Cohomology rings for quantized enveloping algebras

Drupieski, Christopher

doi:10.1090/S0002-9939-2013-11659-X

Cohomology rings for quantized enveloping algebras
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by Christopher M. Drupieski PDF

Proc. Amer. Math. Soc. 141 (2013), 3739-3753 Request permission

Abstract:

We compute the structure of the cohomology ring for the quantized enveloping algebra (quantum group) $U_q$ associated to a finite-dimensional simple complex Lie algebra $\mathfrak {g}$. We show that the cohomology ring is generated as an exterior algebra by homogeneous elements in the same odd degrees as those that generate the cohomology ring for the Lie algebra $\mathfrak {g}$. Partial results are also obtained for the cohomology rings of the non-restricted quantum groups obtained from $U_q$ by specializing the parameter $q$ to a non-zero value $\varepsilon \in \mathbb {C}$.

References

H. H. Andersen and V. Mazorchuk, Category $\mathcal {O}$ for quantum groups, preprint, 2011.
H. H. Andersen, P. Polo, and K. Wen, Representations of quantum algebras, Invent. Math. 104 (1991), no. 1, 1–59. MR 1094046 (92e:17011)
D. W. Barnes, Spectral sequence constructors in algebra and topology, Mem. Amer. Math. Soc. 53 (1985), no. 317, viii+174. MR 776177 (86e:55032)
Nicolas Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. Translated from the French; Reprint of the 1989 English translation. MR 1727221
—, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, Translated from the 1968 French original by Andrew Pressley. MR 1890629 (2003a:17001)
K. A. Brown and K. R. Goodearl, Homological aspects of Noetherian PI Hopf algebras of irreducible modules and maximal dimension, J. Algebra 198 (1997), no. 1, 240–265. MR 1482982, DOI 10.1006/jabr.1997.7109
K. A. Brown and J. J. Zhang, Dualising complexes and twisted Hochschild (co)homology for Noetherian Hopf algebras, J. Algebra 320 (2008), no. 5, 1814–1850. MR 2437632, DOI 10.1016/j.jalgebra.2007.03.050
S. Chemla, Rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators, J. Algebra 276 (2004), no. 1, 80–102. MR 2054387 (2005e:17022)
C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MR 0024908 (9,567a)
C. De Concini and C. Procesi, Quantum groups, $D$-modules, representation theory, and quantum groups (Venice, 1992), Lecture Notes in Math., vol. 1565, Springer, Berlin, 1993, pp. 31–140. MR 1288995 (95j:17012)
C. De Concini and V. G. Kac, Representations of quantum groups at roots of $1$, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 471–506. MR 1103601 (92g:17012)
Christopher M. Drupieski, Daniel K. Nakano, and Nham V. Ngo, Cohomology for infinitesimal unipotent algebraic and quantum groups, Transform. Groups 17 (2012), no. 2, 393–416. MR 2921071, DOI 10.1007/s00031-012-9181-x
Eric M. Friedlander and Brian J. Parshall, Cohomology of Lie algebras and algebraic groups, Amer. J. Math. 108 (1986), no. 1, 235–253 (1986). MR 821318, DOI 10.2307/2374473
Werner Greub, Stephen Halperin, and Ray Vanstone, Connections, curvature, and cohomology, Pure and Applied Mathematics, Vol. 47-III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Volume III: Cohomology of principal bundles and homogeneous spaces. MR 0400275
N. Kowalzig and U. Krähmer, Duality and products in algebraic (co)homology theories, J. Algebra 323 (2010), no. 7, 2063–2081. MR 2594665
George Lusztig, Quantum groups at roots of $1$, Geom. Dedicata 35 (1990), no. 1-3, 89–113. MR 1066560, DOI 10.1007/BF00147341
S. Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1975 edition. MR 1344215 (96d:18001)
A. Masuoka, Abelian and non-abelian second cohomologies of quantized enveloping algebras, J. Algebra 320 (2008), no. 1, 1–47. MR 2417975 (2009f:16016)
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons Ltd., Chichester, 1987, With the cooperation of L. W. Small, A Wiley-Interscience Publication. MR 934572 (89j:16023)
M. Rosso, Analogues de la forme de Killing et du théorème d’Harish-Chandra pour les groupes quantiques, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 3, 445–467. MR 1055444 (93e:17026)
Joseph J. Rotman, An introduction to homological algebra, 2nd ed., Universitext, Springer, New York, 2009. MR 2455920, DOI 10.1007/b98977
Hans Samelson, Topology of Lie groups, Bull. Amer. Math. Soc. 58 (1952), 2–37. MR 45129, DOI 10.1090/S0002-9904-1952-09544-6
Hirosi Toda and Takashi Watanabe, The integral cohomology ring of $\textbf {F}_{4}/\textbf {T}$ and $\textbf {E}_{6}/\textbf {T}$, J. Math. Kyoto Univ. 14 (1974), 257–286. MR 358847, DOI 10.1215/kjm/1250523239
Takashi Watanabe, The integral cohomology ring of the symmetric space EVII, J. Math. Kyoto Univ. 15 (1975), no. 2, 363–385. MR 380857, DOI 10.1215/kjm/1250523069