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), 8599
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 , 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 FriedlanderParshall and PoloTilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.
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 [BNPP]
 Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, and Cornelius Pillen, Cohomology for quantum groups via the geometry of the nullcone, preprint, 2007.
 [BNW]
 Brian D. Boe, Daniel K. Nakano, and Emilie Wiesner, Category for the Virasoro algebra: cohomology and Koszulity, Pacific J. Math. 234 (2008), no. 1, 121. MR 2375311
 [DCK]
 Corrado De Concini and Victor G. Kac, Representations of quantum groups at roots of , Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), Progress in Mathematics, vol. 92, Birkhäuser, Boston, MA, 1990, pp. 471506. MR 1103601 (92g:17012)
 [FP]
 Eric M. Friedlander and Brian J. Parshall, Cohomology of infinitesimal and discrete groups, Math. Ann. 273 (1986), no. 3, 353374. MR 824427 (87e:22026)
 [GK]
 Victor Ginzburg and Shrawan Kumar, Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), no. 1, 179198. MR 1201697 (94c:17026)
 [GW]
 Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831 (99b:20073)
 [HeK]
 István Heckenberger and Stefan Kolb, On the BernsteinGelfandGelfand resolution for KacMoody algebras and quantized enveloping algebras, Transform. Groups 12 (2007), no. 4, 647655. MR 2365438 (2008k:17033)
 [Hoc]
 Gerald Hochschild, Relative homological algebra, Trans. Amer. Math. Soc. 82 (1956), 246269. MR 0080654 (18:278a)
 [HK]
 Jin Hong and SeokJin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971 (2002m:17012)
 [Hum]
 James E. Humphreys, Representations of semisimple Lie algebras in the BGG category , Graduate Studies in Mathematics, vol. 94, American Mathematical Society, Providence, RI, 2008. MR 2428237 (2009f:17013)
 [Jan1]
 Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532 (96m:17029)
 [Jan2]
 , Representations of algebraic groups, second ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057 (2004h:20061)
 [Jan3]
 , Nilpotent orbits in representation theory, Lie theory, Progress in Mathematics, vol. 228, Birkhäuser, Boston, MA, 2004, pp. 1211. MR 2042689 (2005c:14055)
 [Kna]
 Anthony W. Knapp, Lie groups, Lie algebras, and cohomology, Mathematical Notes, vol. 34, Princeton University Press, Princeton, NJ, 1988. MR 938524 (89j:22034)
 [Kum]
 Shrawan Kumar, KacMoody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser, Boston, MA, 2002. MR 1923198 (2003k:22022)
 [Mal]
 Fyodor Malikov, Quantum groups: singular vectors and BGG resolution, Infinite analysis, Parts A and B (Kyoto, 1991), Adv. Ser. Math. Phys., vol. 16, World Scientific, River Edge, NJ, 1992, pp. 623643. MR 1187567 (93k:17034)
 [PW]
 Brian Parshall and Jian Pan Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), no. 439. MR 1048073 (91g:16028)
 [PT]
 Patrick Polo and Jacques Tilouine, BernsteinGelfandGelfand complexes and cohomology of nilpotent groups over for representations with small weights, Astérisque (2002), no. 280, Cohomology of Siegel varieties, 97135. MR 1944175 (2003j:17027)
 [RC]
 Alvany RochaCaridi, Splitting criteria for modules induced from a parabolic and the BerňsteĭnGel fandGel fand resolution of a finitedimensional, irreducible module, Trans. Amer. Math. Soc. 262 (1980), no. 2, 335366. MR 586721 (82f:17006)
 [UGA]
 University of Georgia VIGRE Algebra Group, On Kostant's theorem for Lie algebra cohomology, Cont. Math. (2009), no. 478, 3960.
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Additional Information
University of Georgia VIGRE algebra group
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
DOI:
http://dx.doi.org/10.1090/S0002993909100394
PII:
S 00029939(09)100394
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, JaeHo 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.
