Séminaires : Séminaire d'Algèbre

The Bethe subalgebras of the Yangian Y(gl(n)) are a family of maximal commutative subalgebras that generalize the Gelfand-Tsetlin algebras. Moreover, they are indexed by points of the Deligne-Mumford compactification of the moduli space M(0,n+2). We consider points C in the real locus of this parameter space. For a fixed tame representation of Y(gl(n)), the Bethe subalgebra B(C) corresponding to any real point C acts with simple spectrum, resulting in an unramified covering of the parameter space whose fiber over C is the set of eigenlines for the action of B(C). I will discuss how to identify each fiber with a collection of Gelfand-Tsetlin keystone patterns, carrying a gl(n)-crystal structure, as well as the monodromy action for the covering realized by the mirabolic cactus group. Large parts of this construction are expected to generalize to arbitrary semisimple Lie algebras. This is joint work with Anfisa Gurenkova and Leonid Rybnikov.

This talk will take place in hybrid mode at the Institut Henri Poincaré.