Homotopy theory of Hopf Galois extensions
Annales de l'Institut Fourier, Volume 55 (2005) no. 7, pp. 2521-2550.

We introduce the concept of homotopy equivalence for Hopf Galois extensions and make a systematic study of it. As an application we determine all H-Galois extensions up to homotopy equivalence in the case when H is a Drinfeld-Jimbo quantum group.

Nous étudions le concept d’équivalence d’homotopie pour les extensions H-galoisiennes où H désigne une algèbre de Hopf. Ceci nous permet de classifier les extensions H-galoisiennes à homotopie près lorsque H est un groupe quantique de Drinfeld-Jimbo.

DOI: 10.5802/aif.2169
Classification: 16W30, 17B37, 55R10, 58B34, 81R50, 81R60
Keywords: Galois extension, Hopf algebra, quantum group, homotopy, noncommutative geometry, principal fibre bundle, Galois extension, Hopf algebra, quantum group, homotopy, noncommutative geometry, principal fibre bundle
Mot clés : extension galoisienne, algèbre de Hopf, groupe quantique, homotopie, géométrie non commutative, fibré principal

Kassel, Christian 1; Schneider, Hans-Jürgen 2

1 Université Louis Pasteur, Institut de Recherche Mathématique Avancée, CNRS, 7 rue René Descartes, 67084 Strasbourg (France)
2 Universität München, Mathematisches Institut, Theresienstr. 39, 80333 Munich (Allemagne)
@article{AIF_2005__55_7_2521_0,
     author = {Kassel, Christian and Schneider, Hans-J\"urgen},
     title = {Homotopy theory of {Hopf} {Galois} extensions},
     journal = {Annales de l'Institut Fourier},
     pages = {2521--2550},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {7},
     year = {2005},
     doi = {10.5802/aif.2169},
     zbl = {1090.16019},
     mrnumber = {2207392},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2169/}
}
TY  - JOUR
AU  - Kassel, Christian
AU  - Schneider, Hans-Jürgen
TI  - Homotopy theory of Hopf Galois extensions
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 2521
EP  - 2550
VL  - 55
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2169/
DO  - 10.5802/aif.2169
LA  - en
ID  - AIF_2005__55_7_2521_0
ER  - 
%0 Journal Article
%A Kassel, Christian
%A Schneider, Hans-Jürgen
%T Homotopy theory of Hopf Galois extensions
%J Annales de l'Institut Fourier
%D 2005
%P 2521-2550
%V 55
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2169/
%R 10.5802/aif.2169
%G en
%F AIF_2005__55_7_2521_0
Kassel, Christian; Schneider, Hans-Jürgen. Homotopy theory of Hopf Galois extensions. Annales de l'Institut Fourier, Volume 55 (2005) no. 7, pp. 2521-2550. doi : 10.5802/aif.2169. https://aif.centre-mersenne.org/articles/10.5802/aif.2169/

[AJS] Andersen, H.H.; Jantzen, J.; Soergel, W. Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: Independence of p (Astérisque), Volume 220, Soc. Math. France, Paris, 1994 | MR | Zbl

[AS] Andruskiewitsch, N.; Schneider, H.-J. Finite quantum groups over abelian groups of prime exponent, Ann. Sci. Ec. Norm. Supér., Volume 35 (2002), pp. 1-26 | Numdam | MR | Zbl

[B] Bass, H. Algebraic K-theory, Benjamin, New York (1968) | MR | Zbl

[BH] Brzezinski, T.; Hajac, P.M. Galois type extensions and non-commutative geometry (2003) (preprint)

[C] Caenepeel, S. Brauer Groups, Hopf Algebras and Galois Theory, Kluwer Acad. Publ., Dordrecht (1998) | MR | Zbl

[CP] Concini, C. de; Procesi, C. Quantum Groups, D-modules, Representation Theory and Quantum Groups (Lecture Notes in Mathematics), Volume 1565, Springer-Verlag, Berlin, 1993, pp. 31-140 | Zbl

[D] Doi, Y. Braided bialgebras and quadratic bialgebras, Comm. Algebra, Volume 17 (1989), pp. 3053-3085 | DOI | MR | Zbl

[Di] Didt, D. Linkable Dynkin diagrams and quasi-isomorphisms for finite dimensional pointed Hopf algebras, Ludwig-Maximilians-Universität München (2002) (PhD thesis)

[DT] Doi, Y.; Takeuchi, M. Multiplication alteration by two-cocycles. The quantum version, Comm. Algebra, Volume 22 (1994), pp. 5715-5732 | DOI | MR | Zbl

[G] Gersten, S.M. On Mayer-Vietoris functors and algebraic K-theory, J. Algebra, Volume 18 (1971), pp. 51-88 | DOI | MR | Zbl

[Ha] Hasse, H. Die Multiplikationsgruppe der abelschen Körper mit fester Galois-Gruppe, Abh. Math. Sem. Univ. Hamburg,, Volume 16 (1949), pp. 29-40 | MR | Zbl

[Hu] Husemoller, D. Fibre Bundles, Second Edition (Grad. Texts in Math.), Volume 20, Springer-Verlag, New York-Heidelberg, 1975 | MR | Zbl

[J] Jantzen, J.C. Lectures on Quantum Groups (Graduate Studies in Mathematics), Volume 6, Amer. Math. Soc., Providence, RI, 1995 | MR | Zbl

[K] Kassel, C.; O. A. Laudal, R. Piene (eds) Quantum principal bundles up to homotopy equivalence, The legacy of Niels Henrik Abel (2004), pp. 737-748 | MR | Zbl

[KS] Klimyk, A.; Schmüdgen, K. Quantum Groups and Their Representations, Texts and Monographs in Physics (1997) | MR | Zbl

[KT] Kreimer, H.F.; Takeuchi, M. Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J., Volume 30 (1981), pp. 675-692 | DOI | MR | Zbl

[L] Lam, T.Y. Lectures on modules and rings (Grad. Texts in Math.), Volume 189 (1999) | MR | Zbl

[M] Montgomery, S. Hopf Algebras and Their Actions on Rings, CBMS Conf. Series in Math., 82, Amer. Math. Soc., Providence, RI, 1993 | MR | Zbl

[Ma1] Masuoka, A. Cleft extensions for a Hopf algebra generated by a nearly primitive element, Comm. Algebra, Volume 22 (1994), pp. 4537-4559 | DOI | MR | Zbl

[Ma2] Masuoka, A. Defending the negated Kaplansky conjecture, Proc. Amer. Math. Soc., Volume 129 (2001), pp. 3185-3192 | DOI | MR | Zbl

[MS] Montgomery, S.; Schneider, H.-J. Krull relations in Hopf Galois extensions: lifting and twisting, J. Algebra, Volume 288 (2005), pp. 364-383 | DOI | MR | Zbl

[P] Pedrini, C. On the K 0 of certain polynomial extensions (Lect. Notes Math.), Volume 342 (1973), pp. 92-108 | MR | Zbl

[S] Schneider, H.-J. Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math., Volume 72 (1990), pp. 167-195 | DOI | MR | Zbl

[Sch] Schauenburg, P. Hopf-Galois and bi-Galois extensions (Fields Institute Communications), Volume 43 (2004) | MR | Zbl

[Sw1] Swan, R.G. Some relations between higher K-functors, J. Algebra, Volume 21 (1972), pp. 113-136 | DOI | MR | Zbl

Cited by Sources: