V (λ) = {V (λ), H}.…”