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arXiv · 1703.04147

Degeneration of Bethe subalgebras in the Yangian of $\mathfrak{gl}_n$

Aleksei Ilin·Leonid Rybnikov
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Abstract

We study degenerations of Bethe subalgebras $B(C)$ in the Yangian $Y(\mathfrak{gl}_n)$, where $C$ is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parametrizes all possible degenerations, is the Deligne-Mumford moduli space of stable rational curves $\overline{M_{0,n+2}}$. All subalgebras corresponding to the points of $\overline{M_{0,n+2}}$ 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 space $\overline{M_{0,n+2}}$ 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.

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Aleksei Ilin, Leonid Rybnikov. 2017-12-05. Degeneration of Bethe subalgebras in the Yangian of $\mathfrak{gl}_n$. https://doi.org/10.1007/s11005-017-1031-2

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