Abstract
After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre-braided just as in the case of bialgebroids, and is embedded into the one-sided center of the comodule category. We proceed to define braided cocommutative coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of Brzeziński and Militaru (J Algebra 251:279–294, 2002) and Bálint and Szlachányi (J Algebra 296:520–560, 2006), originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule category over a bicoalgebroid with the category of coalgebras of the associated comonad, we obtain a comonadic (weakened) version of Schauenburg’s theorem. Finally, we take a look at the scalar extension and braided cocommutative coalgebras from a (co-)monadic point of view.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bálint, I., Szlachányi, K.: Finitary Galois extensions over noncommutative bases. J. Algebra 296, 520–560 (2006)
Brzeziński, T., Militaru, G.: Bialgebroids, × A -bialgebras and duality. J. Algebra 251, 279–294 (2002)
Brzeziński, T., Wisbauer, R.: Corings and Comodules. London Math. Soc. LNS 309, Cambridge Univ. Press (2003)
Caenepeel, S., Wang, D., Yin, Y.: Yetter-Drinfeld modules over weak Hopf algebras and the center construction. Ann. Univ. Ferrara Sez. VII 51, 69–98 (2005)
Joyal, A., Street, R.H.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)
Joyal, A., Street, R.H.: Tortile Yang-Baxter operators in tensor categories. J. Pure Appl. Algebra 71, 43–51 (1991)
Johnstone, P.T.: Adjoint lifting theorems for categories of modules. Bull. London Mat. Soc. 7, 294–297 (1975)
Kadison, L., Szlachányi, K.: Bialgebroid actions on depth two extensions and duality. Adv. Math. 179, 75–121 (2003)
Kadison, L.: Co–depth two and related topics, math.QA/0601001
Kassel, C.: Quantum Groups. Grad. Textsin Math. vol. 155, Springer, Berlin (1995)
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. GTM 5, Springer-Verlag New-York Inc (1998)
Majid, S.: Representations, duals and quantum doubles of monoidal categories, Rend. Circ. Mat. Palermo 2(26), 197–206 (1991)
Moerdijk, I.: Monads on tensor categories. J. Pure Appl. Algebra 168, 189–208 (2002)
Montgomery, S.: Hopf Algebras and Their Actions on Rings. CBMS Lecture Notes, vol. 82, Am. Math. Soc., Providence, RI (1993)
Schauenburg, P.: Duals and doubles of quantum groupoids. In: New Trends in Hopf Algebra theory, Contemporary Mathematics, vol. 267, p. 273, AMS (2000)
Schneider, H.-J.: Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math. 72(1–2), (1990)
Szlachányi, K.: The monoidal Eilenberg–Moore construction and bialgebroids. J Pure Appl Algebra 182, 287–315 (2003)
Takeuchi, M.: Morita theorems for categories of comodules. J. Fac. Sci. Univ. Tokyo, Sec. IA, 24, 629–644 (1977)
Torrecillas, B., van Oystaeyen, F., Zhang, Y.H.: The Brauer group of a cocommutative coalgebra. J. Algebra, 177, 536–568 (1995)
Wisbauer, R.: Algebras versus coalgebras. Appl. Categ. Structures (in press) doi:10.1007/s10485-007-9076-5
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bálint, I. Scalar Extension of Bicoalgebroids. Appl Categor Struct 16, 29–55 (2008). https://doi.org/10.1007/s10485-007-9098-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-007-9098-z