Division, adjoints, and dualities of bilinear maps

TL;DR

This paper explores the structure of bilinear maps through adjoint-morphisms, establishing a duality and categorical framework that connects algebraic and geometric perspectives, especially in the context of division maps and nonassociative rings.

Contribution

It introduces a categorical framework for bilinear maps using adjoint-morphisms, revealing dualities and geometric interpretations, and clarifies the structure of division maps and nonassociative rings.

Findings

Adjoint-morphisms form a complete abelian category with duality.

Bilinear division maps are characterized as simple bimaps.

The framework relates algebraic objects to geometric structures in linear geometry.

Abstract

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The bilinear division maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic objects with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism; hence, nonassociative division rings can be studied within this framework. This also corrects an error in an earlier pre-print; see Remark 2.11.