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The algebraic and geometric classification of noncommutative Jordan algebras

Hani Abdelwahab, Kobiljon Abdurasulov, Ivan Kaygorodov

TL;DR

This work develops a framework to algebraically classify finite-dimensional noncommutative Jordan algebras by leveraging the Jordan algebra classification, via the correspondence with generic Poisson--Jordan structures. It delivers detailed algebraic classifications in low dimensions: 3D complex noncommutative Jordan, Kokoris, and standard algebras, plus 4D nilpotent cases, accompanied by explicit families and rigid instances. The geometric classification analyzes varieties of these algebras, identifying their dimensions, irreducible components, rigid algebras, and concrete degeneration patterns, thereby describing the orbit closures generating each component. Overall, the paper provides a unified approach to both algebraic and geometric aspects of noncommutative Jordan-type algebras and related Poisson-like structures, highlighting the rich landscape of degenerations and rigidity in small dimensions.

Abstract

In this paper, we develop a method to obtain the algebraic classification of noncommutative Jordan algebras from the classification of Jordan algebras of the same dimension. We use this method to obtain the algebraic classification of complex $3$-dimensional noncommutative Jordan algebras. As a byproduct, we obtain the classification of complex $3$-dimensional Kokoris, standard, generic Poisson, and generic Poisson--Jordan algebras; and also complex $4$-dimensional nilpotent Kokoris and standard algebras. In addition, we consider the geometric classification of varieties of cited algebras, that is the description of its irreducible components.

The algebraic and geometric classification of noncommutative Jordan algebras

TL;DR

This work develops a framework to algebraically classify finite-dimensional noncommutative Jordan algebras by leveraging the Jordan algebra classification, via the correspondence with generic Poisson--Jordan structures. It delivers detailed algebraic classifications in low dimensions: 3D complex noncommutative Jordan, Kokoris, and standard algebras, plus 4D nilpotent cases, accompanied by explicit families and rigid instances. The geometric classification analyzes varieties of these algebras, identifying their dimensions, irreducible components, rigid algebras, and concrete degeneration patterns, thereby describing the orbit closures generating each component. Overall, the paper provides a unified approach to both algebraic and geometric aspects of noncommutative Jordan-type algebras and related Poisson-like structures, highlighting the rich landscape of degenerations and rigidity in small dimensions.

Abstract

In this paper, we develop a method to obtain the algebraic classification of noncommutative Jordan algebras from the classification of Jordan algebras of the same dimension. We use this method to obtain the algebraic classification of complex -dimensional noncommutative Jordan algebras. As a byproduct, we obtain the classification of complex -dimensional Kokoris, standard, generic Poisson, and generic Poisson--Jordan algebras; and also complex -dimensional nilpotent Kokoris and standard algebras. In addition, we consider the geometric classification of varieties of cited algebras, that is the description of its irreducible components.
Paper Structure (17 sections, 19 theorems, 12 equations)

This paper contains 17 sections, 19 theorems, 12 equations.

Table of Contents

  1. The algebraic classification of non-associative algebras
  2. The classification method of noncommutative Jordan algebras
  3. The algebraic classification of noncommutative Jordan algebras
  4. The algebraic classification of complex $3$-dimensional noncommutative Jordan algebras
  5. The algebraic classification of complex $4$-dimensional nilpotent noncommutative Jordan algebras
  6. The algebraic classification of Kokoris algebras
  7. The algebraic classification of complex $3$-dimensional Kokoris algebras
  8. The algebraic classification of complex $3$-dimensional flexible power-associative algebras
  9. The algebraic classification of complex $4$-dimensional nilpotent Kokoris algebras
  10. The algebraic classification of standard algebras
  11. The algebraic classification of complex $3$-dimensional standard algebras
  12. The algebraic classification of complex $4$-dimensional nilpotent standard algebras
  13. The geometric classification of non-associative algebras
  14. Definitions and notation
  15. The geometric classification of Kokoris algebras
  16. ...and 2 more sections

Key Result

Proposition 2

$({\bf A},\cdot)$ is a noncommutative Jordan algebra if and only if $({\bf A},\circ ,[\cdot,\cdot])$ is a generic Poisson--Jordan algebra.

Theorems & Definitions (30)

  • Definition 1
  • Proposition 2
  • proof
  • Definition 3
  • Proposition 4
  • Definition 5
  • Lemma 6
  • Lemma 7
  • proof
  • Remark 8
  • ...and 20 more