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$δ$-Leibniz algebras and related $δ$-type algebras

Jobir Adashev, Ivan Kaygorodov

Abstract

This paper introduces and investigates the structure of $δ$-Leibniz algebras, which serve as a parametric generalization of classical Leibniz algebras defined by a scalar $δ$. The authors define $δ$-Lie algebras, $δ$-Lie dialgebras, and $δ$-Zinbiel algebras via a standard procedure and study their fundamental properties. Furthermore, the research describes symmetric $δ$-Leibniz algebras and algebras of $δ$-biderivation type, establishing their connections with nilalgebras. Finally, these results provide a unified framework for understanding various classes of non-associative algebras through the lens of the $δ$ parameter.

$δ$-Leibniz algebras and related $δ$-type algebras

Abstract

This paper introduces and investigates the structure of -Leibniz algebras, which serve as a parametric generalization of classical Leibniz algebras defined by a scalar . The authors define -Lie algebras, -Lie dialgebras, and -Zinbiel algebras via a standard procedure and study their fundamental properties. Furthermore, the research describes symmetric -Leibniz algebras and algebras of -biderivation type, establishing their connections with nilalgebras. Finally, these results provide a unified framework for understanding various classes of non-associative algebras through the lens of the parameter.
Paper Structure (14 sections, 43 theorems, 18 equations)

This paper contains 14 sections, 43 theorems, 18 equations.

Table of Contents

  1. $\delta$-Lie algebras
  2. $\delta$-Lie dialgebras
  3. Antiassociative anti-right-commutative algebras
  4. $\delta$-Leibniz algebras
  5. Anti-Leibniz algebras
  6. $\delta$-Leibniz algebras
  7. $\delta$-Leibniz algebras via $\delta$-associative algebras and dialgebras
  8. Symmetric $\delta$-Leibniz algebras
  9. $\delta$-Zinbiel algebras
  10. $\delta$-Zinbiel algebras
  11. Anti-Zinbiel algebras
  12. Symmetric $\delta$-Leibniz admissible algebras
  13. Algebras of $\delta$-biderivation-type
  14. Examples of $2$- and $3$-dimensional algebras.

Key Result

Lemma 3

Let $L$ be a $\delta$-Lie algebra $(\delta\neq1).$ Then $L$ is antiassociative $($i.e., $(x,y,z)_{-1}=0$ and it is a dual mock Lie algebra Z17$).$

Theorems & Definitions (93)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 83 more