Notes on Leibniz $n$-algebras

Technical Summary: Notes on Leibniz n-Algebras

Problem Statement
The paper investigates the structural and homological properties of Leibniz $n$-algebras, which generalize Lie $n$-algebras, Leibniz algebras, and Nambu algebras. Specifically, the authors analyze the behavior of two fundamental functors acting between categories of Leibniz algebras with different arities: the generalized forgetful functor ($U_n^p$) and the Daletskii-Takhtajan functor ($D_q^p$). The central problem is to determine whether these functors preserve "perfect" objects (algebras equal to their own derived subalgebra), actions, and crossed modules. Furthermore, the paper seeks to establish relationships between the homology and universal central extensions of Leibniz $n$-algebras and their associated Leibniz $p$-algebras (where $p = k(n-1) + 1$).

Methodology
The authors employ category-theoretic methods and explicit algebraic constructions within the framework of non-skew-symmetric $n$-ary algebras.

1. Functorial Analysis: The study defines and analyzes the generalized forgetful functor $U_n^p: n\text{Lb} \to p\text{Lb}$, which assigns to a Leibniz $n$-algebra $L$ a Leibniz $p$-algebra with the same underlying vector space but an iterated $p$-ary bracket. Similarly, the generalized Daletskii-Takhtajan functor $D_q^p: q\text{Lb} \to p\text{Lb}$ is defined, mapping a Leibniz $q$-algebra to a Leibniz $p$-algebra via tensor powers ($L^{\otimes \kappa}$).
2. Preservation Properties: The authors rigorously test whether these functors preserve the property of being a perfect object (defined as $L = [L^n]$), as well as the structures of actions and crossed modules.
3. Counter-Example Construction: To address the limitations of the Daletskii-Takhtajan functor, the authors construct specific counter-examples using finite-dimensional Lie and Leibniz algebras to demonstrate the failure of preservation of perfectness.
4. Homological Connections: The paper constructs chain complex morphisms between the Leibniz homology of a Leibniz algebra $L$ and the Leibniz $n$-homology of $U_n(L)$.
5. Universal Central Extensions: Using the non-abelian tensor power construction ($L^*_n$), the authors analyze the universal central extensions of perfect Leibniz $n$-algebras and compare them with those in the category of Leibniz $p$-algebras.

Key Contributions and Results

• Preservation by Forgetful Functors: The paper proves that the generalized forgetful functor $U_n^p$ is an exact functor that preserves and reflects perfect objects (Theorem 3.5). Consequently, if a Leibniz $n$-algebra is perfect, its image under $U_n^p$ is a perfect Leibniz $p$-algebra.
• Extension to Crossed Modules: The authors demonstrate that $U_n^p$ extends to a functor between categories of crossed modules ($X_n\text{Lb} \to X_p\text{Lb}$). They prove that if $\mu: L \to P$ is a crossed module of Leibniz $n$-algebras, then $U_n^p(\mu)$ is a crossed module of Leibniz $p$-algebras (Proposition 4.5). This includes the preservation of actions (Lemma 4.2).
• Failure of Daletskii-Takhtajan Preservation: In contrast to the forgetful functor, the paper establishes that the Daletskii-Takhtajan functor $D_n$ (and its generalization $D_q^p$) does not preserve perfect objects in general. The authors provide explicit counter-examples (Example 5.1) involving a perfect Lie 3-algebra and specific perfect Leibniz algebras where the resulting tensor product algebra is not perfect.
• Homological Morphisms: A morphism of chain complexes $h_*$ is constructed between the homology of a Leibniz algebra $L$ and the homology of the Leibniz $n$-algebra $U_n(L)$ with coefficients in a trivial co-representation. The authors show this map is an isomorphism in dimension 0 and an epimorphism if $L$ is perfect (Proposition 6.1).
• Universal Central Extensions: The paper analyzes the relationship between the universal central extension of a perfect Leibniz $n$-algebra $L$ and its universal central extension in the category of Leibniz $p$-algebras. It is shown that applying $U_n^p$ to the universal central extension of $L$ yields a central extension in $p\text{Lb}$, leading to a unique surjective homomorphism $\phi: L^*_p \to L^*_n$ (Section 7).

Significance and Claims
The paper claims to establish further structural properties of the functors connecting categories of Leibniz algebras of varying arities. By proving that the forgetful functor preserves perfectness and crossed module structures, the authors provide a robust tool for transferring results from lower arity (e.g., Leibniz algebras) to higher arity Leibniz $n$-algebras. Conversely, by demonstrating the failure of the Daletskii-Takhtajan functor to preserve perfectness, the paper delineates the boundaries of homological transfer via tensor powers.

The significance lies in the application of these functorial properties to homology and universal central extensions. The results clarify how the non-abelian tensor power (a key tool for describing universal central extensions) behaves under changes in arity. Specifically, the paper provides a concrete description of the relationship between the universal central extensions of a perfect Leibniz $n$-algebra in the categories of $n$-algebras and $p$-algebras, offering a commutative diagram that links their respective homology groups and tensor powers. The work serves to unify the structure theory of Leibniz $n$-algebras with established results in Leibniz algebras and Lie $n$-algebras, while highlighting distinct behaviors arising from the non-skew-symmetric nature of Leibniz structures.