论文标题
反尼亚人的贝斯亚抗物节
Bethe subalgebras in antidominantly shifted Yangians
论文作者
论文摘要
一个简单的复杂谎言组$ g $的循环组$ g（（z^{ - 1}））$具有天然的泊松结构。我们介绍了一个自然家族的Poisson托管子代理$ \ edline {\ MathBf {b}}（c）\ subset \ Mathcal \ Mathcal \ Mathcal {o}（g（（（（z^{ - 1}}））$，取决于g $ in Gu $ in Gu $ in Classical bethe bethe bethe subalgebras.tormant $ co的参数$ c \在$ t \子集g $一个人可以关联封闭的泊松子空间$ \ mathcal {w}_μ$ of $ g（（（z^{ - 1}）））$（poisson algebra $ \ mathcal {o} $y_μ（\ Mathfrak {g}）$）。贝特（Bethe）的Yangians中的贝斯（Bethe）。美元\ subset {\ mathbf {y}}}^{\ mathrm {rtt}}}}（\ mathfrak {gl} _n）$。使用$y_μ（\ Mathfrak {gl} _n）$的RTT实现（由Frassek，Pestun和Tsymbaliuk发明），我们获得了自然的过滤$ {\ Mathbf {y}}}} y_μ（\ mathfrak {gl} _n）$量化嵌入$ \ mathcal {w}_μ\ subset \ operatorname {mat} _n（（（z^{ - 1}}）））$）。拍摄$ {\ mathbf {b}}（c）$ in $y_μ（\ mathfrak {gl} _n）$的图像。
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.
