Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The algebraic and geometric classification of right alternative and semi-alternative algebras

Hani Abdelwahab, Ivan Kaygorodov, Roman Lubkov

TL;DR

The paper delivers a complete algebraic and geometric classification of complex $3$-dimensional right alternative and semi-alternative algebras. It develops a cohomology-based framework using $Z^2(A,A)$, automorphism orbits, and one-parameter twists to construct all $3$-dimensional right alternative and semi-alternative algebras (Theorems A1 and A2), from which corollaries for related varieties such as $\,\mathfrak{perm}$, binary $\,\mathfrak{perm}$, associative, and $(-1,1)$-types follow. The geometric classification identifies rigid algebras and irreducible components across these varieties and documents degeneration graphs and chains, revealing the structure of the moduli spaces for $3$-dimensional algebras and providing insights into higher-dimensional nuances. The results include the first explicit nonassociative examples in dimension $3$ (right alternative and assosymmetric) and several dimension-4 phenomena, as well as a simpler resolution to a problem from the Dniester Notebook. These classifications offer rigorous foundations for studying nonassociative algebra varieties and their degenerations, with implications for related algebraic systems and deformation theory.

Abstract

The algebraic and geometric classifications of complex $3$-dimensional right alternative and semi-alternative algebras are given. As corollaries, we have the algebraic and geometric classification of complex $3$-dimensional $\mathfrak{perm}$, binary $\mathfrak{perm}$, associative, $(-1,1)$-, binary $(-1,1)$-, and assosymmetric algebras. In particular, we proved that the first example of non-associative right alternative algebras appears in dimension $3;$ the first example of non-associative assosymmetric algebras appears in dimension $3;$ the first example of non-assosymmetric semi-alternative algebras appears in dimension $4;$ the first example of binary $(-1,1)$-algebras, which is non-$(-1,1)$-, appears in dimension $4;$ the first example of right alternative algebras, which is not binary $(-1,1)$-, appears in dimension $4;$ the first example of binary $\mathfrak{perm}$ non-$\mathfrak{perm}$ algebras appears in dimension $4.$ As a byproduct, we give a more easy answer to problem 2.109 from the Dniester Notebook, previously resolved by Shestakov and Arenas.

The algebraic and geometric classification of right alternative and semi-alternative algebras

TL;DR

The paper delivers a complete algebraic and geometric classification of complex -dimensional right alternative and semi-alternative algebras. It develops a cohomology-based framework using , automorphism orbits, and one-parameter twists to construct all -dimensional right alternative and semi-alternative algebras (Theorems A1 and A2), from which corollaries for related varieties such as , binary , associative, and -types follow. The geometric classification identifies rigid algebras and irreducible components across these varieties and documents degeneration graphs and chains, revealing the structure of the moduli spaces for -dimensional algebras and providing insights into higher-dimensional nuances. The results include the first explicit nonassociative examples in dimension (right alternative and assosymmetric) and several dimension-4 phenomena, as well as a simpler resolution to a problem from the Dniester Notebook. These classifications offer rigorous foundations for studying nonassociative algebra varieties and their degenerations, with implications for related algebraic systems and deformation theory.

Abstract

The algebraic and geometric classifications of complex -dimensional right alternative and semi-alternative algebras are given. As corollaries, we have the algebraic and geometric classification of complex -dimensional , binary , associative, -, binary -, and assosymmetric algebras. In particular, we proved that the first example of non-associative right alternative algebras appears in dimension the first example of non-associative assosymmetric algebras appears in dimension the first example of non-assosymmetric semi-alternative algebras appears in dimension the first example of binary -algebras, which is non--, appears in dimension the first example of right alternative algebras, which is not binary -, appears in dimension the first example of binary non- algebras appears in dimension As a byproduct, we give a more easy answer to problem 2.109 from the Dniester Notebook, previously resolved by Shestakov and Arenas.
Paper Structure (17 sections, 24 theorems, 7 equations)

This paper contains 17 sections, 24 theorems, 7 equations.

Table of Contents

  1. The algebraic classification of algebras
  2. Preliminaries: the algebraic classification
  3. Right alternative algebras
  4. The classification Theorem A1
  5. Proof of Theorem A1
  6. Corollaries from Theorem A1
  7. Semi-alternative algebras
  8. The classification Theorem A2
  9. Proof of Theorem A2
  10. Corollaries from Theorem A2
  11. The geometric classification of algebras
  12. Preliminaries: definitions and notation
  13. The geometric classification of algebras
  14. The geometric classification of $\mathfrak{perm}$ algebras
  15. The geometric classification of associative algebras
  16. ...and 2 more sections

Key Result

Lemma 3

Let $({\rm A},\cdot)$ be a Jordan (resp., Malcev) algebra and $\theta \in {\rm Z}^2({\rm A},{\rm A})$. Then $( {\rm A},\cdot_{\theta})$ is a right alternative (resp., semi-alternative) algebra, where $x\cdot_{\theta} y := x \cdot y + x *_{\theta} y.$

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Remark 9
  • Theorem 10
  • ...and 33 more