Vidinli algebras
Alberto Elduque, Javier Rández-Ibáñez
TL;DR
The paper introduces Vidinli algebras as conic unital algebras with commutators lying in the scalar multiples of the unity, joining them to Jordan algebras of Clifford type in characteristic not two while revealing a markedly richer structure in characteristic two. It provides a complete structural framework over fields with $\operatorname{char}\mathbb{F}\neq 2$, showing that such algebras are determined by a bilinear form $B$ on a complement $V$ of $\mathbb{F}1$, with multiplication $(\alpha 1+u)(\beta 1+v)=(\alpha\beta-B(u,v))1+(\alpha v+\beta u)$ and a corresponding norm $q$, and it analyzes the radical, simplicity, automorphisms, derivations, and the multiplication algebra in detail. The characteristic-two case is completely classified as $A \cong \mathcal{A}(V,*,\varphi)$, governed by an anticommutative product $*$ on $V$ and a bilinear form $\varphi$, with isomorphisms captured by a combination of a linear form on $V$ and a linear isomorphism, including the unitization of anticommutative algebras as a notable instance. These results establish precise connections to Jordan/Q-form structures, clarify how automorphisms and derivations arise from the bilinear data, and provide a comprehensive taxonomy of Vidinli algebras across characteristics.
Abstract
A new class of nonassociative algebras, Vidinli algebras, is defined based on recent work of Coşkun and Eden. These algebras are conic (or quadratic) algebras with the extra restriction that the commutator of any two elements is a scalar multiple of the unity. Over fields of characteristic not 2, Vidinli algebras may be considered as generalizations of the Jordan algebras of Clifford type. However, in characteristic 2, the class of Vidinli algebras is much larger and include the unitizations of anticommutative algebras.