## Séminaire de Géométrie et Quantification |

**Pavel
Saponov **

(Inst. for High Energy Physic and Higher School of Economics, Moscou)

Lundi
6 février à
15h30, IHP salle **01**,

In my
talk, I shall consider a q-deformation of the so-called Yangian
Y(gl(m)), associated
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.

.