Abstract
Based on the novel notion of ‘weakly counital fusion morphism’, regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields (Böhm et al. Trans. Amer. Math. Soc. 367, 8681–8721, 2015) and multiplier bimonoids in braided monoidal categories (Böhm and Lack, J. Algebra 423, 853–889, 2015). Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Böhm, G.: Comodules over weak multiplier bialgebras. Int. J. Math. 25, 1450037 (2014)
Böhm, G., Caenepeel, S., Janssen, K.: Weak bialgebras and monoidal categories. Comm. Algebra 39(12), 4584–4607 (2011). (special volume dedicated to Mia Cohen)
Böhm, G., Gómez-Torrecillas, J.: Firm Frobenius monads and firm Frobenius algebras. Bull. Math. Soc. Sci. Math. Roumanie 56(104)(3), 281–298 (2013)
Böhm, G., Gómez-Torrecillas, J., López-Centella, E.: Weak multiplier bialgebras. Trans. Amer. Math. Soc. 367, 8681–8721 (2015)
Böhm, G., Lack, S.: Multiplier bialgebras in braided monoidal categories. J. Algebra 423, 853–889 (2015)
Böhm, G., Lack, S.: A category of multiplier bimonoids, Applied Categorical Structures, available online, doi:10.1007/s10485-016-9429-z, Preprint available at arXiv:1509.07171
Böhm, G., Lack, S.: A simplicial approach to multiplier bimonoids, Bull. Belgian Math. Soc. in press, Preprint available at arXiv:1512.01259
Böhm, G., Lack, S.: Multiplier Hopf monoids, Algebras and Representation Theory, in press, Preprint available at arXiv:1511.03806
Böhm, G., Lack, S., Street, R.: Weak bimonads and weak Hopf monads. J. Algebra 328, 1–30 (2011)
Böhm, G., Nill, F., Szlachányi, K.: Weak Hopf algebras. I. Integral theory and C*-structure. J. Algebra 221(2), 385–438 (1999)
Böhm, G., Szlachányi, K.: Weak Hopf algebras. II. Representation theory, dimensions, and the Markov trace. J. Algebra 233, 156–212 (2000)
Böhm, G., Vercruysse, J.: Morita theory for comodules over corings. Comm. Algebra 37(9), 3207–3247 (2009)
Brzeziński, T., Kadison, L., Wisbauer, R.: On coseparable and biseparable corings. In: Caenepeel, S., Van Oystaeyen, F. (eds.) Hopf Algebras in Noncommutative Geometry and Physics, Monographs on pure and applied mathematics, vol. 239, pp. 71–88. Marcel Dekker, New York (2004)
Dauns, J.: Multiplier rings and primitive ideals. Trans. Amer. Math. Soc. 145, 125–158 (1969)
Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. (2) 162, 581–642 (2005)
Hogbe-Nlend, H.: Bornologies and functional analysis. Translated from the French by V. B. Moscatelli. North-Holland Mathematics Studies, Vol. 26 Notas de Matemática, No. 62. North-Holland Publishing Co., Amsterdam-New York-Oxford (1977)
Johnstone, P. T.: Adjoint lifting theorems for categories of algebras. Bull. London Math. Soc. 7(3), 294–297 (1975)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras, vol. I. Academic Press, New York (1983)
Meyer, R.: Local and analytic cyclic homology. EMS Tracts in Mathematics 3 (2007)
Nill, F.: Axioms for weak bialgebras, preprint available at arXiv:math/9805104
Pastro, C., Street, R.: Weak Hopf monoids in braided monoidal categories. Algebra and Number Theory 3(2), 149–207 (2009)
Quillen, D.: Module theory over nonunital rings. Unpublished Notes (1997)
Van Daele, A.: Multiplier Hopf algebras. Trans. Amer. Math. Soc. 342, 917–932 (1994)
Van Daele, A.: Separability Idempotents and Multiplier Algebras, preprint available at arXiv:1301.4398
Van Daele, A., Wang, S.: Weak Multiplier Hopf Algebras. The main theory. Journal für die reine und angewandte Mathematik (Crelles Journal) 705, 155–209 (2015)
Van Daele, A., Wang, S.: Weak Multiplier Hopf Algebras. Preliminaries, motivation and basic examples. In: Pusz, W., Sotan, P.M. (eds.) Operator Algebras and Quantum Groups, Banach Center Publ., vol. 98, pp. 367-415 (2012)
Voigt, C.: Bornological quantum groups. Pac. J. Math. 235(1), 93–135 (2008)
Zimmermann-Huisgen, B.: Pure submodules of direct products of free modules. Math. Ann. 224(2), 233–245 (1976)
Acknowledgments
We thank Ralf Meyer and Christian Voigt for highly enlightening discussions about bornological vector spaces. We gratefully acknowledge the financial support of the Hungarian Scientific Research Fund OTKA (grant K108384), of ‘Ministerio de Economía y Competitividad’ and ‘Fondo Europeo de Desarrollo Regional FEDER’ (grant MTM2013-41992-P), as well as the Australian Research Council Discovery Grant (DP130101969) and an ARC Future Fellowship (FT110100385). GB expresses thanks for the kind invitations and the warm hospitality that she experienced visiting the University of Granada in Nov 2014, Feb 2015 and Oct 2015. SL is grateful for the warm hospitality of his hosts during visits to the Wigner Research Centre in Sept-Oct 2014 and Aug-Sept 2015.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Böhm, G., Gómez-Torrecillas, J. & Lack, S. Weak Multiplier Bimonoids. Appl Categor Struct 26, 47–111 (2018). https://doi.org/10.1007/s10485-017-9481-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-017-9481-3