Bethe subalgebras in antidominantly shifted Yangians

TL;DR

This paper introduces classical and quantum Bethe subalgebras within shifted Yangians and their Poisson structures, providing explicit computations and realizations for the general linear Lie algebra case.

Contribution

It constructs classical universal Bethe subalgebras in shifted Yangians, computes their properties, and establishes their quantizations and realizations for fgl_n, connecting to existing Bethe subalgebra proposals.

Findings

Defined classical Bethe subalgebras fgl_n

Computed Poincare9 series for regular centralizing elements

Established quantizations and embeddings via RTT realization

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 Poincar\'e 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 $Y_\mu(\mathfrak{gl}_n)$ (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 $\mathcal{W}_\mu \subset \operatorname{Mat}_n((z^{-1}))$). 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.