Séminaire de Géométrie et Quantification
(Inst. for High Energy Physic and Higher School of Economics, Moscou)
Lundi 6 février à 15h30, IHP salle 01,
talk, I shall consider a q-deformation of the so-called Yangian
with the Yang R-matrix, which was introduced by V. Drinfeld, and
possesses many interesting properties including applications to the
theory of integrable systems in mathematical physics (for example,
the non-linear Schrödinger equation), W-algebras, etc.
Its q-analog, called the q-Yangian, is usually defined as one "half" of a quantum affine group. D. Gurevich and I suggest a new construction for such a q-analog of the Yangian Y(gl(m)), which we call "braided Yangian". We associate braided Yangians with rational and trigonometric quantum R-matrices, depending on a formal parameter. These R-matrices arise from constant involutive or Hecke R-matrices by means of the Baxterization procedure. Our braided Yangians admit an evaluation morphism onto quantum matrix algebras and this property entails that they possess a rich representation theory which we construct. In my talk I also plan to define q-analogs of the symmetric polynomials (full, elementary and power sums) which form a commutative subalgebra in the braided Yangian and to exhibit some noncommutative matrix identities similar to the Newton-Cayley-Hamilton identities in classical matrix analysis.