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 g=sl(n). We also show that Kostant's formula holds when q is
specialized to an ℓ-th root of unity for odd ℓ≥h−1 (where h is
the Coxeter number of 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.Comment: 12 page