Exotic cluster structures on SL_n (Attention : début 13:30)

par Alek Vainshtein (Haifa University)

jeudi 5 oct. 2017, 13:30 → 14:30 Europe/Paris

salle 112 (bât. Braconnier)

Back in 2005, Berenstein, Fomin and Zelevinsky discovered a cluster structure in the ring of regular functions on a double Bruhat cell in a semisimple Lie group, in particular, SL_n. This structure can be easily extended to the whole group. The compatible Poisson bracket is given by the standard r-matrix Poisson-Lie structure on SL_n. The latter is a particular case of Poisson-Lie structures corresponding to quasi-triangular Lie bialgebras. Such structures where classified in 1982 by Belavin and Drinfeld. In 2012, we have conjectured that each Poisson-Lie structure on SL_n gives rise to a cluster structure, and gave several examples of exotic cluster structures corresponding to Poisson-Lie structures distinct from the standard one. In my talk I will tell about the progress in the proof of this conjecture and its modifications. Joint with M.Gekhtman and M.Shapiro.