On classical limits of Bethe subalgebras in Yangians

Abstract

The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Yangian. Bethe subalgebras form a natural family of commutative subalgebras depending on a group element $C$ of the adjoint group $G$. The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain. We describe the associated graded of Bethe subalgebras as subalgebras in $U(\mathfrak{g}[t])$ and in $\mathcal{O}(G_1[[t^{-1}]])$ for all semisimple $C\in G$. We show that associated graded in $U(\mathfrak{g}[t])$ of the Bethe subalgebra assigned to the identity of $G$ is the universal Gaudin subalgebra of $U(\mathfrak{g}[t])$ obtained from the center of the corresponding affine Kac-Moody algebra at the critical level. This generalizes Talalaev's formula for generators of the universal Gaudin subalgebra to $\mathfrak{g}$ of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin-Frenkel center at the critical level. Using our general result on associated graded of Bethe subalgebras, we compute some limits of Bethe subalgebras corresponding to regular semisimple $C\in G$ as $C$ goes to an irregular semisimple group element $C_0$. We show that this limit is the product of the smaller Bethe subalgebra and a quantum shift of argument subalgebra in the universal enveloping algebra of the centralizer of $C_0$ in $\mathfrak{g}$. This generalizes the Nazarov-Olshansky solution of Vinberg's problem on quantization of shift of argument subalgebras.