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The algebraic and geometric classification of derived Jordan and bicommutative algebras

Hani Abdelwahab, Ivan Kaygorodov, Roman Lubkov

TL;DR

This work introduces a novel framework for the algebraic classification of $n$-dimensional derived Jordan algebras, applying it to fully classify $3$-dimensional cases and deriving corresponding classifications for related varieties such as metabelian commutative and derived commutative associative algebras. It then extends these ideas to a complete algebraic classification of $3$-dimensional bicommutative algebras via central extensions and Aut$(A)$-orbit analysis. The geometric side provides a thorough degeneration-based classification across four 3D varieties, identifying dimensions, irreducible components, and rigid algebras, and enumerating key degeneration chains that connect representative algebras. Overall, the paper delivers a comprehensive algebraic and geometric map of small-dimensional algebras in these interrelated varieties, highlighting the structure of their orbit closures and rigidity properties.

Abstract

We developed a new proper method for classifying $n$-dimensional derived Jordan algebras, and apply it to the classification of $3$-dimensional derived Jordan algebras. As a byproduct, we have the algebraic classification of $3$-dimensional metabelian commutative algebras and $3$-dimensional derived commutative associative algebras. After that, we introduced a method of classifying $n$-dimensional bicommutative algebras, based on the classification of $n$-dimensional derived commutative associative algebras, and applied it to the classification of $3$-dimensional bicommutative algebras. The second part of the paper is dedicated to the geometric classification of $3$-dimensional metabelian commutative, derived commutative associative, derived Jordan and bicommutative algebras.

The algebraic and geometric classification of derived Jordan and bicommutative algebras

TL;DR

This work introduces a novel framework for the algebraic classification of -dimensional derived Jordan algebras, applying it to fully classify -dimensional cases and deriving corresponding classifications for related varieties such as metabelian commutative and derived commutative associative algebras. It then extends these ideas to a complete algebraic classification of -dimensional bicommutative algebras via central extensions and Aut-orbit analysis. The geometric side provides a thorough degeneration-based classification across four 3D varieties, identifying dimensions, irreducible components, and rigid algebras, and enumerating key degeneration chains that connect representative algebras. Overall, the paper delivers a comprehensive algebraic and geometric map of small-dimensional algebras in these interrelated varieties, highlighting the structure of their orbit closures and rigidity properties.

Abstract

We developed a new proper method for classifying -dimensional derived Jordan algebras, and apply it to the classification of -dimensional derived Jordan algebras. As a byproduct, we have the algebraic classification of -dimensional metabelian commutative algebras and -dimensional derived commutative associative algebras. After that, we introduced a method of classifying -dimensional bicommutative algebras, based on the classification of -dimensional derived commutative associative algebras, and applied it to the classification of -dimensional bicommutative algebras. The second part of the paper is dedicated to the geometric classification of -dimensional metabelian commutative, derived commutative associative, derived Jordan and bicommutative algebras.
Paper Structure (15 sections, 7 theorems, 15 equations)

This paper contains 15 sections, 7 theorems, 15 equations.

Table of Contents

  1. The algebraic classification of algebras
  2. Metabelian commutative algebras
  3. Derived commutative associative algebras
  4. Derived Jordan algebras
  5. Bicommutative algebras
  6. Preliminaries: the algebraic classification
  7. The classification Theorem
  8. Proof of Theorem A4
  9. The geometric classification of algebras
  10. Preliminaries: definitions and notation
  11. The geometric classification of algebras
  12. Metabelian commutative algebras
  13. Derived commutative associative algebras
  14. Derived Jordan algebras
  15. Bicommutative algebras

Key Result

Proposition 1

Let ${\mathfrak J}$ be a nontrivial complex $2$-dimensional Jordan algebra. Then ${\mathfrak J}$ is isomorphic to one of the following algebras: All algebras, excepting ${\mathfrak J}_{04},$ are associative.

Theorems & Definitions (19)

  • Proposition 1: see aak24
  • Proposition 2: see aak24
  • Proposition 3: see aak24
  • Definition 4
  • proof
  • Definition 5
  • proof
  • Definition 6
  • proof
  • Definition 7
  • ...and 9 more