Jordanian double extensions of a quadratic vector space and symmetric Novikov algebras

TL;DR

This paper explores the structure of Jordan algebras derived from quadratic vector spaces, characterizes certain nilpotent cases, and investigates symmetric Novikov algebras up to dimension 7, contributing to algebra classification.

Contribution

It provides an isomorphic classification of 2-step nilpotent pseudo-Euclidean Jordan algebras and establishes conditions for Novikov algebras to be Jordan-admissible, focusing on symmetric cases.

Findings

Characterization of 2-step nilpotent pseudo-Euclidean Jordan algebras

Jordan-admissible conditions for Novikov algebras

Classification of symmetric Novikov algebras up to dimension 7

Abstract

First, we study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra. We give an isomorphic characterization of 2-step nilpotent pseudo-Euclidean Jordan algebras. Next, we find a Jordan-admissible condition for a Novikov algebra $\N$. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.