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.