Bethe subalgebras in antidominantly shifted Yangians

Abstract

The loop group G((z −1 )) of a simple complex Lie group G has a natural Poisson structure. We introduce a natural family of Poisson commutative subalgebras B(C) ⊂ O(G((z −1 )) depending on the parameter C ∈ G called classical universal Bethe subalgebras. To every antidominant cocharacter µ of the maximal torus T ⊂ G one can associate the closed Poisson subspace Wµ of G((z −1 )) (the Poisson algebra O(Wµ) is the classical limit of so-called shifted Yangian Yµ(g) defined in [BFN, Appendix B]). We consider the images of B(C) in O(Wµ) that we denote by Bµ(C) that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular C centralizing µ, we compute the Poincaré series of these subalgebras. For g = gl n , we define the natural quantization Y(gl n ) rtt of O(gl n ((z −1 )))) and universal Bethe subalgebras B(C) ⊂ Y(gl n ) rtt . Following the ideas of Frassek, Pestun and Tsymbaliuk and using their RTT realization of Yµ(gl n ), we obtain the natural surjections Y(gl n ) rtt ։ Yµ(gl n ) which quantize the embedding Wµ ⊂ gl n ((z −1 ))). Taking the images of B(C) in Yµ(gl n ) we obtain Bethe subalgebras Bµ(C) ⊂ Yµ(gl n ).