Structure of Vidinli Algebras: Simple Nonassociative Jordan-Lie Algebras with Azumaya Properties
Olcay Coskun, Alp Eden
TL;DR
The paper extends Vidinli’s 3D nonassociative product to an infinite family of odd-dimensional simple unital real algebras $\mathbb{V}_{2n+1}$ using a fixed axis $e_1$ and a nondegenerate skew form on $V_0=e_1^{\perp}$, where the symmetric part is a Jordan product and the skew part yields a Heisenberg Lie algebra. It develops three complementary viewpoints—geometric (incidence structures like PG$(k-1,2)$ and Fano plane), categorical (degenerate pushouts assembling lower-dimensional blocks), and analytic (local 2D/3D constraints forcing global structure)—to demonstrate how global non-associativity arises from local constraints. The work shows that each $\mathbb{V}_{2n+1}$ is simple, has multiplication algebra $M_{2n+1}(\mathbb{R})$, possesses an Azumaya-type property, and admits an $\mathrm{Aut}$-action by $U(n)$ while fitting into a Jordan–Lie framework via Vidinli–Jordan algebras and the Heisenberg commutator. The degenerate pushout formalism unifies classical structures (Heisenberg algebras, spin factors) as gluings of primitive blocks and provides a canonical way to realize higher-dimensional Vidinli algebras as amalgamations of 3D components, with $V_7$ distinguished by its Fano-plane and $G_2$ symmetry. Overall, the Vidinli family offers a tractable, richly structured class in the landscape of non-associative algebras, linking geometry, category theory, and analysis in a coherent framework.
Abstract
The Vidinli algebra, originally defined in three dimensions, is extended to arbitrary odd dimensions via symplectic forms. Particular attention is given to the five- and seven-dimensional cases. The symmetric and antisymmetric parts of the multiplication yield a Jordan algebra and a Heisenberg Lie algebra, respectively. A categorical construction, called the degenerate pushout, is introduced to assemble higher-dimensional algebras from lower-dimensional components, even in the presence of degeneracies such as zero divisors. This method applies not only to Vidinli algebras but also recovers classical structures including spin factor Jordan algebras and Heisenberg Lie algebras. The Vidinli family admits three independent characterizations; geometric, categorical, and analytic, demonstrating how global non-associ\-ative structure can arise from local constraints. Together with their variants, the Vidinli algebras form a structured and tractable class within the otherwise wild category of non-associative algebras.