Weak multiplier bialgebras

Böhm, Gabriella; Gómez-Torrecillas, José; López-Centella, Esperanza

doi:10.1090/tran/6308

Weak multiplier bialgebras
HTML articles powered by AMS MathViewer

by Gabriella Böhm, José Gómez-Torrecillas and Esperanza López-Centella PDF

Trans. Amer. Math. Soc. 367 (2015), 8681-8721 Request permission

Abstract:

A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the ‘base algebras’) are shown to carry coseparable co-Frobenius coalgebra structures. Appropriate modules over a nice enough weak multiplier bialgebra are shown to constitute a monoidal category via the (co)module tensor product over the base (co)algebra. The relation to Van Daele and Wang’s (regular and arbitrary) weak multiplier Hopf algebra is discussed.

References

Gabriella Böhm and José Gómez-Torrecillas, Firm Frobenius monads and firm Frobenius algebras, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 56(104) (2013), no. 3, 281–298. MR 3114475
Gabriella Böhm, Florian Nill, and Kornél Szlachányi, Weak Hopf algebras. I. Integral theory and $C^*$-structure, J. Algebra 221 (1999), no. 2, 385–438. MR 1726707, DOI 10.1006/jabr.1999.7984
Gabriella Böhm and Joost Vercruysse, Morita theory for comodules over corings, Comm. Algebra 37 (2009), no. 9, 3207–3247. MR 2554197, DOI 10.1080/00927870902747993
F. Castaño Iglesias, S. Dăscălescu, and C. Năstăsescu, Symmetric coalgebras, J. Algebra 279 (2004), no. 1, 326–344. MR 2078404, DOI 10.1016/j.jalgebra.2004.05.007
John Dauns, Multiplier rings and primitive ideals, Trans. Amer. Math. Soc. 145 (1969), 125–158. MR 257151, DOI 10.1090/S0002-9947-1969-0257151-1
K. Janssen and J. Vercruysse, Multiplier bi- and Hopf algebras, J. Algebra Appl. 9 (2010), no. 2, 275–303. MR 2646665, DOI 10.1142/S0219498810003926
F. Nill, Axioms for weak bialgebras, preprint available at http://arxiv.org/abs/math/9805104.
Kornel Szlachanyi, Adjointable monoidal functors and quantum groupoids, Hopf algebras in noncommutative geometry and physics, Lecture Notes in Pure and Appl. Math., vol. 239, Dekker, New York, 2005, pp. 291–307. MR 2106937
Thomas Timmermann, An invitation to quantum groups and duality, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008. From Hopf algebras to multiplicative unitaries and beyond. MR 2397671, DOI 10.4171/043
A. Van Daele, Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (1994), no. 2, 917–932. MR 1220906, DOI 10.1090/S0002-9947-1994-1220906-5
A. Van Daele, Separability idempotents and multiplier algebras, preprint available at http://arxiv.org/abs/1301.4398.
Alfons Van Daele and Shuanhong Wang, Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples, Operator algebras and quantum groups, Banach Center Publ., vol. 98, Polish Acad. Sci. Inst. Math., Warsaw, 2012, pp. 367–415. MR 3059670, DOI 10.4064/bc98-0-16
A. Van Daele and S. Wang, Weak multiplier Hopf algebras. The main theory, Journal für die reine und angewandte Mathematik (Crelle’s Journal), in press, DOI:10.1515/crelle-2013-0053.