Degeneration of Bethe subalgebras in the Yangian of $gl_n$
We study degenerations of Bethe subalgebras B(C) in the Yangian Y(gln), where C is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parameterizes all possible degenerations, is the Deligne–Mumford moduli space of stable rational curves M0,n+2¯. All subalgebras corresponding to the points of M0,n+2¯are free and maximal commutative. We describe explicitly the “simplest” degenerations and show that every degeneration is the composition of the simplest ones. The Deligne–Mumford spaceM0,n+2¯ generalizes to other root systems as some De Concini–Procesi resolution of some toric variety. We state a conjecture generalizing our results to Bethe subalgebras in the Yangian of arbitrary simple Lie algebra in terms of this De Concini–Procesi resolution.