Morphismes quadratiques entre modules sur un anneau carr\'e

TL;DR

This paper introduces the concept of quadratic maps between modules over a commutative square ring, generalizing existing notions and establishing a categorical framework with internal Hom-functors, connecting algebraic structures via operads.

Contribution

It defines R-quadratic maps and constructs a category of such maps, revealing a right-quadratic structure with an internal Hom, and explores their relation to operads and algebraic generalizations.

Findings

The category of quadratic maps is right-quadratic.

The associated graded of a square ring forms a nilpotent operad of class 2.

Modules over R are algebras over this operad.

Abstract

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.