  We prove the analog of Kostant's Theorem on Lie algebra cohomology in the
context of quantum groups. We prove that Kostant's cohomology formula holds for
quantum groups at a generic parameter $q$, recovering an earlier result of
Malikov in the case where the underlying semisimple Lie algebra $\mathfrak{g} =
\mathfrak{sl}(n)$. We also show that Kostant's formula holds when $q$ is
specialized to an $\ell$-th root of unity for odd $\ell \ge h-1$ (where $h$ is
the Coxeter number of $\mathfrak{g}$) when the highest weight of the
coefficient module lies in the lowest alcove. This can be regarded as an
extension of results of Friedlander-Parshall and Polo-Tilouine on the
cohomology of Lie algebras of reductive algebraic groups in prime
characteristic.
