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The algebraic and geometric classification of nilpotent binary and mono Leibniz algebras

Kobiljon Abdurasulov, Ivan Kaygorodov, Abror Khudoyberdiyev

TL;DR

The article achieves a comprehensive algebraic and geometric classification of complex nilpotent binary Leibniz algebras in dimension 5 and nilpotent mono Leibniz algebras in dimension 4, building on the Skjelbred–Sund central extension framework. By computing second cohomology spaces ${\rm H}^{2}_{\rm BL}$ and ${\rm H}^{2}_{\rm ML}$, identifying Aut‑orbits on Grassmannians, and analyzing annihilator components, the authors construct all non‑split central extensions and produce explicit lists of nonisomorphic algebras, including ${\bf B}_{01},\dots, {\bf B}_{14}$ in 5D and ${\mathbb M}_{01},\dots, {\mathbb M}_{22}$ in 4D. They further perform a geometric (degeneration) analysis, describing the irreducible components and rigid algebras in the corresponding varieties: for 5D nilpotent binary Leibniz algebras there are 10 components with a unique rigid algebra; for 4D nilpotent algebras of nil-index 3 there are two components with no rigid algebras; for 4D nilpotent mono Leibniz algebras there are 3 components with a single rigid algebra. Altogether, the work yields a complete algebraic and geometric taxonomy of these non‑associative algebras, providing a foundation for understanding their deformation theory and orbit closures. The results have implications for the broader study of nilpotent non‑associative algebras and their degenerations in low dimensions.

Abstract

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional nilpotent binary Leibniz and $4$-dimensional nilpotent mono Leibniz algebras. As a corollary, we have the complete algebraic and geometric classification of complex $4$-dimensional nilpotent algebras of nil-index $3$.

The algebraic and geometric classification of nilpotent binary and mono Leibniz algebras

TL;DR

The article achieves a comprehensive algebraic and geometric classification of complex nilpotent binary Leibniz algebras in dimension 5 and nilpotent mono Leibniz algebras in dimension 4, building on the Skjelbred–Sund central extension framework. By computing second cohomology spaces and , identifying Aut‑orbits on Grassmannians, and analyzing annihilator components, the authors construct all non‑split central extensions and produce explicit lists of nonisomorphic algebras, including in 5D and in 4D. They further perform a geometric (degeneration) analysis, describing the irreducible components and rigid algebras in the corresponding varieties: for 5D nilpotent binary Leibniz algebras there are 10 components with a unique rigid algebra; for 4D nilpotent algebras of nil-index 3 there are two components with no rigid algebras; for 4D nilpotent mono Leibniz algebras there are 3 components with a single rigid algebra. Altogether, the work yields a complete algebraic and geometric taxonomy of these non‑associative algebras, providing a foundation for understanding their deformation theory and orbit closures. The results have implications for the broader study of nilpotent non‑associative algebras and their degenerations in low dimensions.

Abstract

This paper is devoted to the complete algebraic and geometric classification of complex -dimensional nilpotent binary Leibniz and -dimensional nilpotent mono Leibniz algebras. As a corollary, we have the complete algebraic and geometric classification of complex -dimensional nilpotent algebras of nil-index .
Paper Structure (21 sections, 4 theorems, 24 equations)

This paper contains 21 sections, 4 theorems, 24 equations.

Table of Contents

  1. The algebraic classification of nilpotent binary and mono Leibniz algebras
  2. Preliminaries and basic definitions
  3. Method of classification of nilpotent algebras
  4. Notations
  5. Classification of 5-dimensional nilpotent binary Leibniz algebras
  6. Central extensions of ${\mathfrak N}_{03}$
  7. The classification theorem
  8. Classification of $4$-dimensional nilpotent mono Leibniz algebras
  9. Central extensions of ${\mathcal{N}}_{01}$
  10. Central extensions of ${\mathcal{N}}_{02}$
  11. Central extensions of ${\mathcal{N}}_{03}^{\alpha\neq0}$
  12. Central extensions of ${\mathcal{N}}_{03}^{0}$
  13. Central extensions of ${\mathcal{N}}_{04}$
  14. The classification theorem
  15. Classification of $4$-dimensional nilpotent algebras with nil-index $3$
  16. ...and 6 more sections

Key Result

Lemma 2

Let ${\bf A}_{\theta },{\bf A}_{\vartheta }\in {\bf E}\left( {\bf A},{\mathbb V}\right)$. Suppose that $\theta \left( x,y\right) = \sum_{i=1}^{s} \theta _{i}\left( x,y\right) e_{i}$ and $\vartheta \left( x,y\right) = \sum_{i=1}^{s} \vartheta _{i}\left( x,y\right) e_{i}$. Then the binary (resp. mon

Theorems & Definitions (9)

  • Definition
  • Definition 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof
  • Lemma 5