[Paper Review] n-Dimensional filiform Leibniz algebras of length (n-1) and their derivations

Sergio Albeverio, Sh. A. Ayupov+2 more|ArXiv.org|Mar 21, 2007

Advanced Topics in Algebra4 references4 citations

TL;DR

This paper classifies complex $n$-dimensional filiform Leibniz algebras of length $n-1$ and fully describes their derivation algebras. By analyzing natural gradations with maximal length $(n-1)$, the authors identify four isomorphism classes ($M_1(k), M_2, M_3, M_4$) and compute the dimensions of their first cohomology and second coboundary spaces, providing a complete structural and cohomological characterization of these algebras.

ABSTRACT

In this work $n$-dimensional filiform Leibniz algebras admitting a gradation of length $(n-1)$ are classified. Derivations of such algebras are also described.

Motivation & Objective

• To classify complex $n$-dimensional filiform Leibniz algebras that admit a connected gradation of length $n-1$.
• To describe the full derivation algebra of such algebras, including inner and outer derivations.
• To compute the dimensions of the first cohomology group $H^1(L,L)$ and the second coboundary space $B^2(L,L)$ for each isomorphism class.
• To extend the classification of naturally graded filiform Leibniz algebras by including those with length $n-1$ rather than maximal length $n$.
• To provide a complete structural and cohomological description of these algebras using homogeneous bases and gradation-induced constraints on structural constants.

Proposed method

• The classification is based on the existence of a connected $\mathbb{Z}$-gradation with $n-1$ non-zero homogeneous components, implying a specific structure of the lower central series and associated graded algebra $grL$.
• The authors use the natural gradation of filiform Leibniz algebras to define a homogeneous basis, reducing the classification problem to solving polynomial equations for structural constants under gradation constraints.
• Derivations are analyzed by decomposing them into components $d_j$ acting on basis elements $y_i$, with the action determined by linear maps $d_j(y_i) = \beta_{j,i} y_{i+j}$, and by introducing auxiliary maps $h_1, h_2, h_0$ to capture non-trivial derivations.
• The derivation space is constructed as a direct sum of maps $h_1, h_2, h_0$ and $d_j$ for $j$ in appropriate ranges, with explicit formulas for their action on basis vectors.
• Cohomological invariants are computed via the quotient $H^1(L,L) = \text{Der}(L)/\text{Inn}(L)$, and the dimension of $B^2(L,L)$ is derived from the image of the coboundary map.
• The classification is completed by distinguishing cases based on the value of $k$ in the multiplication rule $[y_1, y_k] = \gamma_{1,k} y_n$, leading to four distinct isomorphism classes.

Experimental results

Research questions

• RQ1 Which $n$-dimensional complex filiform Leibniz algebras admit a connected gradation of length $n-1$?
• RQ2 What is the complete isomorphism class structure of filiform Leibniz algebras of length $n-1$?
• RQ3 What is the dimension of the derivation algebra $\text{Der}(L)$ for such algebras?
• RQ4 How do the first cohomology group $H^1(L,L)$ and the second coboundary space $B^2(L,L)$ vary across the isomorphism classes?
• RQ5 What is the precise structure of the derivation algebra, including the roles of inner, outer, and homogeneous derivations?

Key findings

• The classification yields four isomorphism classes: $M_1(k)$, $M_2$, $M_3$, and $M_4$, depending on the value of $k$ in the multiplication rule $[y_1, y_k] = \gamma_{1,k} y_n$.
• For $M_1(k)$ with $2k-2 \leq n-1$, $\dim H^1(M_1(k), M_1(k)) = n-2$ and $\dim B^2(M_1(k), M_1(k)) = n^2 - n + 2$.
• For $M_1(k)$ with $2k-2 \geq n$, $\dim H^1(M_1(k), M_1(k)) = n-1$ and $\dim B^2(M_1(k), M_1(k)) = n^2 - n + 1$.
• For $M_2$ and $M_3$, $\dim H^1(M_i, M_i) = n-2$ and $\dim B^2(M_i, M_i) = n^2 - n + 2$.
• For $M_4$, $\dim H^1(M_4, M_4) = n-3$ and $\dim B^2(M_4, M_4) = n^2 - n + 3$, indicating a more complex derivation structure.
• The derivation algebra of each algebra is spanned by a basis of maps $h_1, h_2, h_0$ and $d_j$, with explicit formulas for their action on the homogeneous basis vectors.