Abstract
The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid ℋ = (ℋ L , ℋ R )) is cleft if and only if it is ℋ R -Galois and has a normal basis property relative to the base ring L of ℋ L . Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the equivalence classes of crossed products and gauge transformations is established. Strong connections in cleft extensions are classified and sufficient conditions are derived for the Chern–Galois characters to be independent on the choice of strong connections. The results concerning cleft extensions and crossed product are then extended to the case of weak cleft extensions of Hopf algebroids hereby defined.
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Abuhlail, J.: Morita contexts for corings and equivalences. In: Caenepeel, S., Van Oystaeyen, F. (eds.) Hopf Algebras in Noncommutative Geometry and Physics. Lecture Notes in Pure and Appl. Math., vol. 239, pp. 1–19. Marcel Dekker, New York (2005)
Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R., Rodríguez Raposo, A.B.: Weak C-cleft extensions, weak entwining structures and weak Hopf algebras. J. Algebra 284, 679–704 (2005)
Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R., Rodríguez Raposo, A.B.: Weak C-cleft extensions and weak Galois extensions. J. Algebra 299, 276–293 (2006)
Bálint, I., Szlachányi, K.: Finitary Galois extensions over non-commutative bases. J. Algebra 296, 520–560 (2006)
Blattner, R.J., Cohen, M., Montgomery, S.: Crossed products and inner actions of Hopf algebras. Trans. Amer. Math. Soc. 298, 671–711 (1986)
Blattner, R.J., Montgomery, S.: Crossed products and Galois extensions of Hopf algebras. Pacific J. Math. 137, 37–54 (1989)
Böhm, G.: Integral theory for Hopf algebroids. Alg. Rep. Theory 8(4), 563–599 (2005)
Böhm, G.: Galois theory for Hopf algebroids. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 51, 233–262 (2005)
Böhm, G., Brzeziński, T.: Strong connections and the relative Chern–Galois character for corings. Internat. Math. Res. Notices 42, 2579–2625 (2005)
Böhm, G., Szlachányi, K.: Hopf algebroids with bijective antipodes: axioms, integrals and duals. J. Algebra 274, 708–750 (2004)
Böhm, G., Vercruysse, J.: Morita theory for coring extensions and cleft bicomodules. arxiv:math.RA/0601464 DOI:10.1016/J.AIM.2006.05.010
Brzeziński, T.: On modules associated to coalgebra Galois extensions. J. Algebra 215, 290–317 (1999)
Brzeziński, T.: A note on coring extensions. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 51, 15–27 (2005)
Brzeziński, T., Caenepeel, S., Militaru, G.: Doi-Koppinen modules for quantum groupoids. J. Pure Appl. Algebra 175, 45–62 (2002)
Brzeziński, T., Hajac, P.M.: Coalgebra extensions and algebra coextensions of Galois type. Comm. Algebra 27, 1347–1367 (1999)
Brzeziński, T., Majid, S.: Coalgebra bundles. Comm. Math. Phys. 191, 467–492 (1998)
Brzeziński, T., Wisbauer, R.: Corings and Comodules. Cambridge University Press, Cambridge, UK (2003)
Caenepeel, S., De Groot, E.: Modules over weak entwining structures. In: Andruskiewitsch, N., Ferrer Santos, W.R., Schneider, H.-J. (eds) New Trends in Hopf Algebra Theory. Contemp. Math.- Amer. Math. Soc., vol 267, pp. 31–54. Providence, RI (2000)
Caenepeel, S., Vercruysse, J., Wang, S.: Morita theory for corings and cleft entwining structures. J. Algebra 276, 210–235 (2004)
Doi, Y., Takeuchi, M.: Cleft comodule algebras for a bialgebra. Comm. Algebra 14, 801–817 (1986)
Kadison, L.: Depth two and the Galois coring. In: Fuchs, J., Stolin, A.A. (eds.) Noncommutative geometry and representation theory in mathematical physics. Contemp. Math.-Amer. Math. Soc., vol. 391 pp. 149–156. Providence, RI (2005)
Kadison, L.: An endomorphism ring theorem for Galois and D2 extensions. J. Algebra 305, 163–184 (2006)
Kadison, L., Szlachányi, K.: Bialgebroid actions on depth two extensions and duality. Adv. Math. 179, 75–121 (2003)
Lu, J.-H.: Hopf algebroids and quantum groupoids. Internat. J. Math. 7, 47–70 (1996)
Schauenburg, P.: Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules. Appl. Categ. Structures 6, 193–222 (1998)
Sweedler, M.E.: Hopf Algebras. Benjamin, New York (1969)
Takeuchi, M.: Groups of algebras over A⊗Ā. J. Math. Soc. Japan 29, 459–492 (1977)
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Dedicated to Stef Caenepeel on the occasion of his 50th birthday.
An erratum to this article is available at http://dx.doi.org/10.1007/s10485-008-9168-x.
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Böhm, G., Brzeziński, T. Cleft Extensions of Hopf Algebroids. Appl Categor Struct 14, 431–469 (2006). https://doi.org/10.1007/s10485-006-9043-6
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DOI: https://doi.org/10.1007/s10485-006-9043-6