Representation Theory and Mathematical Physics seminar, UC Berkeley

Leonid Rybnikov, HSE University

Bethe subalgebras in Yangians and the wonderful compactification

Organizer's time: 2020-06-05 09:30 America/Los_Angeles

Duration: 1 hour

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Bethe subalgebras in the Yangian $Y(\mathfrak{g})$ are maximal commutative subalgebras depending on a parameter from the corresponding adjoint Lie group $G$. For $\mathfrak{g}=\mathfrak{sl}_2$, these subalgebras are generated by the integrals of the XXX Heisenberg magnet chain. We extend the family of Bethe subalgebras to the De Concini--Procesi wonderful compactification $\overline{G}\supset G$ and describe the subalgebras corresponding to generic points of any stratum in $\overline{G}$ as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in $\mathfrak{g}$. We expect that monodromy of solutions of Bethe ansatz equations along some particular paths in this extended parameter space gives rise to a group of piecewise linear transformations generalizing the cactus group action on Kashiwara crystals (due to Henriques and Kamnitzer).

Joint work with Aleksei Ilin, https://arxiv.org/abs/1810.07308

Submitted by: Alexander Shapiro