Bethe subalgebras in antidominantly shifted Yangians
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910014689509376 |
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| author | Krylov, Vasily Rybnikov, Leonid |
| author_facet | Krylov, Vasily Rybnikov, Leonid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04700 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bethe subalgebras in antidominantly shifted Yangians Krylov, Vasily Rybnikov, Leonid Representation Theory Algebraic Geometry Quantum Algebra 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. |
| title | Bethe subalgebras in antidominantly shifted Yangians |
| topic | Representation Theory Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2205.04700 |