Notes on Extensions of Hopf Algebras
Canadian journal of mathematics, Tome 48 (1996) no. 1, pp. 3-42

Voir la notice de l'article provenant de la source Cambridge University Press

This article contains examples and applications of the notion of exact sequences of Hopf algebras.
DOI : 10.4153/CJM-1996-001-8
Mots-clés : 17B37, 16W30, Quantum groups, Hopf algebras
Andruskiewitsch, Nicolás; Witsch, Ruskie. Notes on Extensions of Hopf Algebras. Canadian journal of mathematics, Tome 48 (1996) no. 1, pp. 3-42. doi: 10.4153/CJM-1996-001-8
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[A] Andruskiewitsch, N., Some exceptional compact matrix pseudogroups, Bull. Soc. Math. France 120(1992), 297–325. Google Scholar

[AD] Andruskiewitsch, N. and Devoto, J., Extensions of Hopf algebras, Algebra i Analiz (1) 7(1995), 22–61. Google Scholar

[BCM] Blattner, R.J., Cohen, M. and Montgomery, S., Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298(1986), 671–711. Google Scholar

[BM] Blattner, R.J. and Montgomery, S., Crossed products and Galois extensions of Hopf algebras, Pacific J. Math. 137(1989), 37–54. Google Scholar

[Bou] Bourbaki, N., Groupes et algébres de Lie. Chapitres 4, 5 et 6 Hermann, Paris, 1968. Google Scholar

[Br] Brown, K.S., Cohomology of groups. Graduate Texts in Math. 87, Springer, Berlin, Heidelberg, New York, 1982. Google Scholar

[By] Byott, N.P., Cleft extensions of Hopf algebras, J. Algebra 157(1993), 405–429. Google Scholar

[By2] Byott, N.P., Cleft extensions of Hopf algebras II, Proc. London Math. Soc. (3) 67(1993), 277–304. Google Scholar

[CE] Cartan, H. and Eilenberg, S., Homological algebra. Princeton Univ. Press, 1956. Google Scholar

[CM] Chin, W. and Musson, I., The coradical filtration for quantized enveloping algebras, 1994. preprint. Google Scholar

[Ci] Cibils, C., A quiver quantum group, Comm. Math. Phys. 157(1993), 459–477. Google Scholar

[dCKP] de Concini, C., Kac, V.G. and Procesi, C., Quantum coadjoint action, J. Amer. Math. Soc. 5, 151–189. Google Scholar

[dCL] de Concini, C. and Lyubashenko, V., Quantum Function algebra at roots of, Adv. Math. 108(1994), 205–261. Google Scholar

[dCP] de Concini, C. and Procesi, C., Quantum Groups, Matem. 6, Scuola Norm. Sup. Pisa, (1993), preprint. Google Scholar

[DG] Demazure, M. and Gabriel, P., Groupes algèbriques. Masson & Cie, Paris, 1970. Google Scholar

[DT] Doi, Y. and Takeuchi, M., Cleft comodule algebras for a bialgebra, Comm. Algebra 14(1986), 801–818. Google Scholar

[Dr] Drinfeld, V.G., Quantum groups. Proc. of the ICM, Berkeley, 1986. 798–820. Google Scholar

[Gr] Greither, C., Extensions of finite group schemes, and Hopf Galois extensions over a complete discrete valuation ring, Math. Z. 210(1992), 37–67. Google Scholar

[H] Hall, M., The theory of groups. MacMillan, New York, 1959. Google Scholar

[Hf] Hofstetter, I., Erweiterungen von HopfAlgebren und ihre kohomologische Beschreibung, Dissertation Universität Munchen, 1990. J. Algebra 164(1994), 264–298. Google Scholar

[Ho] Holt, V.G., An interpretation of the cohomology groups Hn(G,M), J. Algebra 60(1979), 307–320. Google Scholar

[J] Jimbo, M., A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10(1985), 63–69. Google Scholar

[K] Kostant, B., Groups over Z, Proc. Sympos. Pure Math. 9(1966), 90–98. Google Scholar

[Kc] Kac, V., Infinite dimensional Lie algebras. Cambridge Univ. Press, Cambridge, 1985. Google Scholar

[Li] Lin, Z., Induced representations of Hopf algebras: applications to quantum groups at roots of 1,J. Algebra 154(1993), 152–187. Google Scholar

[Lo] Loday, V.G., Cohomologie et groupes de Steinberg relatives, J. Algebra 54(1978), 178–202. Google Scholar

[LI] Lusztig, G., Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70(1988), 237–249. Google Scholar

