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26 April 2024

» arxiv » 0809.0838

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Cohomology of quantum groups: An analog of Kostant's Theorem
University of Georgia VIGRE Algebra Group ; Irfan Bagci ; Brian D. Boe ; Leonard Chastkofsky ; Benjamin Connell ; Benjamin Jones ; Wenjing Li ; Daniel K. Nakano ; Kenyon J. Platt ; Jae-Ho Shin ; Caroline B. Wright ;
Date:	4 Sep 2008
Abstract:	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.
Source:	arXiv, 0809.0838
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