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COALTERNATIVE COALGEBRAS

Part of: Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  20 August 2019

HELENA ALBUQUERQUE ,
ELISABETE BARREIRO ,
JOSÉ M. SÁNCHEZ  and
CARLOS SONEIRA CALVO
HELENA ALBUQUERQUE
Affiliation:
CMUC, Department of Mathematics, University of Coimbra Apartado 3008, 3001-454 Coimbra, Portugal e-mail: lena@mat.uc.pt, mefb@mat.uc.pt, txema.sanchez@mat.uc.pt
ELISABETE BARREIRO
Affiliation:
CMUC, Department of Mathematics, University of Coimbra Apartado 3008, 3001-454 Coimbra, Portugal e-mail: lena@mat.uc.pt, mefb@mat.uc.pt, txema.sanchez@mat.uc.pt
JOSÉ M. SÁNCHEZ
Affiliation:
CMUC, Department of Mathematics, University of Coimbra Apartado 3008, 3001-454 Coimbra, Portugal e-mail: lena@mat.uc.pt, mefb@mat.uc.pt, txema.sanchez@mat.uc.pt
CARLOS SONEIRA CALVO
Affiliation:
Department of Pedagogy and Didactics A Coruña, University of A Coruña, Spain e-mail: carlos.soneira@udc.gal
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Abstract

In this paper, we define a Cayley–Dickson process for k-coalgebras proving some results that describe the properties of the final coalgebra, knowing the properties of the initial one. Namely, after applying the Cayley–Dickson process for k-coalgebras to a coassociative coalgebra, we obtain a coalternative one. Moreover, the first coalgebra is cocommutative if and only if the final coalgebra is coassociative. Finally we extend these results to a more general approach of D-coalgebras, where D is a k-coalgebra, presenting a class of examples of coalternative (non-coassociative) coalgebras obtained from group D-coalgebras.

MSC classification

Primary: 16T15: Coalgebras and comodules; corings
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Special Issue S1: Workshop on Nonassociative algebras and their applications , December 2020 , pp. S142 - S164
Copyright
© Glasgow Mathematical Journal Trust 2019

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References

REFERENCES

Abe, E., Hopf algebras (Cambridge University Press, Cambridge, New York, 1980).Google Scholar
Albuquerque, H., Elduque, A. and M. Pérez-Izquierdo, J., Alternative quasialgebras, Bull. Austral. Math. Soc. 63 (2001), 257–268.CrossRefGoogle Scholar
Albuquerque, H. and Majid, S., Quasialgebra structure of the octonions, J. Algebra 220 (1999), 188–224.CrossRefGoogle Scholar
Brzezinski, T. and Wisbauer, R., Corings and comodules, London Mathematical Society Lecture Note Series, 309. (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Bulacu, D., The weak braided Hopf algebra structure of some Cayley–Dickson algebras, J. Algebra 322 (2009), 2404–2427.CrossRefGoogle Scholar
Cuadra, J., García Rozas, J. R. and Torrecillas, B., Picard groups and strongly graded coalgebras, J. Pure Appl. Algebra 165(30) (2001), 267–289.CrossRefGoogle Scholar
Kassel, C., Quantum groups, Graduate Texts in Mathematics, vol. 155 (Springer-Verlag, Berlin, 1995).Google Scholar
Sweedler, M., Hopf algebras, Mathematics Lecture Note Series (W.A. Benjamin, Inc., New York, 1969).Google Scholar