We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) an equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra $A$, there exists a unique universal Hopf algebra $H$ together with an $H$-(co)module structure on $A$ such that any other equivalent (co)module algebra structure on $A$ factors through the action of $H$. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf–Galois extensions, actions of algebraic groups, and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. We apply support equivalence in the study of the asymptotic behavior of codimensions of $H$-identities and, in particular, to the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary polynomial identities. As an example, we prove this analogue for all unital $H$-module structures on the algebra $F[x]/(x^2)$ of dual numbers.

中文翻译：

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Hopf 代数上的（共）模代数结构的等价

我们引入了支持等价的概念对于（co）模代数（在 Hopf 代数上），它以自然的方式（弱）概括了等价的等级。我们证明，对于给定代数 $A$ 上的（共）模代数结构的每个等价类，存在唯一的通用 Hopf 代数 $H$ 以及 $A$ 上的 $H$-（共）模结构，使得通过 $H$ 的作用在 $A$ 因子上的任何其他等效（共）模代数结构。我们研究支持等价和上面提到的通用 Hopf 代数，用于群分级、Hopf-Galois 扩展、代数群的动作和互易 Hopf 代数。我们展示了如何使用支持等价的概念来减少 Hopf 代数（协同）动作的分类问题。我们在 $H$-恒等式的余维的渐近行为研究中应用支持等价，特别是，与 Amitsur 猜想的类比（由 Yu. A. Bahturin 提出），该猜想最初与普通多项式恒等式有关。作为一个例子，我们证明了对偶数代数 $F[x]/(x^2)$ 上所有单位 $H$-模结构的类比。