  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.

Representation Theory;Rings and Algebras