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Derivations in Dialgebras Derivations and Biderivations in Dialgebras

Gabriel Gustavo Restrepo-Sánchez, José Gregorio Rodríguez-Nieto, Olga Patricia Salazar-Díaz, Andrés Sarrazola-Alzate, Raúl Velásquez

TL;DR

This work develops a systematic operator framework for dialgebras by introducing diderivations and studying their interaction with classical derivations and inner operators. It proves that derivations form a Lie subalgebra and that inner derivations form an ideal, while diderivations are characterized by multiplicative-operator identities and interact with derivations in a controlled way, yielding a Leibniz algebra structure on biderivations. The paper provides explicit complete classifications of diderivations for all 2D and 3D complex dialgebras, with Dias$_3^{16}$ requiring a detailed parameter-dependent analysis that can yield up to six independent directions. It then constructs the Leibniz algebra $\pazocal{B}ider(\mathcal{D})$ and analyzes its ideal structure, before presenting a thorough treatment of the dialgebra $K[x,y]$, giving exact formulas for diderivations and identifying inner ones via $Ad_{p(x,y)}$. The results unify and extend known derivation-like operators in the dialgebra setting and pave the way for higher-dimensional and cohomological investigations, including potential deformation-theoretic applications.

Abstract

The concepts of derivations and right derivations for Leibniz algebras and $K$-B quasi-Jordan algebras naturally arise from the inner derivations determined by their algebraic structures. In this paper we introduce the corresponding analogues for dialgebras, which we call diderivations, and examine their properties in relation to antiderivations and right derivations. Our approach is based on the study of multiplicative operators and on the construction of the Leibniz algebra generated by biderivations, thereby providing a systematic framework that unifies several types of derivation-like operators. In addition to the general theory, we present a complete classification of the spaces of diderivations for dialgebras of dimensions two and three, obtained through explicit computations. These low-dimensional results not only exemplify the general constructions but also reveal structural patterns that inform possible extensions to higher dimensions and more intricate algebraic contexts.28

Derivations in Dialgebras Derivations and Biderivations in Dialgebras

TL;DR

This work develops a systematic operator framework for dialgebras by introducing diderivations and studying their interaction with classical derivations and inner operators. It proves that derivations form a Lie subalgebra and that inner derivations form an ideal, while diderivations are characterized by multiplicative-operator identities and interact with derivations in a controlled way, yielding a Leibniz algebra structure on biderivations. The paper provides explicit complete classifications of diderivations for all 2D and 3D complex dialgebras, with Dias requiring a detailed parameter-dependent analysis that can yield up to six independent directions. It then constructs the Leibniz algebra and analyzes its ideal structure, before presenting a thorough treatment of the dialgebra , giving exact formulas for diderivations and identifying inner ones via . The results unify and extend known derivation-like operators in the dialgebra setting and pave the way for higher-dimensional and cohomological investigations, including potential deformation-theoretic applications.

Abstract

The concepts of derivations and right derivations for Leibniz algebras and -B quasi-Jordan algebras naturally arise from the inner derivations determined by their algebraic structures. In this paper we introduce the corresponding analogues for dialgebras, which we call diderivations, and examine their properties in relation to antiderivations and right derivations. Our approach is based on the study of multiplicative operators and on the construction of the Leibniz algebra generated by biderivations, thereby providing a systematic framework that unifies several types of derivation-like operators. In addition to the general theory, we present a complete classification of the spaces of diderivations for dialgebras of dimensions two and three, obtained through explicit computations. These low-dimensional results not only exemplify the general constructions but also reveal structural patterns that inform possible extensions to higher dimensions and more intricate algebraic contexts.28

Paper Structure

This paper contains 14 sections, 24 theorems, 103 equations, 3 tables.

Table of Contents

  1. Preliminaries
  2. Derivations and Diderivations over Dialgebras
  3. Derivations over Dialgebras
  4. Diderivations over Dialgebras
  5. Diderivations of Low Dimensional Dialgebras
  6. Diderivations of Two Dimensional Dialgebras
  7. Diderivations of Three Dimensional Dialgebras
  8. Compact case table $Dias_3^{16}$
  9. The Leibniz Algebra $\pazocal{B}ider(\mathcal{D})$
  10. Derivations and Diderivations of the Dialgebra $K[x,y]$
  11. Conclusion and Further Work
  12. Python Code
  13. Three-Dimensional Dialgebras
  14. Diderivations over $Dias_3^{16}$

Key Result

Proposition 1.15

Let $\mathcal{D}$ be a dialgebra and $\mathcal{D}_{as}$ be its canonical associative algebra. There exist a $\mathcal{D}_{as}$-bimodule structure over $\mathcal{D}$ and a morphism of $\mathcal{D}_{as}$-bimodules $\varphi:\mathcal{D}\rightarrow\mathcal{D}_{as}$ such that the dialgebra structure of $\

Theorems & Definitions (56)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • Example 1.9
  • Example 1.10
  • ...and 46 more