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

DOI:
https://doi.org/10.1090/S0002-9939-09-10039-4

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 , recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra . We also show that Kostant's formula holds when is specialized to an -th root of unity for odd (where is the Coxeter number of ) 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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Additional Information

**University of Georgia VIGRE algebra group**

Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602

DOI:
https://doi.org/10.1090/S0002-9939-09-10039-4

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.