Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Author: University of Georgia VIGRE algebra group
Journal: Proc. Amer. Math. Soc. 138 (2010), 85-99
MSC (2000): Primary 20G42
Published electronically: August 25, 2009
MathSciNet review: 2550172
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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.

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University of Georgia VIGRE algebra group
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Received by editor(s): September 8, 2008
Received by editor(s) in revised form: September 28, 2008, and May 14, 2009
Published electronically: August 25, 2009
Additional Notes: The members of the UGA VIGRE Algebra Group are Irfan Bagci, Brian D. Boe, Leonard Chastkofsky, Benjamin Connell, Benjamin Jones, Wenjing Li, Daniel K. Nakano, Kenyon J. Platt, Jae-Ho Shin, and Caroline B. Wright.
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.