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Séminaire Algèbre ICJ

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

by Alek Vainshtein (Haifa University)

jeudi 5 octobre 2017 de au (Europe/Paris)
at bât. Braconnier ( salle 112 )
ICJ, UCBL - La Doua
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.