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Notes on Leibniz $n$-algebras

José Manuel Casas, Emzar Khmaladze, Manuel Ladra

TL;DR

The paper analyzes generalized forgetful functors between Leibniz n-algebras and p-algebras and the Daletskii-Takhtajan functor, proving that forgetful functors preserve perfect objects and crossed modules while the DT functor does not always preserve perfection. It develops a homology framework relating Leibniz n-algebra homology to classical Leibniz homology and studies how universal central extensions transfer across these categories for perfect algebras. Through explicit constructions and counterexamples, it delineates when central extension data can be transported along these functors and highlights the interaction between arity and homological invariants. These results provide structural tools for comparing higher arity and central extension theory in related algebraic categories.

Abstract

We analyze behaviors of generalized forgetful and Daletskii-Takhtajan's functors on perfect objects and crossed modules of Leibniz $n$-algebras. Then we give applications to homology and universal central extensions of Leibniz $n$-algebras.

Notes on Leibniz $n$-algebras

TL;DR

The paper analyzes generalized forgetful functors between Leibniz n-algebras and p-algebras and the Daletskii-Takhtajan functor, proving that forgetful functors preserve perfect objects and crossed modules while the DT functor does not always preserve perfection. It develops a homology framework relating Leibniz n-algebra homology to classical Leibniz homology and studies how universal central extensions transfer across these categories for perfect algebras. Through explicit constructions and counterexamples, it delineates when central extension data can be transported along these functors and highlights the interaction between arity and homological invariants. These results provide structural tools for comparing higher arity and central extension theory in related algebraic categories.

Abstract

We analyze behaviors of generalized forgetful and Daletskii-Takhtajan's functors on perfect objects and crossed modules of Leibniz -algebras. Then we give applications to homology and universal central extensions of Leibniz -algebras.
Paper Structure (7 sections, 8 theorems, 48 equations)

This paper contains 7 sections, 8 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Leibniz $n$-algebras
  3. $\operatorname{\frak{U}}_p^q$ functors and perfect objects
  4. $\operatorname{\frak{U}}_p^q$ functors and crossed modules
  5. ${\frak D}_q^p$ functors and perfect objects
  6. A bit about the homology of Leibniz $n$-algebras
  7. $\operatorname{\frak{U}}_n^p$ functors and universal central extensions

Key Result

Proposition 3.2

Let $A$ be a vector space and $\omega_n : A^{\otimes n} \to A$ be an $n$-linear operation. Let $\omega_p :A^{\otimes p} \to A$ be given by Then, a derivation $f:A\to A$ with respect to $\omega_n$ is a derivation with respect to $\omega_p$.

Theorems & Definitions (29)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • ...and 19 more