Green's theorem states that the Hall algebra of the category of representations of a quiver over a finite field is a twisted bialgebra. Considering instead categories of orthogonal or symplectic quiver representations leads to a class of modules over the Hall algebra, called Hall modules, which are also comodules. A module theoretic analogue of Green's theorem, describing the compatibility of the module and comodule structures, is not known. In this paper we prove module theoretic analogues of Green's theorem in the degenerate settings of finitary Hall modules of $\mathsf{Rep}_{\mathbb{F}_1}(Q)$ and constructible Hall modules of $\mathsf{Rep}_{\mathbb{C}}(Q)$. The result is that the module and comodule structures satisfy a compatibility condition reminiscent of that of a Yetter-Drinfeld module.

中文翻译：

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霍尔模块格林定理的退化版本

格林定理指出，有限域上颤动的表示范畴的霍尔代数是扭曲双代数。相反，考虑正交或辛颤动表示的类别会导致霍尔代数上的一类模块，称为霍尔模块，它们也是协模块。格林定理的模理论模拟，描述模和协模结构的兼容性，是未知的。在本文中，我们在 $\mathsf{Rep}_{\mathbb{F}_1}(Q)$ 的有限霍尔模和 $\mathsf{Rep} 的可构造霍尔模的退化设置中证明了格林定理的模理论类似物_{\mathbb{C}}(Q)$。结果是模块和协模块结构满足让人联想到 Yetter-Drinfeld 模块的兼容性条件。