We introduce a construction that produces from each bialgebra H an operad \(\mathsf {Ass}_H\) controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of H and the Koszul model of H. This operad is seldom quadratic—and hence does not fall within the reach of Koszul duality theory—so our work provides a new rich family of examples where an explicit minimal model of an operad can be obtained. As an application, we observe that if we take H to be the mod-2 Steenrod algebra \({\mathscr {A}}\), then this notion of an associative H-algebra coincides with the usual notion of an \(\mathscr {A}\)-algebra considered by homotopy theorists. This makes available to us an operad \(\mathsf {Ass}_{{\mathscr {A}}}\) along with its minimal model that controls the category of associative \({\mathscr {A}}\)-algebras, and the notion of strong homotopy associative \({\mathscr {A}}\)-algebras.

我们引入了一个构造，它从每个双代数H产生一个操作数\(\mathsf {Ass}_H\)控制H - 模的单曲面类别中的关联代数，或者简单地说，H -代数。当这个双代数的基础代数是 Koszul 时，我们给出这个操作数的最小模型的显式公式，仅取决于H的余积和H的 Koszul 模型。这个算子很少是二次的——因此不属于 Koszul 对偶理论的范围——所以我们的工作提供了一个新的丰富的例子系列，其中可以获得算子的显式最小模型。作为一个应用程序，我们观察到，如果我们取H是 mod-2 Steenrod 代数\({\mathscr {A}}\)，那么这个联想H -代数的概念与同伦考虑的\(\mathscr {A}\) -代数的通常概念一致理论家。这使我们可以使用操作符\(\mathsf {Ass}_{{\mathscr {A}}}\)及其控制关联\({\mathscr {A}}\)类别的最小模型 -代数，以及强同伦结合\({\mathscr {A}}\) -代数的概念。