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

Abstract

The aim of this work is to study a very special family of odd-quadratic Lie superalgebras g=g0¯⊕g1¯ such that g1¯ is a weak filiform g0¯-module (weak filiform type). We introduce this concept after having proved that the unique non-zero odd-quadratic Lie superalgebra (g,B) with g1¯ a filiform g0¯-module is the abelian 2-dimensional Lie superalgebra g=g0¯⊕g1¯ such that dimg0¯=dimg1¯=1. Let us note that in this context the role of the center of g 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.