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The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras

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Abstract

We consider the factorization problem for bialgebras. Let L and H be algebras and coalgebras (but not necessarily bialgebras) and consider two maps R : H ⊗ L → L ⊗ H and W : L ⊗ H → H ⊗ L. We introduce a product K = L W ⋈ R H and we give necessary and sufficient conditions for K to be a bialgebra. Our construction generalizes products introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed by Beattie, Dăscălescu and Grünenfelder appear as special cases.

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References

  1. Abe, E.: Hopf Algebras, Cambridge Univ. Press, Cambridge, 1977.

    Google Scholar 

  2. Beattie, M., Dăascăalescu, S. and Grünenfelder, L.: Constructing pointed Hopf algebras by Ore extensions, J. Algebra, to appear.

  3. Bespalov, Y. and Drabant, B.: Cross product bialgebras, Part I, J. Algebra 219 (1999), 466-505.

    Google Scholar 

  4. Brzezinski, T.: On modules associated to coalgebra Galois extensions, J. Algebra 215 (1999), 290-317.

    Google Scholar 

  5. Brzezinski, T. and Majid, S.: Coalgebra bundles, Comm. Math. Phys. 191 (1998), 467-492.

    Google Scholar 

  6. Caenepeel, S. and D¡asc¡alescu, S.: On pointed Hopf algebras of dimension 2n, Bull. London Math. Soc. 31 (1999), 17-24.

    Google Scholar 

  7. Caenepeel, S. and D¡asc¡alescu, S.: Classifying pointed Hopf algebras of dimension 16, Comm. Algebra, in press.

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

    Google Scholar 

  9. Doi, Y.: Unifying Hopf modules, J. Algebra 153 (1992), 373-385.

    Google Scholar 

  10. Kassel, C.: Quantum Groups, Springer-Verlag, New York, 1995.

    Google Scholar 

  11. Koppinen, M.: Variations on the smash product with applications to group-graded rings, J. Pure Appl. Algebra 104 (1995), 61-80.

    Google Scholar 

  12. Long, F. W.: The Brauer group of dimodule algebras, J. Algebra 31 (1974), 559-601.

    Google Scholar 

  13. Majid, S.: Physics for algebrists: Non-commutative and non-cocommutative Hopf algebras by a bycrossproduct construction, J. Algebra 130 (1990), 17-64.

    Google Scholar 

  14. Majid, S.: Algebras and Hopf algebras in braided categories, In: J. Bergen and S. Montgomery (eds), Advances in Hopf Algebras, Marcel Dekker, New York, 1994, pp. 55-106.

    Google Scholar 

  15. Majid, S.: Foundations of Quantum Groups Theory, Cambridge Univ. Press, Cambridge, 1995.

    Google Scholar 

  16. Militaru, G.: The Long dimodule category and nonlinear equations, Algebr. Represent. Theory 2 (1999), 177-200.

    Google Scholar 

  17. Molnar, R. K.: Semi-direct products of Hopf algebras, J. Algebra 47 (1977), 29-51.

    Google Scholar 

  18. Montgomery, S.: Hopf Algebras and their Actions on Rings, Amer. Math. Soc., Providence, 1993.

  19. Podles, P. and Woronowicz, S. L.: Quantum deformation of Lorentz group, Comm. Math. Phys. 130 (1990), 381-431.

    Google Scholar 

  20. Radford, D.: The structure of Hopf algebras with projection, J. Algebra 92 (1985), 322-347.

    Google Scholar 

  21. Radford, D.: Minimal quasitriangular Hopf algebras, J. Algebra 157 (1993), 285-315.

    Google Scholar 

  22. Radford, D.: On Kauffman's knot invariants arising from finite-dimensional Hopf algebras, In: J. Bergen and S. Montgomery (eds), Advances in Hopf Algebras, Marcel Dekker, New York, 1994, pp. 205-266.

    Google Scholar 

  23. Radford, D. and Towber, J.: Yetter-Drinfel'd categories associated to an arbitrary bialgebra, J. Pure Appl. Algebra 87 (1993), 259-279.

    Google Scholar 

  24. Sweedler, M. E.: Hopf Algebras, Benjamin, New York, 1969.

    Google Scholar 

  25. Taft, E.: The order of the antipode of a finite-dimensional Hopf algebra, Proc. Nat. Acad. Sci. USA 68 (1971), 2631-2633.

    Google Scholar 

  26. Skandalis, G.: Operator algebras and duality, In: Proc. ICM Kyoto, 1990, pp. 997-1009.

  27. Takeuchi, M.: Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981), 841-882.

    Google Scholar 

  28. Takeuchi, M.: Ext ad(SpR; υA) ≅ Br(A/k), J. Algebra 67 (1980), 436-475.

    Google Scholar 

  29. van Daele, A. and van Keer, S.: The Yang-Baxter and pentagon equation, Compositio Math. 91 (1994), 201-221.

    Google Scholar 

  30. Yetter, D.: Quantum groups and representations of monoidal categories, Math. Proc. Cambridge Philos. Soc. 108 (1990), 261-290.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Applied Sciences, University of Brussels, VUB, Pleinlaan 2, B-1050, Brussels, Belgium

    S. Caenepeel

  2. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, U.S.A.

    Bogdan Ion

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

    G. Militaru

  4. Institute of Mathematics, Fudan University, Shanghai, 200433, China

    Shenglin Zhu

Authors
  1. S. Caenepeel
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  2. Bogdan Ion
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  3. G. Militaru
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  4. Shenglin Zhu
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Caenepeel, S., Ion, B., Militaru, G. et al. The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras. Algebras and Representation Theory 3, 19–42 (2000). https://doi.org/10.1023/A:1009917210863

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