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COQUASITRIANGULAR STRUCTURES FOR EXTENSIONS OF HOPF ALGEBRAS. APPLICATIONS

Published online by Cambridge University Press:  02 August 2012

A. L. AGORE
A. L. AGORE*
Affiliation:
Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium e-mail: ana.agore@vub.ac.be, ana.agore@gmail.com
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Abstract

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Let A ⊆ E be an extension of Hopf algebras such that there exists a normal left A-module coalgebra map π : E → A that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra E in terms of the datum (A, E, π) as follows: first, any such extension E is isomorphic to a unified product A ⋉ H, for some unitary subcoalgebra H of E (A. L. Agore and G. Militaru, Unified products and split extensions of Hopf algebras, to appear in AMS Contemp. Math.). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product A ⋉ H and a certain set of datum (p, τ, u, v) related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite-dimensional quantum double Dλ(A, H) = A ⋈τH to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.

Keywords

16T1016T0516S40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 1 , January 2013 , pp. 201 - 215
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Agore, A. L. and Militaru, G., Extending structures II: The quantum version, J. Algebra 336 (2011), 321–341.Google Scholar
2.Agore, A. L. and Militaru, G., Unified products and split extensions of Hopf algebras, AMS Contemp. Math. (to appear) arXiv:1105.1474v3.Google Scholar
3.Agore, A. L., Crossed product of Hopf algebras, Comm. Algebra. (to appear) arXiv:1203.2454v1.Google Scholar
4.Caenepeel, S., Militaru, G. and Zhu, S., Frobenius and separable functors for generalized module categories and nonlinear equations, Lecture Notes in Mathematics, vol. 1787 (Springer Verlag, Berlin, 2002).CrossRefGoogle Scholar
5.Doi, Y. and Takeuchi, M., Multiplication alteration by two-cocycles, Commun. Algebra. 22 (14) (1994), 5715–5732.CrossRefGoogle Scholar
6.Drinfeld, V., Quantum groups, Proc. Int. Congr. Math. Berkeley. I (1987), 798–820Google Scholar
7.Jiao, Z. and Wisbauer, R., The braided structures for ω-smash coproduct Hopf algebras, J. Algebra 287 (2) (2005), 474–495.Google Scholar
8.Jiao, Z. and Wisbauer, R., The braided structures for T-smash product Hopf algebras, Int. Electronical J. Algebra 1 (2007), 30–45.Google Scholar
9.Kassel, C., Quantum groups, Graduate texts in mathematics, vol. 155 (Springer-Verlag, New York, 1995).Google Scholar
10.Larson, R. and Towber, J., Two dual classes of bialgebras related to the concepts of “quantum groups” and “quantum Lie algebras,” Commun. Algebra 19 (1991), 3295–3345.Google Scholar
11.Ma, T., Li, H. and Zhao, W., On the braided structures of Radford's biproduct, Acta Math. Sci. B Engl. 31 (2), (2011), 701–715.Google Scholar
12.Majid, S., Quantum groups and quantum probability, in Quantum probability and related topics, vol. 6 (World Scienntific, River Edge, NJ, 1991), 333–358.Google Scholar
13.Majid, S., Foundations of Quantum group theory (Cambridge Univ. Press, Cambridge, UK, 1995).Google Scholar
14.Takeuchi, M., Representations of the Hopf algebra U(n), in Hopf algebras and generalizations, Contemporary Mathematics, vol. 441, (Amer. Math. Soc., Providence, RI, 2007), 155–174.Google Scholar