[Paper Review] On the classification of complex Leibniz superalgebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and nilindex $n+m$

Sergio Albeverio, B. A. Omirov+1 more|ArXiv.org|Jun 29, 2008

Advanced Topics in Algebra8 references3 citations

TL;DR

This paper classifies complex Leibniz superalgebras with characteristic sequence $(n-1,1|m_1,\dots,m_k)$ and nilindex $n+m$. It proves that if $m_2 \neq 0$, such superalgebras cannot achieve nilindex $n+m$, reducing the classification to the case where the odd part has a single Jordan block, which was previously solved. The result simplifies the classification problem by eliminating non-filiform cases.

ABSTRACT

In this work we investigate the complex Leibniz superalgebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m.$ We prove that such superalgebras with the condition $m_2 eq0$ have nilindex less than $n+m$. Therefore the complete classification of Leibniz algebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m$ is reduced to the classification of filiform Leibniz superalgebras of nilindex equal to $n+m,$ which was provided in \cite{AOKh} and \cite{GKh}.

Motivation & Objective

• To classify complex Leibniz superalgebras with characteristic sequence $(n-1,1|m_1,\dots,m_k)$ and nilindex $n+m$.
• To determine whether such superalgebras can achieve maximal nilindex when the odd part has more than one Jordan block.
• To reduce the classification problem to previously solved cases by proving that $m_2 \neq 0$ leads to nilindex less than $n+m$.
• To clarify the structural constraints imposed by the characteristic sequence and nilindex in Leibniz superalgebras.

Proposed method

• Analysis of the descending central series of Leibniz superalgebras to determine nilindex.
• Construction of adapted bases to simplify multiplication tables and analyze nilpotency.
• Use of the Leibniz superidentity to derive constraints on structure constants.
• Application of the right annihilator ideal to identify elements that vanish in higher powers.
• Comparison of coefficients in nested bracket expressions to derive contradictions when $m_2 \neq 0$, leading to nilindex reduction.
• Reduction of the problem to the filiform case $(n-1,1|m)$, already classified in prior works [3] and [7].

Experimental results

Research questions

• RQ1 Can Leibniz superalgebras with characteristic sequence $(n-1,1|m_1,\dots,m_k)$ and $m_2 \neq 0$ achieve nilindex $n+m$?
• RQ2 What structural constraints arise from the condition $m_2 \neq 0$ in such superalgebras?
• RQ3 Does the nilindex of a Leibniz supergebra with characteristic sequence $(n-1,1|m_1,\dots,m_k)$ drop below $n+m$ when $m_2 \neq 0$?
• RQ4 Is the classification of such superalgebras with maximal nilindex reducible to the filiform case?
• RQ5 What role do the right annihilator and descending central series play in determining nilindex?

Key findings

• Leibniz superalgebras with characteristic sequence $(n-1,1|m_1,\dots,m_k)$ and $m_2 \neq 0$ have nilindex strictly less than $n+m$.
• The condition $m_2 \neq 0$ leads to a contradiction in the structure constants when assuming nilindex $n+m$, as shown by coefficient comparison in nested brackets.
• The nilindex reduction arises from the non-vanishing condition $(\beta_{2,2}, \gamma_{2,2}) \neq (0,0)$, which cannot be satisfied under the assumed nilindex.
• The right annihilator ideal contains elements of the form $[a,b] + (-1)^{ab}[b,a]$, which are essential in deriving nilpotency constraints.
• The classification of Leibniz superalgebras with nilindex $n+m$ and characteristic sequence $(n-1,1|m_1,\dots,m_k)$ reduces to the filiform case $(n-1,1|m)$, which was previously classified.
• The maximal nilindex $n+m$ is only achievable when the odd part has a single Jordan block, i.e., $k=1$ and $m_1 = m$.