Cohomology of quantum groups: An analog of Kostant's Theorem

Details Cohomology of quantum groups: An analog of Kostant's Theorem Cohomology of quantum groups: An analog of Kostant's Theorem

2008-09-04

arxiv.org/abs/0809.0838v1

Proc. Amer. Math. Soc. 138 (2010), 85-99

Mathematics

Representation Theory

12 pages

Scientific paper

10.1090/S0002-9939-09-10039-4

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.

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