[Paper Review] Emergent algebras as generalizations of differentiable algebras, with applications

Marius Buliga|arXiv (Cornell University)|Jul 9, 2009

Geometric and Algebraic Topology14 references7 citations

TL;DR

This paper introduces emergent algebras as a generalization of differentiable algebras by replacing the differential structure with a uniform idempotent right quasigroup (irq). It shows that algebraic operations and relations emerge uniformly from the irq structure, establishing a bijection between contractible groups and distributive uniform irqs, and linking symmetric spaces to distributive uniform quasigroups.

ABSTRACT

We propose a generalization of differentiable algebras, where the underlying differential structure is replaced by a uniform idempotent right quasigroup (irq). Then a emergent algebra is a algebra A over the uniform irq X such that all operations and algebraic relations from A can be constructed or deduced from combinations of operations in the uniform irq, possibly by taking limits which are uniform with respect to a set of parameters. In this approach, the usual compatibility condition between algebraic information A and differential information D, expressed as the differentiability of operations from A with respect to D, is replaced by the ”emergence ” of algebraic operations and relations from the minimal structure of a uniform irq. Two applications are considered: we prove a bijection between contractible groups and distributive uniform irqs and that symmetric spaces in the sense of Loos may be seen as uniform quasigroups with a distributivity property.

Motivation & Objective

• To generalize differentiable algebras by replacing differential structure with a uniform idempotent right quasigroup (irq).
• To establish a framework where algebraic operations and relations emerge from the minimal structure of a uniform irq through uniform limits.
• To explore connections between emergent algebras and known algebraic structures such as contractible groups and symmetric spaces.
• To demonstrate that the traditional differentiability condition between algebraic and differential structures is replaced by structural emergence from the irq.

Proposed method

• Define a uniform idempotent right quasigroup (irq) as the foundational structure replacing differentiability.
• Construct emergent algebras as algebras over a uniform irq where operations and relations arise from combinations of irq operations and uniform limits.
• Use uniform convergence with respect to parameters to ensure the stability and consistency of emergent algebraic structures.
• Establish a bijection between contractible groups and distributive uniform irqs by leveraging the algebraic properties of the irq.
• Characterize symmetric spaces in the sense of Loos as uniform quasigroups with a distributivity property.
• Demonstrate that the emergence of algebraic structure from the irq is independent of traditional differentiability, relying instead on uniformity and quasigroup axioms.

Experimental results

Research questions

• RQ1 How can differentiable algebras be generalized by replacing differential structure with a uniform idempotent right quasigroup?
• RQ2 In what way do algebraic operations and relations emerge uniformly from the structure of a uniform irq?
• RQ3 What is the relationship between contractible groups and distributive uniform irqs in this framework?
• RQ4 How do symmetric spaces in the sense of Loos relate to the proposed uniform quasigroup structure?
• RQ5 Can the compatibility condition between algebraic and differential structures be redefined through emergence from a minimal algebraic system?

Key findings

• A bijection is established between contractible groups and distributive uniform irqs, showing a structural equivalence.
• Symmetric spaces in the sense of Loos are characterized as uniform quasigroups endowed with a distributivity property.
• Algebraic operations and relations in emergent algebras are derived from combinations of irq operations and uniform limits.
• The differentiability condition between algebraic and differential structures is replaced by the emergence of algebraic information from the uniform irq.
• The framework provides a new foundation for differential geometry and algebraic structures by replacing smoothness with uniform quasigroup axioms.
• The emergent algebra framework unifies contractible groups and symmetric spaces under a common algebraic-geometric structure based on uniform irqs.