Axial identities

TL;DR

This work develops a unified framework of idempotental and axial identities for axial algebras, enabling a structural reinterpretation of solid-subalgebra theorems and providing a bridge between fusion rules and Frobenius-form identities. It introduces generalized Frobenius polynomials and axial Frobenius identities, yielding tools to derive axial properties from idempotent substitutions and to analyze Miyamoto involutions in a broad setting. The paper proves key results for Jordan-type $\frac{1}{2}$ algebras, including solidity criteria and a Jordan-type conclusion for 3-generated PAJ algebras, and constructs generic (universal) axial algebras over Noetherian rings, producing examples that are Jordan, Matsuo, or neither, with both singular and nonsingular Frobenius forms. It also explores Matsuo versus Jordan structures, and analyzes trace-admissibility and nilpotence, contributing to a deeper classification framework for axial algebras and their subalgebras. Overall, the work advances the algebraic and geometric understanding of axial algebras by connecting idempotent/axial identities, universal constructions, and detailed case studies (notably Jordan type $\frac{1}{2}$) with potential implications for related areas such as group theory and vertex-operator algebra theory.

Abstract

The notions of idempotental identities and axial identities of axial algebras are introduced, in order to understand better some theorems of J.~Desmet, I.~Gorshkov, S.~Shpectorov, and A.~Staroletov about solid subalgebras; this approach produces generic examples, including an example of an axial algebra of Jordan type 1/2 with a Frobenius form having radical 0, which is neither Jordan nor a homomorphic image of a Matsuo algebra.

Theorems & Definitions (94)

• Remark 1.2
• Lemma 1.3
• Remark 1.4
• Definition 1.5
• Lemma 1.7
• proof
• Remark 1.8
• Example 1.9
• Lemma 1.10
• proof