On First and Second Cohomology Groups for BBW Parabolics for Classical Lie Superalgebras

Abstract

Let ${\mathfrak g}$ be a classical simple Lie superalgebra. In this paper, the author studies the cohomology groups for the subalgebra $\mathfrak{n}^{+}$ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu. These classical Lie superalgebras have a triangular decomposition ${\mathfrak g}={\mathfrak n}^{-}\oplus {\mathfrak f} \oplus {\mathfrak n}^{+}$ where $\mathfrak f$ is a detecting subalgebra as introduced by Boe, Kujawa and Nakano. It is shown that there exists a Hochschild-Serre spectral sequence that collapses for all infinite families of classical simple Lie superalgebras. This enables the author to explicitly compute the first and second cohomologies for ${\mathfrak n}^{+}$. The paper concludes with tables listing the weight space decompositions and dimension formulas for these cohomology groups.