Idempotent geometry in generic algebras

TL;DR

This paper characterizes the structure of two-dimensional real generic algebras, revealing three homotopic types and relating these findings to quadratic ODEs, with a focus on the importance of genericity conditions.

Contribution

It introduces a classification of two-dimensional real generic algebras into three homotopic types using the syzygy method, extending understanding of idempotent geometry.

Findings

Exactly three homotopic types of such algebras exist

The classification relates to qualitative theory of quadratic ODEs

Genericity condition is essential for the classification

Abstract

Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.