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 | References | Similar Articles | Additional Information

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.

**[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, 1-21. 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. 471-506. MR**1103601 (92g:17012)****[FP]**Eric M. Friedlander and Brian J. Parshall,*Cohomology of infinitesimal and discrete groups*, Math. Ann.**273**(1986), no. 3, 353-374. 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, 179-198. 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 Bernstein-Gelfand-Gelfand resolution for Kac-Moody algebras and quantized enveloping algebras*, Transform. Groups**12**(2007), no. 4, 647-655. MR**2365438 (2008k:17033)****[Hoc]**Gerald Hochschild,*Relative homological algebra*, Trans. Amer. Math. Soc.**82**(1956), 246-269. MR**0080654 (18:278a)****[HK]**Jin Hong and Seok-Jin 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. 1-211. 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,*Kac-Moody 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. 623-643. 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,*Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over for representations with -small weights*, Astérisque (2002), no. 280, Cohomology of Siegel varieties, 97-135. MR**1944175 (2003j:17027)****[RC]**Alvany Rocha-Caridi,*Splitting criteria for -modules induced from a parabolic and the Berňsteĭn-Gel fand-Gel fand resolution of a finite-dimensional, irreducible -module*, Trans. Amer. Math. Soc.**262**(1980), no. 2, 335-366. MR**586721 (82f:17006)****[UGA]**University of Georgia VIGRE Algebra Group,*On Kostant's theorem for Lie algebra cohomology*, Cont. Math. (2009), no. 478, 39-60.

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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.