[Paper Review] Metagroups and their smashed twisted wreath products

S. V. Ludkowski|arXiv (Cornell University)|Sep 8, 2018

Algebraic and Geometric Analysis25 references3 citations

TL;DR

This paper introduces and investigates nonassociative metagroups—generalizations of groups with weakened associativity governed by a central twist function. It develops smashed twisted wreath products of metagroups, proving they yield metagroups under specific conditions, and establishes new results on extensions of central metagroups, providing a framework for nonassociative algebraic structures in noncommutative geometry and mathematical physics.

ABSTRACT

In this article nonassociative metagroups are studied. Different types of smashed products and smashed twisted wreath products are scrutinized. Extensions of central metagroups are studied.

Motivation & Objective

• To formalize and study nonassociative metagroups as generalizations of groups with weakened associativity.
• To investigate the structure and properties of smashed products and smashed twisted wreath products of metagroups.
• To analyze extensions of central metagroups, particularly through smashed splitting extensions.
• To establish conditions under which smashed twisted wreath products yield metagroups rather than just loops.
• To provide foundational results applicable to nonassociative algebras, cohomology, and mathematical physics.

Proposed method

• Uses a generalized associativity condition: (ab)c = t(a,b,c)a(bc), where t(a,b,c) ∈ Z(G), the center of G.
• Applies left and right division operations (a\b and b/a) to define inverses and solve equations in metagroups.
• Introduces the concept of a central metagroup via the commutation identity ab = t₂(a,b)ba with t₂(a,b) ∈ Z(G).
• Constructs smashed twisted wreath products using group actions and twist functions, ensuring compatibility with metagroup axioms.
• Employs quotient structures and normal submetagroups to analyze extensions and splitting conditions.
• Applies identities from Lemmas 2.2, 2.3, and 2.6 to derive structural properties in Theorems 3.3, 3.4, 4.8, and 4.10.

Experimental results

Research questions

• RQ1 Under what conditions does a smashed twisted wreath product of metagroups yield a metagroup rather than a loop?
• RQ2 How can central metagroups be extended via smashed splitting extensions, and what are the structural constraints?
• RQ3 What role do twist functions t(a,b,c) and t₂(a,b) play in preserving metagroup axioms under product constructions?
• RQ4 Can metagroups be generated from submetagroups and specific elements under the smashed twisted wreath product construction?
• RQ5 How do the centers and commutativity properties of submetagroups influence the structure of the resulting metagroup?

Key findings

• Theorem 4.10 establishes that if Zₘ(D) ⊆ Z₁ and Zₘ(D) ⊆ Z₀, then the smashed twisted wreath product C* is a metagroup.
• Theorem 4.14 proves that the smashed splitting extension C* of a central metagroup F* by D satisfies F* ⊆ [F*, C*]Z(F*) and C* is generated by F* and elements c_j with d_j\e ∈ F*c_j.
• Corollary 4.15 shows that if D is generated by Zₘ(D) and at least two elements d₁, d₂ with [d₂\e, d₁\e] = e, then C* is generated by Z(F*) and elements c₁, c₂, ... such that d_j\e ∈ F*c_j.
• The construction ensures that F* is contained in the commutator subgroup [F*, C*]Z(F*), implying F* is deeply embedded in the structure of C*.
• The identity f₀^{γ} = f₀ for all γ ∈ Z implies that central elements stabilize the generating functions in F* under conjugation.
• The twist functions t and ξ are shown to take values in the center, ensuring consistency and closure in the metagroup product operations.