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 $\mathfrak{g}={\mathfrak{g}}_{\bar{0}}\oplus {\mathfrak{g}}_{\bar{1}}$ such that ${\mathfrak{g}}_{\bar{1}}$ is a weak filiform ${\mathfrak{g}}_{\bar{0}}$-module (weak filiform type). We introduce this concept after having proved that the unique non-zero odd-quadratic Lie superalgebra $(\mathfrak{g},B)$ with ${\mathfrak{g}}_{\bar{1}}$ a filiform ${\mathfrak{g}}_{\bar{0}}$-module is the abelian $2$-dimensional Lie superalgebra $\mathfrak{g}={\mathfrak{g}}_{\bar{0}}\oplus {\mathfrak{g}}_{\bar{1}}$ such that $\mbox{\rm dim }{\mathfrak g}_{\bar 0}=\mbox{\rm dim }{\mathfrak g}_{\bar 1}=1$. Let us note that in this context the role of the center of $\mathfrak{g}$ 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.

References in corpus (2)

• On the structure of symmetric self-dual Lie algebras
• Wess-Zumino-Novikov-Witten Models Based on Lie Superalgebras