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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 triangulardecomposition ${\mathfrak g}={\mathfrak n}^{-}\oplus {\mathfrak f} \oplus{\mathfrak n}^{+}$ where $\mathfrak f$ is a detecting subalgebra as introducedby Boe, Kujawa and Nakano. It is shown that there exists a Hochschild-Serrespectral sequence that collapses for all infinite families of classical simpleLie superalgebras. This enables the author to explicitly compute the first andsecond cohomologies for ${\mathfrak n}^{+}$. The paper concludes with tableslisting the weight space decompositions and dimension formulas for thesecohomology groups.

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