模间二次态射
Morphismes quadratiques entre modules sur un anneau carré
摘要：
引入了交换方环$R$和$R$上模间二次映射的概念，称之为$R$ -二次映射.这个概念推广了文献中代数对象之间的二次映射的各种概念。我们构造了一个$R$ -模之间的二次映射范畴，并证明了它是一个右二次范畴，具有一个内部Hom-函子.一路上，我们回顾了一般的广场环$R$和一个模块上的$R$的概念，并讨论了他们的基本性质的一些细节，采用一个操作的观点。特别地，证明了方环$R$的相关分次对象是类2的幂零操作数，并且$R$ -模的相关分次对象是这个操作数上的代数，以函子的方式。这推广了群和分次李代数之间的关系（在幂零类2的情况下）。我们还将群论中的一些基本概念推广到了方环上的模。
英文摘要：
We introduce the notions of a commutative square ring $R$ and of a quadratic map between modules over $R$, called $R$-quadratic map. This notion generalizes various notions of quadratic maps between algebraic objects in the literature. We construct a category of quadratic maps between $R$-modules and show that it is a right-quadratic category and has an internal Hom-functor. Along our way, we recall the notions of a general square ring $R$ and of a module over $R$, and discuss their elementary properties in some detail, adopting an operadic point of view. In particular, it turns out that the associated graded object of a square ring $R$ is a nilpotent operad of class 2, and the associated graded object of an $R$-module is an algebra over this operad, in a functorial way. This generalizes the well-known relation between groups and graded Lie algebras (in the case of nilpotency class 2). We also generalize some elementary notions from group theory to modules over square rings.
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