主丛模空间上的重合泊松结构
Coincident Poisson structures on principal-bundle moduli spaces
AI总结：
研究配备线性约化仿射群概型作用的约化复代数群，通过双线性形式诱导扩张，将椭圆曲线丛扩张与模扩张关联，确定主丛模空间光滑轨迹上的泊松结构，验证费金 - 奥杰斯基的相关识别。
AI中文摘要：
考虑一个配备线性约化仿射群概型$\mathbb{K}$作用的约化复代数群$\mathbb{G}$。对于$\mathbb{K}$不变的抛物子群$\mathbb{P}\leq\mathbb{G}$，由$\mathfrak{g}:=Lie(\mathbb{G})$上的$(\mathbb{K},\mathfrak{g})$不变对称非退化双线性形式诱导的$\mathfrak{p}^*$由$\mathfrak{p}$的扩张，与通过与$\mathbb{K}$不变的嘉当/博雷尔对$\mathbb{H}\leq\mathbb{B}\leq\mathbb{P}\leq\mathbb{G}$相关的标准双代数结构和相同双线性形式得到的扩张是$\mathbb{K}$等变同构的。将椭圆曲线$E$上的丛扩张与上述$\mathfrak{p}$模扩张相关联，这确定了分别由巴尔杜齐（使用前一种扩张）和费金 - 奥杰斯基（通过标准双代数结构）定义的$E$上主$\mathbb{P}$丛模空间光滑轨迹上的泊松结构。这特别验证了费金 - 奥杰斯基关于双代数诱导的辛叶与沿$\mathbb{P}\leq\mathbb{G}$忘记结构后相互同构的丛的轨迹的识别。
英文摘要：
Consider a reductive complex algebraic group $\mathbb{G}$ equipped with an action by a linearly reductive affine group scheme $\mathbb{K}$. The extension of $\mathfrak{p}^*$ by $\mathfrak{p}$ induced by an $(\mathbb{K},\mathfrak{g})$-invariant symmetric non-degenerate bilinear form on $\mathfrak{g}:=Lie(\mathbb{G})$, for a $\mathbb{K}$-invariant parabolic $\mathbb{P}\le \mathbb{G}$, is $\mathbb{K}$-equivariantly isomorphic to the extension obtained via the standard bialgebra structure attached to a $\mathbb{K}$-invariant Cartan/Borel pair $\mathbb{H}\le \mathbb{B}\le \mathbb{P}\le \mathbb{G}$ and the same bilinear form. Associating bundle extensions on an elliptic curve $E$ to said $\mathfrak{p}$-module extensions, this identifies Poisson structures on the smooth locus of the principal-$\mathbb{P}$-bundle moduli space over $E$ respectively defined by Balduzzi (using the former extension) and Feigin-Odesskii (via the standard bialgebra structure). This in particular verifies Feigin-Odesskii's identification of the bialgebra-induced symplectic leaves with the loci of bundles mutually isomorphic after forgetting structure along $\mathbb{P}\le \mathbb{G}$.
