Addendum: Generalized Spencer Cohomology and Filtered Deformations of $\mathbb{Z}$-graded Lie Superalgebras

Abstract

In [K] all the possibilities for the non-positive part $\mathfrak{g}_{\le 0}=\bigoplus_{j=-h}^{0}\mathfrak{g}_{j}$ of the associated graded Lie superalgebra $\mathfrak{g}=\bigoplus_{j\ge -h}\mathfrak{g}_j$ of a simple linearly compact Lie superalgebra $L$, for a "good" choice of its filtration, were obtained. In [CK2] the transitive $\mathbb{Z}$-graded $\mathfrak{g}$ with those $\mathfrak{g}_{\le 0}$ were classified, and in [CK1] and [K], in order to reconstruct $L$ from $\mathfrak{g}$, their filtered deformations were classified. However, some cases had been inadvertently omitted. In this note we take the opportunity to amend this.