[L2] Lusztig, G., Modular representations and quantum groups, Contemp. Math. 82(1989), 59–77. Google Scholar

[L3] Lusztig, G., Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. (1) 3,257–296. Google Scholar

[L4] Lusztig, G., Quantum groups at roots of\, Geom. Dedicata 35(1990), 89–114. Google Scholar

[L5] Lusztig, G., Introduction to quantized enveloping algebras, In: Proc. of the Third Workshop on Lie Groups Representations and its applications, Carlos Paz, 1989. Progr. Math. 105, Birkhäuser. Google Scholar

[LR] Larson, A.R.G. and Radford, D.E., Semisimple Hopf algebras, (1989), preprint. Google Scholar

[McL] Mac Lane, S., Categories for the Working Mathematician. Springer- Verlag, 1971. Google Scholar

[Mj] Majid, S., More examples of bicrossproduct and double crossproduct Hopf algebras, Israel J. Math. 72(1990), 133–148. Google Scholar

[Ma] Majid, S., Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 130(1990), 17–64. Google Scholar

[MjS] Majid, S. and Ya. Soibelman, Bicrossproduct structure of the quantum Weyl group, J. Algebra 163(1994), 68–87. Google Scholar

[M] Manin, Y.I., Quantum groups and non-commutative geometry. Les Publications du CRM 1561, Montreal, 1988. Google Scholar

[Ms] Masuoka, A., Semisimple Hopf algebras of dimension 6, 8, preprint. Google Scholar

[NZ] Nichols, W.D. and Zoeller, M.B., A Hopf algebra freeness theorem, Amer. J. Math. 111(1989), 381–385. Google Scholar

[OS] Oberst, U. and Schneider, H.-J., Untergruppen formeller Gruppen von endlichen Index, J. Algebra 31 (1974), 1(M4) Google Scholar

[R] Radford, D., Hopf algebras with projection, J. Algebra 92(1985), 322–347. Google Scholar

[Se] Serre, J.-P., Cohomologie galoisienne. Lecture Notes in Math. 5, Springer-Verlag, 1964. Google Scholar

[Si] Singer, W., Extension theory for connected Hopf algebras, J. Algebra 21(1972), 1—16. Google Scholar

[Sch] Schneider, H.-J., Some remarks on exact sequences of quantum groups, Comm. Algebra (9) 21(1993), 3337–3358. Google Scholar

[Sch2] Schneider, H.-J., Zerlegbare Erweiterungen qffiner Gruppen, J. Algebra 66(1980), 569–593. Google Scholar

[Sch3] Schneider, H.-J., Zerlegbare Untergruppen affiner Gruppen, Math. Ann. 255(1981), 139–158. Google Scholar

[Sch4] Schneider, H.-J., Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152(1992), 289–312. Google Scholar

[Sch5] Schneider, H.-J., Hopf Galois extensions, crossed products and Clifford theory, Proc. of the Conference on Hopf algebras and their actions on rings, 1992. to appear. Google Scholar

[Sw] Sweedler, M., Hopf algebras. Benjamin, New York, 1969. Google Scholar

[Sw2] Sweedler, M., Cohomology of algebras over Hopf algebras, Trans. Amer. Math. Soc. 133(1968), 205–239. Google Scholar

[Tl] Takeuchi, M., Finite dimensional representations of the quantum Lorentz group, Comm. Math. Phys. 144(1992), 557–580. Google Scholar

[T2] Takeuchi, M., Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra (8) 9(1981), 841–882. Google Scholar

[T3] Takeuchi, M., Relative Hopf modules—equivalences andfreeness criteria, J. Algebra 60(1979), 452—471. Google Scholar

[T4] Takeuchi, M., Some topics on GLq(n), J. Algebra 147(1992), 379–410. Google Scholar

[T5] Takeuchi, M., Quotient spaces for Hopf algebras, Comm. Algebra (7) 22(1994), 2503–2523. Google Scholar

[Tf] Taft, E.J., The order of the antipode of a finite dimensional Hopf algebra, Proc. Nat. Acad. Sci. U.S.A. 68(1971), 2631–2633. Google Scholar

[TO] Tate, J. and Oort, F., Group schemes of prime order, Ann. Sci. École Norm. Sup. 3(1970), 1—21. Google Scholar

[W] Woronowicz, S.L., Compact matrixpseudogroups, Comm. Math. Phys. 111(1987), 613–665. Google Scholar

[Z] Yongchang Zhu, Hopf algebras of prime dimension, Internat. Math. Res. Notices 1(1994), 53–59. Google Scholar

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