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The algebraic and geometric classification of $δ$-Novikov algebras

Hani Abdelwahab, Ivan Kaygorodov, Roman Lubkov

Abstract

The notion of $δ$-Novikov algebras was introduced recently as a generalization of Novikov and bicommutative algebras. It looks like $δ$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have a $2$-dimensional simple algebra for $δ=-1.$ The present paper is dedicated to the study of $3$-dimensional $δ$-Novikov algebras for $δ\notin \big\{0,1\big\}.$ The algebraic and geometric classifications of complex $3$-dimensional $δ$-Novikov algebras are given. As a corollary, we prove that there are no simple $3$-dimensional $δ$-Novikov algebras.

The algebraic and geometric classification of $δ$-Novikov algebras

Abstract

The notion of -Novikov algebras was introduced recently as a generalization of Novikov and bicommutative algebras. It looks like -Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have a -dimensional simple algebra for The present paper is dedicated to the study of -dimensional -Novikov algebras for The algebraic and geometric classifications of complex -dimensional -Novikov algebras are given. As a corollary, we prove that there are no simple -dimensional -Novikov algebras.

Paper Structure

This paper contains 9 sections, 6 theorems, 3 equations.

Table of Contents

  1. The algebraic classification of $\delta$-Novikov algebras
  2. Preliminaries: the algebraic classification
  3. The algebraic classification of anti-Novikov algebras
  4. The algebraic classification of $\delta$-Novikov algebras $(\delta \notin \{0,\pm 1\})$
  5. The geometric classification of algebras
  6. Preliminaries: definitions and notation
  7. The geometric classification of $\cap$-Novikov algebras
  8. The geometric classification of $\delta$-Novikov algebras $(\delta \notin \{0,\pm 1\})$
  9. The geometric classification of anti-Novikov algebras

Key Result

Lemma 2

Let $({\rm A},\cdot)$ be a Lie algebra and $\theta \in {\rm Z}^2_\delta({\rm A},{\rm A})$. Then $( {\rm A},\cdot_{\theta})$ is a $\delta$-Novikov algebra, where

Theorems & Definitions (12)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Proposition 4
  • Proposition 5
  • proof
  • proof
  • Corollary 6
  • Lemma 7
  • proof
  • ...and 2 more