An analog of Kostant’s theorem for the cohomology of quantum groups

Abstract:

We prove the analog of Kostant’s Theorem on Lie algebra cohomology in the context of quantum groups. In particular, it is shown 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 analog of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.