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Cohomology of quantum groups: An analog of Kostants Theorem

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 Added by Benjamin Jones
 Publication date 2008
  fields
and research's language is English




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We prove the analog of Kostants Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostants 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 Kostants 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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