Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T05:32:13.743Z Has data issue: false hasContentIssue false
  • English
  • Français

The braided monoidal structures on the category of vector spaces graded by the Klein group

Part of: Hopf algebras, quantum groups and related topics Categories with structure

Published online by Cambridge University Press:  14 June 2011

D. Bulacu ,
S. Caenepeel  and
B. Torrecillas
D. Bulacu
Affiliation:
Faculty of Mathematics and Informatics, University of Bucharest, Str. Academiei 14, 010014 Bucharest 1, Romania (daniel.bulacu@fmi.unibuc.ro) Faculty of Engineering, Vrije Universiteit Brussel, 1050 Brussels, Belgium (scaenepe@vub.ac.be)
S. Caenepeel
Affiliation:
Faculty of Engineering, Vrije Universiteit Brussel, 1050 Brussels, Belgium (scaenepe@vub.ac.be)
B. Torrecillas
Affiliation:
Departamento Álgebra y Análisis Matemático, Universidad de Almería, Ctra. Sacramento S/N, La Cañada de San Urbano, 04120 Almería, Spain (btorreci@ual.es)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let k be a field, let k* = k \ {0} and let C2 be a cyclic group of order 2. We compute all of the braided monoidal structures on the category of k-vector spaces graded by the Klein group C2 × C2. For the monoidal structures we compute the explicit form of the 3-cocycles on C2 × C2 with coefficients in k*, while, for the braided monoidal structures, we compute the explicit form of the abelian 3-cocycles on C2 × C2 with coefficients in k*. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras with underlying vector space k[C2 × C2].

Keywords

braided monoidal categoryKlein groupgroup cohomologyquasi-Hopf algebra

MSC classification

Secondary: 16T05: Hopf algebras and their applications 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 3 , October 2011 , pp. 613 - 641
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Albuquerque, H. and Majid, S., Quasialgebra structure of the octonions, J. Alg. 220 (1999), 247–264.Google Scholar
2.Albuquerque, H. and Majid, S., ℤn-quasialgebras, in Matrices and group representations (ed. Santana, A. P., Duarte, A. L. and Queiró, J. F.), Textos de Matemática Série B, Volume 19, pp. 57–64 (University of Coimbra, 1999).Google Scholar
3.Albuquerque, H. and Majid, S., Clifford algebras obtained by twisting of group algebras, J. Pure Appl. Alg. 171 (2002), 133–148.CrossRefGoogle Scholar
4.Álvarez, J. N. Alonso, Vilaboa, J. M. Fernández and Rodríguez, R. González, Weak braided Hopf algebras, Indiana Univ. Math. J. 57(5) (2008), 2423–2458.CrossRefGoogle Scholar
5.Böhm, G., Nill, F. and Szlachányi, K., Weak Hopf algebras, I, Integral theory and C*-structure, J. Alg. 221 (1999), 385–438.CrossRefGoogle Scholar
6.Bulacu, D., The weak braided Hopf algebra structures of some Cayley.Dickson algebras, J. Alg. 322 (2009), 2404–2427.CrossRefGoogle Scholar
7.Bulacu, D., Weak braided Hopf algebras obtained from Clifford processes for algebras and coalgebras, J. Alg. 322 (2009), 2404–2427.CrossRefGoogle Scholar
8.Caenepeel, S., Brauer groups, Hopf algebras and Galois theory, K-Monographs in Mathematics, Volume 4 (Kluwer, Dordrecht, 1998).CrossRefGoogle Scholar
9.Dăscălescu, S., Năstăsescu, C. and Raianu, Ş., Hopf algebras: an introduction, Monographs and Textbooks in Pure and Applied Mathematics, Volume 235 (Dekker, New York, 2001).Google Scholar
10.Eilenberg, S. and Mac Lane, S., Cohomology theory of abelian groups and homotopy theory, I, Proc. Natl Acad. Sci. USA 36 (1950), 443–447.Google Scholar
11.Eilenberg, S. and Mac Lane, S., Cohomology theory of abelian groups and homotopy theory, II, Proc. Natl Acad. Sci. USA 36 (1950), 657–663.CrossRefGoogle ScholarPubMed
12.Eilenberg, S. and Mac Lane, S., Cohomology theory of abelian groups and homotopy theory, III, Proc. Natl Acad. Sci. USA 37 (1951), 307–310CrossRefGoogle ScholarPubMed
13.Eilenberg, S. and Mac Lane, S., Cohomology theory of abelian groups and homotopy theory, IV, Proc. Natl Acad. Sci. USA 38 (1952), 325–329.CrossRefGoogle ScholarPubMed
14.Etingof, P. and Gelaki, S., Finite dimensional quasi-Hopf algebras with radical of codimension 2, Math. Res. Lett. 11 (2004), 685–696.CrossRefGoogle Scholar
15.Joyal, A. and Street, R., Braided monoidal categories, Mathematics Reports 86008, Macquarie University (1986).Google Scholar
16.Joyal, A. and Street, R., Braided tensor categories, Adv. Math. 102 (1993), 20–78.Google Scholar
17.Kassel, C., Quantum groups, Graduate Texts in Mathematics, Volume 155 (Springer, 1995).CrossRefGoogle Scholar
18.Mac Lane, S., Cohomology theory of abelian groups, in Proceedings of the International Congress of Mathematicians 1950, Volume 2, pp. 8–14 (American Mathematical Society, Providence, RI, 1952).Google Scholar
19.Majid, S., Foundations of quantum group theory (Cambridge University Press, 1995).Google Scholar
20.Rotman, J., An introduction to homological algebra (Academic Press, New York, 1979).Google Scholar
21.Weibel, C., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Volume 38 (Cambridge University Press, 1994).Google Scholar