Emergent algebras

TL;DR

This paper introduces emergent algebras based on uniform idempotent right quasigroups, providing an alternative to differentiable algebras with applications in sub-Riemannian geometry and knot theory.

Contribution

It proposes emergent algebras as a new algebraic framework and establishes connections with contractible groups and symmetric spaces.

Findings

Bijection between contractible groups and distributive uniform irqs

Emergent algebras generalize differential calculus in sub-Riemannian geometry

Some symmetric spaces can be modeled as uniform quasigroups

Abstract

Inspired from research subjects in sub-riemannian geometry and metric geometry, we propose uniform idempotent right quasigroups and emergent algebras as an alternative to differentiable algebras. Idempotent right quasigroups (irqs) are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). To any uniform idempotent right quasigroup can be associated an approximate differential calculus, with Pansu differential calculus in sub-riemannian geometry as an example. An emergent algebra A over a uniform idempotent right quasigroup X is a collection of operations such that each operation emerges from X, meaning that it can be realized as a combination of the operations of the uniform irq X, possibly by taking limits which are uniform with respect to a set of parameters. Two applications are considered: we prove a bijection between contractible groups and distributive uniform irqs (uniform quandles) and that some symmetric spaces in the sense of Loos may be seen as uniform quasigroups with a distributivity property.