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The algebraic and geometric classification of right alternative superalgebras

Hani Abdelwahab, Ivan Kaygorodov, Abror Khudoyberdiyev

TL;DR

The study delivers a complete algebraic and geometric classification of complex $3$-dimensional right alternative superalgebras. It develops a general method based on the Jordan superalgebra framework and the cohomology space $Z^{2}(A,A)$ to enumerate all right alternative structures of a fixed type, then specializes to types $(1,2)$ and $(2,1)$ to produce explicit lists of nonisomorphic algebras. Geometrically, it analyzes degenerations, orbit closures, and irreducible components across several principal subvarieties (associative, perm, binary perm, and $(-1,1)$ variants, plus right alternative), identifying rigid algebras and providing new degeneration results. Together, the algebraic and geometric classifications yield a comprehensive picture of the structure and deformation behavior of $3$-dimensional right alternative superalgebras and their principal subvarieties, with broad implications for nonassociative algebraic geometry. The work also offers a computational framework applicable to similar classifications in related algebraic varieties.

Abstract

The algebraic and geometric classifications of complex $3$-dimensional right alternative superalgebras are given. As a byproduct, we have the algebraic and geometric classification of the variety of $3$-dimensional $\mathfrak{perm}$, binary $\mathfrak{perm}$, associative, binary associative, $\big(-1,1\big)$-, and binary $\big(-1,1\big)$-superalgebras.

The algebraic and geometric classification of right alternative superalgebras

TL;DR

The study delivers a complete algebraic and geometric classification of complex -dimensional right alternative superalgebras. It develops a general method based on the Jordan superalgebra framework and the cohomology space to enumerate all right alternative structures of a fixed type, then specializes to types and to produce explicit lists of nonisomorphic algebras. Geometrically, it analyzes degenerations, orbit closures, and irreducible components across several principal subvarieties (associative, perm, binary perm, and variants, plus right alternative), identifying rigid algebras and providing new degeneration results. Together, the algebraic and geometric classifications yield a comprehensive picture of the structure and deformation behavior of -dimensional right alternative superalgebras and their principal subvarieties, with broad implications for nonassociative algebraic geometry. The work also offers a computational framework applicable to similar classifications in related algebraic varieties.

Abstract

The algebraic and geometric classifications of complex -dimensional right alternative superalgebras are given. As a byproduct, we have the algebraic and geometric classification of the variety of -dimensional , binary , associative, binary associative, -, and binary -superalgebras.
Paper Structure (23 sections, 33 theorems, 10 equations)

This paper contains 23 sections, 33 theorems, 10 equations.

Table of Contents

  1. The algebraic classification of superalgebras
  2. Preliminaries: the method for algebraic classification
  3. Right alternative superalgebras of type $(1,2)$
  4. The classification Theorem A1
  5. The proof of Theorem A1
  6. Right alternative superalgebras of type $(2,1)$
  7. The classification Theorem A2
  8. The proof of Theorem A2
  9. Corollaries: $\Omega$- and $($binary-$\Omega)$-superalgebras
  10. $\mathfrak{perm}$ and binary $\mathfrak{perm}$ superalgebras
  11. Associative and binary associative superalgebras
  12. $(-1,1)$- and binary $(-1,1)$-superalgebras
  13. The geometric classification of superalgebras
  14. Preliminaries: degenerations and geometric classification
  15. The geometric classification
  16. ...and 8 more sections

Key Result

Lemma 3

Let $\left( \rm{A},\cdot \right)$ be a Jordan superalgebra and $\theta ,\vartheta \in {\rm Z}^{2}\left( \rm{A},\rm{A}\right)$. Then $\left( \rm{A},\ast _{\theta }\right)$ and $\left( \rm{A},\ast _{\vartheta }\right)$ are isomorphic if and only if there is a $\phi \in \rm{Aut}\left( \rm{A}\right)$ wi

Theorems & Definitions (52)

  • Definition 1
  • Definition 2
  • Lemma 3
  • Proposition 4
  • Proposition 5
  • Corollary 6
  • Corollary 7
  • Definition 8
  • Definition 9: see KS
  • Corollary 10
  • ...and 42 more