2022/12/22 by Krylov, Vasily, Mashanova-Golikova, Inna, Rybnikov, Leonid · 1 citation
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2212.11995
Let \mathfrakg be a complex simple finite dimensional Lie algebra and G be the adjoint Lie group with the Lie algebra \mathfrakg. To every C ∈ G one can associate a commutative subalgebra B(C) in the Yangian Y(\mathfrakg), which is responsible for the integrals of the (generalized) XXX Heisenberg magnet chain. Using the approach of arXiv:1708.05105, we construct a natural structure of affine crystals on spectra of B(C) in Kirillov-Reshetikhin Y(\mathfrakg)-modules in type A. We conjecture that such a construction exists for arbitrary \mathfrakg and gives Kirillov-Reshetikhin crystals. Our main technical tool is the degeneration of Bethe subalgebras in the Yangian to commutative subalgebras Aχu in the universal enveloping of the current Lie algebra, U(\mathfrakg[t]), which depend on the parameter χ from the Lie algebra \mathfrakg (and are of independent interest). We show that these subalgebras come from the Feigin-Frenkel center on the critical level as described by Feigin, Frenkel and Toledano Laredo in arXiv:math/0612798. This allows to prove that our affine crystals in type A are indeed Kirillov-Reshetikhin by reducing to the crystal structure on the spectra of inhomogeneous Gaudin model which is already known (arXiv:1708.05105).