Work

Zappa–Szép Skew Braces: A Unified Framework for Mutual Interactions in Noncommutative Algebra

Year: 2026

Type: article

Abstract: This paper introduces and systematically develops the theory of Zappa–Szép skew braces, a novel algebraic structure that provides a unified framework for bidirectional group interactions, thereby generalizing the classical constructions of semidirect skew braces and matched-pair factorizations (ZS1–ZS4, BC1–BC2). We establish the complete axiomatic foundation for these objects, characterizing them through necessary and sufficient compatibility conditions that encode mutual actions between two digroups. Central results include a semidirect embedding theorem, explicit constructions of nontrivial examples—notably a fully mutual brace of order 12 built from V4 and C3—and a detailed analysis of key structural invariants such as the socle, center, and automorphism groups. The framework is further elucidated via universal properties and categorical adjunctions, positioning Zappa–Szép skew braces as fundamental objects within noncommutative algebra. Applications to representation theory, cohomology, and the construction of set-theoretic solutions to the Yang–Baxter equation are derived, demonstrating both the generality and utility of the theory.

Source: Mathematics

DOI: 10.3390/math14020215

Authors: Suha Wazzan, David A. Oluyori

Institutions: King Abdulaziz University, Federal University of Technology, Babura

Language: English

Topic: Advanced Topics in Algebra

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: King Abdulaziz University