Odd-quadratic Lie superalgebras with a weak filiform module as an odd part

The aim of this work is to study a very special family of odd-quadratic Lie superalgebras such that is a weak filiform -module (weak filiform type). We introduce this concept after having proved that the unique non-zero odd-quadratic Lie superalgebra with a filiform -module is the abelian 2-dimensional Lie superalgebra such that . Let us note that in this context the role of the center of is crucial. Thus, we obtain an inductive description of odd-quadratic Lie superalgebras of weak filiform type via generalized odd double extensions. Moreover, we obtain the classification, up to isomorphism, for the smallest possible dimensions, that is, six and eight.The first author was supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Third author was supported by Junta de Extremadura and Fondo Europeo de Desarrollo Regional (GR21005 and IB18032) and by Agencia Estatal de Investigación (Spain), grant PID2020-115155GB-I00 (European FEDER support included, UE). Fourth author was supported by the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’ and by the PAI with project number FQM298