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Generalized Diagonal Crossed Products and Smash Products for Quasi-Hopf Algebras. Applications

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Abstract

In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may then be applied to H *-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the “generating matrix” formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products.

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References

  1. Akrami S.E., Majid S. (2004). Braided cyclic cocycles and nonassociative geometry. J. Math. Phys. 45:3883–3911

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Albuquerque H., Majid S. (1999). Quasialgebra structure of the octonions. J. Algebra 220:188–224

    Article  MATH  MathSciNet  Google Scholar 

  3. Altschuler D., Coste A. (1992). Quasi-quantum groups, knots, three-manifolds and topological field theory. Commun. Math. Phys. 150:83–107

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Beggs, E. J., Majid, S.: Quantization by cochain twists and nonassociative differentials. http://arxiv.org/listmath.QA/0506450, 2005

  5. Bulacu D., Panaite F., Van Oystaeyen F. (2000). Quasi-Hopf algebra actions and smash products. Comm. Algebra 28: 631–651

    Article  MATH  MathSciNet  Google Scholar 

  6. Bulacu D., Nauwelaerts E. (2000). Relative Hopf modules for (dual) quasi-Hopf algebras. J. Algebra 229:632–659

    Article  MATH  MathSciNet  Google Scholar 

  7. Bulacu D., Caenepeel S. (2003). Two-sided two-cosided Hopf modules and Doi-Hopf modules for quasi-Hopf algebras. J. Algebra 270:55–95

    Article  MATH  MathSciNet  Google Scholar 

  8. Caenepeel S., Militaru G., Zhu S. (1997). Crossed modules and Doi-Hopf modules. Israel J. Math. 100:221–247

    Article  MATH  MathSciNet  Google Scholar 

  9. Connes A., Dubois-Violette M. (2002). Noncommutative finite dimensional manifolds I. Spherical manifolds and related examples. Commun. Math. Phys. 230:539–579

    MATH  MathSciNet  Google Scholar 

  10. Dijkgraaf R., Pasquier V., Roche P. (1990). Quasi-Hopf algebras, group cohomology and orbifold models. Nucl. Phys. B Proc. Suppl. 18 B:60–72

    ADS  MathSciNet  Google Scholar 

  11. Drinfeld V.G. (1990). Quasi-Hopf algebras. Leningrad Math. J. 1:1419–1457

    MathSciNet  Google Scholar 

  12. Hausser F., Nill F. (1999). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629

    Article  MATH  MathSciNet  Google Scholar 

  13. Hausser F., Nill F. (1999). Doubles of quasi-quantum groups. Commun. Math. Phys. 199:547–589

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Jara Martínez, P., López Peña, J., Panaite F., Van Oystaeyen, F.: On iterated twisted tensor products of algebras. http://arxiv.org/list/math.QA/0511280, 2005

  15. Jimbo M., Konno H., Odake S., Shiraishi J. (1999). Quasi-Hopf twistors for elliptic quantum groups. Transform. Groups 4:303–327

    Article  MATH  MathSciNet  Google Scholar 

  16. Kassel C. (1995). Quantum Groups. Graduate Texts in Mathematics 155, Springer Verlag, Berlin

    MATH  Google Scholar 

  17. Mack G., Schomerus V. (1992). Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus. Commun. Math. Phys. 149:513–548

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Majid S. (1998). Quantum double for quasi-Hopf algebras. Lett. Math. Phys. 45:1–9

    Article  MATH  MathSciNet  Google Scholar 

  19. Majid S. (1995). Foundations of quantum group theory. Cambridge Univ. Press, Cambridge

    MATH  Google Scholar 

  20. Majid, S.: Gauge theory on nonassociative spaces. J. Math. Phys. 46, 103519, (2005) 23 pp

    Google Scholar 

  21. Panaite F. (2002). Hopf bimodules are modules over a diagonal crossed product algebra. Comm. Algebra 30:4049–4058

    Article  MATH  MathSciNet  Google Scholar 

  22. Schauenburg P. (2002). Hopf modules and the double of a quasi-Hopf algebra. Trans. Amer. Math. Soc. 354:3349–3378

    Article  MATH  MathSciNet  Google Scholar 

  23. Schauenburg P. (2003). Actions of monoidal categories and generalized Hopf smash products. J. Algebra 270:521-563

    Article  MATH  MathSciNet  Google Scholar 

  24. Sweedler M.E. (1969). Hopf algebras. Benjamin, New York

    Google Scholar 

  25. Wang S.-H., Li J. (1998). On twisted smash products for bimodule algebras and the Drinfeld double. Comm. Algebra 26:2435–2444

    Article  MATH  MathSciNet  Google Scholar 

  26. Wang S.-H. (2001). Doi-Koppinen Hopf bimodules are modules. Comm. Algebra 29:4671–4682

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, University of Bucharest, Str. Academiei 14, RO-70109, Bucharest 1, Romania

    Daniel Bulacu

  2. Institute of Mathematics, Romanian Academy, PO-Box 1-764, RO-014700, Bucharest, Romania

    Florin Panaite

  3. Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020, Antwerp, Belgium

    Freddy Van Oystaeyen

Authors
  1. Daniel Bulacu
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  2. Florin Panaite
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  3. Freddy Van Oystaeyen
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Corresponding author

Correspondence to Freddy Van Oystaeyen.

Additional information

Communicated by A. Connes

Research partially supported by the EC programme LIEGRITS, RTN 2003, 505078, and by the bilateral projects “Hopf Algebras in Algebra, Topology, Geometry and Physics” and “New techniques in Hopf algebras and graded ring theory” of the Flemish and Romanian Ministries of Research. The first two authors have been also partially supported by the programme CERES of the Romanian Ministry of Education and Research, contract no. 4-147/2004.

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Bulacu, D., Panaite, F. & Van Oystaeyen, F. Generalized Diagonal Crossed Products and Smash Products for Quasi-Hopf Algebras. Applications. Commun. Math. Phys. 266, 355–399 (2006). https://doi.org/10.1007/s00220-006-0051-z

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