Associative Geometries. I: Torsors, linear relations and Grassmannians

TL;DR

This paper introduces associative geometries, a new geometric framework linked to associative algebras, blending Lie group and projective geometry concepts, with future work planned for involutive cases.

Contribution

It defines associative geometries associated with associative algebras, integrating Lie and Jordan structures, and lays groundwork for further exploration of involutive cases.

Findings

Defined associative geometries from associative algebras

Connected associative geometries to Lie and projective geometries

Established a foundation for future work on involutive associative algebras

Abstract

We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized projective geometries, where the former correspond to the Lie product of an associative algebra and the latter to its Jordan product. A further development of the theory encompassing involutive associative algebras will be given in subsequent work.