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 $\overline{\mathbf{B}}(C)\subset \mathcal{O}(G((z^{-1}))$ depending on the parameter $C\in G$ called classical universal Bethe subalgebras. To every antidominant cocharacter $\mu$ of the maximal torus $T\subset G$ one can associate the closed Poisson subspace ${\mathcal{W}}_{\mu }$ of $G((z^{-1}))$ (the Poisson algebra $\mathcal{O}({\mathcal{W}}_{\mu })$ is the classical limit of so-called shifted Yangian $Y_{\mu }(\mathfrak{g})$). We consider the images of $\overline{\mathbf{B}}(C)$ in $\mathcal{O}({\mathcal{W}}_{\mu })$, that we denote by ${\overline{B}}_{\mu }(C)$, that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular $C$ centralizing $\mu$, we compute the Poincaré series of these subalgebras. For $\mathfrak{g}={\mathfrak{gl}}_n$, we define the natural quantization ${\mathbf{Y}}^{\mathrm{rtt}}({\mathfrak{gl}}_n)$ of $\mathcal{O}({\mathrm{Mat}}_n((z^{-1}))))$ and universal Bethe subalgebras $\mathbf{B}(C)\subset {\mathbf{Y}}^{\mathrm{rtt}}({\mathfrak{gl}}_n)$. Using the RTT realization of $Y_{\mu }({\mathfrak{gl}}_n)$ (invented by Frassek, Pestun, and Tsymbaliuk), we obtain the natural surjections ${\mathbf{Y}}^{\mathrm{rtt}}({\mathfrak{gl}}_n)\twoheadrightarrow Y_{\mu }({\mathfrak{gl}}_n)$ which quantize the embedding ${\mathcal{W}}_{\mu }\subset {\mathrm{Mat}}_n((z^{-1}))$). Taking the images of $\mathbf{B}(C)$ in $Y_{\mu }({\mathfrak{gl}}_n)$ we recover Bethe subalgebras $B_{\mu }(C)\subset Y_{\mu }({\mathfrak{gl}}_n)$ proposed by Frassek, Pestun and Tsymbaliuk.

30 pages; the final published version