For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra \(U_{q}^{-}(\mathfrak {g})\), where \(U_{q}^{-}(\mathfrak {g})\) is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig’s category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra \(U_{q}^{-}(\mathfrak {g})\).

对于非循环箭袋，我们在箭袋格拉斯曼函数的上同调和代数\(U_{q}^{-}(\mathfrak {g})\)的双正则基之间建立联系，其中\(U_{q} ^{-}(\mathfrak {g})\)是与箭袋相关联的量化包络代数的负半部分。为了实现这一目标，我们通过 Lusztig 的范畴研究了 quiver Grassmannians 的上同调。因此，我们根据双正则基的系数明确描述了刚性颤动格拉斯曼函数的庞加莱多项式，它们被视为量子洗牌代数的元素。通过这个结果，我们又一次证明了 quiver Grassmanians 的奇上同调消失定理。同时，对于 Dynkin quivers，我们证明刚性 quiver Grassmannians 的 Poincaré 多项式是代数\(U_{q}^{-}(\mathfrak {g})\)的双 PBW 基的系数。