Abstract
The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele’s definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved — in an appropriate, multiplier-valued sense — which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved — which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.
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Böhm, G., Gómez-Torrecillas, J., Lack, S.: Weak multiplier bimonoids, Preprint available at arXiv:1603.05702
Böhm, G., Gómez-Torrecillas, J., López-Centella, E.: Weak multiplier bialgebras. Trans. Amer. Math. Soc. 367(12), 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, Appl. Categ. Structures, doi:10.1007/s10485-016-9429-z. arXiv:1509.07171
Dauns, J.: Multiplier rings and primitive ideals. Amer. Math. Soc. 145, 125–158 (1969)
De Commer, K.: Galois objects for algebraic quantum groups. J. Algebra 321 (6), 1746–1785 (2009)
Drabant, B., Van Daele, A., Zhang, Y.: Actions of multiplier Hopf algebras. Commun. Algebra 27(9), 4117–4172 (1999)
Hogbe-Nlend, H.: Bornologies and functional analysis. Translated from the French by V. B. Moscatelli. North-Holland Mathematics Studies, Vol. 26 Notas De Matemática, vol. 62. North-Holland Publishing Co., Amsterdam (1977)
Janssen, K., Vercruysse, J.: Multiplier Hopf and bi-algebras. J. Algebra Appl. 9(2), 275–303 (2010)
Joyal, A., Street, R.: The geometry of tensor calculus I. Adv. Math. 88(1), 55–112 (1991)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras, vol. I. Academic, New York (1983)
Kurose, H., Van Daele, A., Zhang, Y.: Corepresentation theory of multiplier Hopf algebras II. Int. J. Math. 11(2), 233–278 (2000)
Larson, R.G., Sweedler, M.E.: An associative orthogonal form for Hopf algebras. Amer. J. Math. 91, 75–94 (1969)
Meyer, R.: Local and analytic cyclic homology. EMS Tracts. Math. 3 (2007)
Timmermann, T.: An invitation to quantum groups and duality EMS textbooks in mathematics (2008)
Van Daele, A.: Multiplier Hopf algebras. Trans. Amer. Math. Soc. 342, 917–932 (1994)
Van Daele, A., Zhang, Y.: Corepresentation theory of multiplier Hopf algebras I. Int. J. Math. 10(4), 503–539 (1999)
Van Daele, A., Zhang, Y.: Galois theory for multiplier hopf algebras with integrals. Alg. Represent. Theory 2(1), 83–106 (1999)
Voigt, C.: Bornological quantum groups. Pacific J. Math. 235(1), 93–135 (2008)
Zimmermann-Huisgen, B.: Pure submodules of direct products of free modules. Math. Annalen 224(2), 233–245 (1976)
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Presented by Susan Montgomery.
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Böhm, G., Lack, S. Multiplier Hopf Monoids. Algebr Represent Theor 20, 1–46 (2017). https://doi.org/10.1007/s10468-016-9630-7
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DOI: https://doi.org/10.1007/s10468-016-9630-7