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The algebraic and geometric classification of antiassociative algebras

Renato Fehlberg Júnior, Ivan Kaygorodov, Crislaine Kuster

Abstract

This paper is devoted to the complete algebraic and geometric classification of complex 4 and 5-dimensional antiassociative algebras. In particular, we proved that the variety of complex 4-dimensional antiassociative algebras has dimension 12 and it is defined by three irreducible components (in particular, there is only 1 rigid algebra in this variety); the variety of complex 5-dimensional antiassociative algebras has dimension 24 and it is defined by 8 irreducible components (in particular, there are only 4 rigid algebras in this variety).

The algebraic and geometric classification of antiassociative algebras

Abstract

This paper is devoted to the complete algebraic and geometric classification of complex 4 and 5-dimensional antiassociative algebras. In particular, we proved that the variety of complex 4-dimensional antiassociative algebras has dimension 12 and it is defined by three irreducible components (in particular, there is only 1 rigid algebra in this variety); the variety of complex 5-dimensional antiassociative algebras has dimension 24 and it is defined by 8 irreducible components (in particular, there are only 4 rigid algebras in this variety).
Paper Structure (18 sections, 3 theorems, 14 equations)

This paper contains 18 sections, 3 theorems, 14 equations.

Table of Contents

  1. The algebraic classification of antiassociative algebras
  2. Method of classification of nilpotent algebras
  3. Notations
  4. The algebraic classification of $4$-dimensional non-2-step nilpotent antiassociative algebras
  5. $1$-dimensional central extensions of ${\mathcal{A}}_{3,1}$
  6. The algebraic classification of $5$-dimensional non-2-step nilpotent antiassociative algebras
  7. $2$-dimensional central extensions of ${\mathcal{A}}_{3,1}$
  8. $1$-dimensional central extensions of $4$-dimensional $2$-step nilpotent algebras
  9. Central extensions of ${\mathcal{A}}_{4,1}$
  10. Central extensions of ${\mathcal{A}}_{4,2}$
  11. Central extensions of ${\mathcal{A}}_{4,7}$
  12. Central extensions of ${\mathcal{A}}_{4,12}$
  13. Central extensions of ${\mathcal{A}}_{4,14}^{\lambda}$
  14. Classification theorem
  15. The geometric classification of nilpotent algebras
  16. ...and 3 more sections

Key Result

Lemma 1

Let ${\bf A}$ be an $n$-dimensional antiassociative algebra such that $\dim (\operatorname{Ann}({\bf A}))=m\neq0$. Then there exists, up to isomorphism, a unique $(n-m)$-dimensional antiassociative algebra ${\bf A}'$ and a bilinear map $\theta \in {\rm Z^2}({\bf A'}, {\mathbb V})$ with $\operatornam

Theorems & Definitions (6)

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