标题： Fukaya范畴在Koszul对偶理论中 显示英文标题

标题： Fukaya categories in Koszul duality theory

摘要： 在本文中，我们通过其$A_{\infty}$-导出范畴中的扭转来定义定向$A_{\infty}$-范畴的$A_{\infty}$-Koszul对偶。然后，我们计算了具有定向$A_n$-型Gabriel图的路径代数的$A_{\infty}$-Koszul对偶的具体公式。 为了计算这样的代数$A$的$A_\infty$-Koszul 对偶，我们构造一个Fukaya范畴的定向子范畴，该子范畴与$A$-模的范畴$A_\infty$-导出等价，并将Dehn扭转计算为扭转。 该公式揭示了parh 代数的简单模的所有ext群及其高复合结构。 显示英文摘要

摘要： In this paper, we define $A_{\infty}$-Koszul duals for directed $A_{\infty}$-categories in terms of twists in their $A_{\infty}$-derived categories. Then, we compute a concrete formula of $A_{\infty}$-Koszul duals for path algebras with directed $A_n$-type Gabriel quivers. To compute an $A_\infty$-Koszul dual of such an algebra $A$, we construct a directed subcategory of a Fukaya category which are $A_\infty$-derived equivalent to the category of $A$-modules and compute Dehn twists as twists. The formula unveils all the ext groups of simple modules of the parh algebras and their higher composition structures.