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.

我们引入一种构造，从每个双代数H产生一个操作数\(\mathsf {Ass}_H\)控制H模的幺半群范畴中的关联代数，或者简称为H代数。当该双代数的基础代数为 Koszul 时，我们仅根据H的余积和H的 Koszul 模型给出该运算的最小模型的显式公式。该运算很少是二次的，因此不属于科祖尔对偶理论的范围，因此我们的工作提供了一个新的丰富示例系列，在这些示例中可以获得运算的显式最小模型。作为一个应用，我们观察到，如果我们将H视为 mod-2 Steenrod 代数\({\mathscr {A}}\)，那么结合H代数的概念与通常的\(\ mathscr {A}\) - 同伦理论家所考虑的代数。这为我们提供了一个操作数\(\mathsf {Ass}_{{\mathscr {A}}}\)及其控制关联\({\mathscr {A}}\)代数类别的最小模型，以及强同伦关联\({\mathscr {A}}\) -代数的概念。