Associative Geometries. II: Involutions, the classical torsors, and their homotopes

TL;DR

This paper introduces a geometric construction of contractions called homotopes for classical groups and their analogs, utilizing involutions of associative geometries, and establishes their canonical semigroup completions.

Contribution

It provides a new geometric method to construct and analyze homotopes of classical groups, extending to infinite dimensions and general base fields or rings.

Findings

Construction of homotopes for classical groups using involutions

Existence of canonical semigroup completions for groups and homotopes

Applicability to infinite-dimensional and generalized base field cases

Abstract

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion.