Bethe subalgebras in antidominantly shifted Yangians
Vasily Krylov, Leonid Rybnikov
TL;DR
This work constructs and analyzes Bethe subalgebras in both classical and quantum Yangians, focusing on antidominant shifts. It defines a universal classical family $\overline{\mathbf{B}}(C)$ in ${\mathcal{O}}(G((z^{-1})))$ and its images in shifted settings ${\mathcal{O}}({\mathcal{W}}_{\mu})$, proving Poisson commutativity and explicit graded isomorphisms to Levi components for regular parameters. In type A, it develops a corresponding RTT quantization ${\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n)$ with universal subalgebras ${\mathbf{B}}(C)$ and shows the graded image recovers the classical ${\overline{\mathbf{B}}}(C)$, thereby connecting classical and quantum Bethe algebras. The paper also introduces shifted Yangians $Y_{\mu}(\mathfrak{gl}_n)$ (standard and RTT realizations), proves PBW-type results, and proves that Bethe subalgebras $B_{\mu}(C)$ are commutative with graded sizes controlled by Levi subgroups, matching the corresponding classical Bethe subalgebras $B_{L}(C)$. This furnishes a unified framework for Bethe subalgebras across classical and quantum, as well as their geometric incarnations via generalized transversal slices and truncated 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 $μ$ of the maximal torus $T \subset G$ one can associate the closed Poisson subspace $\mathcal{W}_μ$ of $G((z^{-1}))$ (the Poisson algebra $\mathcal{O}(\mathcal{W}_μ)$ is the classical limit of so-called shifted Yangian $Y_μ(\mathfrak{g})$). We consider the images of $\overline{\mathbf{B}}(C)$ in $\mathcal{O}(\mathcal{W}_μ)$, that we denote by $\overline{B}_μ(C)$, 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 $\mathfrak{g}=\mathfrak{gl}_n$, we define the natural quantization ${\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n)$ of $\mathcal{O}(\operatorname{Mat}_n((z^{-1}))))$ and universal Bethe subalgebras ${\mathbf{B}}(C) \subset {\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n)$. Using the RTT realization of $Y_μ(\mathfrak{gl}_n)$ (invented by Frassek, Pestun, and Tsymbaliuk), we obtain the natural surjections ${\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n) \twoheadrightarrow Y_μ(\mathfrak{gl}_n)$ which quantize the embedding $\mathcal{W}_μ\subset \operatorname{Mat}_n((z^{-1}))$). Taking the images of ${\mathbf{B}}(C)$ in $Y_μ(\mathfrak{gl}_n)$ we recover Bethe subalgebras $B_μ(C) \subset Y_μ(\mathfrak{gl}_n)$ proposed by Frassek, Pestun and Tsymbaliuk.