We report the explicit form of the quantumR matrix in the fundamental representation for the generalized Toda system associated with non-exceptional affine Lie algebras.

/pdf/quantum-r-matrix-for-the-generalized-toda-system-3sxeqky330.pdf

Quantum $R$ matrix for the generalized Toda system

On explique les proprietes de Poisson des transformations d'habillage en theorie des solitons

/pdf/dressing-transformations-and-poisson-group-actions-1ihxpbuffk.pdf

Dressing transformations and Poisson group actions

On montre comment on peut obtenir des groupes quantiques multiparametres a partir des algebres de Hopf torsadees

Multiparameter quantum groups and twisted quasitriangular Hopf algebras

We study the deformation, defined by a Nijenhuis operator, and the dualization, defined by a Poisson bivector, of the Lie bracket of vector fields on a manifold and, more generally, of the Lie bracket on a differential Lie algebra over a commutative algebra. Requiring that the two processes commute, one obtains hierarchies of pairwise compatible Lie brackets on the module and on its dual. Each differential Lie algebra-structure on a module gives rise to a cohomology operator on the algebra of forms over the module, as well as to a graded Lie algebra-structure on the algebra of multivectors (the Schouten algebra). We study the deformation and the dualization of the derivations of the algebra of forms and of the Schouten bracket of multivectors, thus obtaining generalizations of the preceding differential geometric results together with new proofs. Section 2 comprises the study of the Nijenhuis operators on the twilled Lie algebras, an «N-matrix version» of the Kostant-Symes theorem, and an application to Hamiltonian systems of Toda type on semisimple Lie algebras Nous etudions la deformation, definie par un tenseur de Nijenhuis, et la dualisation, definie par un bivecteur de Poisson, du crochet de Lie des champs de vecteurs sur une variete et, plus generalement, du crochet de Lie sur une algebre de Lie differentielle sur une algebre commutative

/pdf/poisson-nijenhuis-structures-1x644yh94e.pdf

Poisson-Nijenhuis structures

We describe Hopf algebras which are central extensions of quantum current groups. For a special value of the central charge, we describe Casimir elements in these algebras. New types of generators for quantum current algebra and its central extension for quantum simple Lie groups, are obtained. The application of our construction to the elliptic case is also discussed.

M. A. Semenov-Tyan-Shanskii

Papers

What is a classical r-matrix?

What is a Classical R-Matrix