Duke Mathematical Journal

Cohomology of quantum groups at roots of unity

Victor Ginzburg and Shrawan Kumar

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Article information

Duke Math. J. Volume 69, Number 1 (1993), 179-198.

First available in Project Euclid: 20 February 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
Secondary: 17B56: Cohomology of Lie (super)algebras


Ginzburg, Victor; Kumar, Shrawan. Cohomology of quantum groups at roots of unity. Duke Math. J. 69 (1993), no. 1, 179--198. doi:10.1215/S0012-7094-93-06909-8. http://projecteuclid.org/euclid.dmj/1077293430.

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  • [APW] H. H. Andersen, P. Polo, and K. Wen, Representations of quantum algebras, Invent. Math. 104 (1991), no. 1, 1–59.
  • [APW2] H. H. Andersen, P. Polo, and W. Kexin, Injective modules for quantum algebras, Amer. J. Math. 114 (1992), no. 3, 571–604.
  • [AW] H. H. Andersen and K. Wen, Representations of quantum algebras—the mixed case, preprint, March 1991.
  • [BGS] A. Beilinson, V. Ginsburg, and W. Soergel, Koszul duality patterns in representation theory, preprint to appear in J. Amer. Math. Soc.
  • [B] D. J. Benson, Representations and Cohomology I. Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Stud. Adv. Math., vol. 30, Cambridge Univ. Press, Cambridge, 1991.
  • [Bo] N. Bourbaki, Éléments de Mathématique, Masson, Paris, 1981, Groupes et algèbres de Lie, Chapitres 4, 5, et 6.
  • [CE] H. Cartan and S. Eilenberg, Homological Algebra, Princeton Math. Ser., vol. 19, Princeton Univ. Press, Princeton, 1956.
  • [Ca] L. Casian, Kazhdan-Lusztig conjecture in the negative level case (Kac-Moody algebras of affine type), preprint.
  • [DK] 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.
  • [FP] E. M. Friedlander and B. J. Parshall, Cohomology of Lie algebras and algebraic groups, Amer. J. Math. 108 (1986), no. 1, 235–253.
  • [G] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288.
  • [Gi] V. Ginsburg, Perverse sheaves on a loop group and Langlands duality, to appear in Inst. Hautes Études Sci. Publ. Math.
  • [Gi2] V. Ginsburg, Perverse sheaves and $\mathbbC^ *$-actions, J. Amer. Math. Soc. 4 (1991), no. 3, 483–490.
  • [GS] M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theory, Deformation theory of algebras and structures and applications (Il Ciocco, 1986) eds. M. Hazewinkel and M. Gerstenhaber, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 247, Kluwer Acad. Publ., Dordrecht, 1988, pp. 11–264.
  • [J] J. C. Jantzen, Representations of Algebraic Groups, Pure Appl. Math., vol. 131, Academic Press, Boston, 1987.
  • [K] S. Kumar, Representations of quantum groups at roots of unity, preprint, March 1991.
  • [KL] D. Kazhdan and G. Lusztig, Affine Lie algebras and quantum groups, Internat. Math. Res. Notices (1991), no. 2, 21–29.
  • [KV] D. Kazhdan and M. Verbitsky, Cohomology of restricted quantized universal enveloping algebras, preprint, 1992.
  • [L1] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc. 3 (1990), no. 1, 257–296.
  • [L2] G. Lusztig, Quantum groups at roots of $1$, Geom. Dedicata 35 (1990), no. 1-3, 89–114.
  • [P] S. B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60.
  • [S] J. P. Serre, Algèbres de Lie semi-simples complexes, W. A. Benjamin, New York-Amsterdam, 1966.