Invertible skew pairings and crossed products for weak Hopf algebras

Year 2022, , 1600 - 1620, 01.12.2022

José Nicanor Alonso álvarez J.m. Fernadez Vılaboa , Ramon Gonzalez Rodriguez

https://doi.org/10.15672/hujms.913738

Abstract

In this paper we work with invertible skew pairings for weak bialgebras in a symmetric monoidal category where every idempotent morphism splits. We prove that this kind of skew pairings induces examples of weak distributive laws and therefore they provide weak wreath products. Also we will show that they define weakly comonoidal mutually weak inverse pairs of weak distributive laws and, by the results proved by G. Böhm and J. Gómez-Torrecillas, we obtain weak wreath products that become weak bialgebras with respect to the tensor product coalgebra structure. As an application, we will show that the Drinfel'd double of a finite weak Hopf algebra can be constructed using the weak wreath product associated to an invertible $1$-skew pairing.

Keywords

weak Hopf algebra, skew pairing, monoidal category, distributive law, wreath product, Drinfel’d double

Supporting Institution

Ministerio de Economía, Industria y Competitividad of Spain. Agencia Estatal de Investigación. Unión Europea - Fondo Europeo de Desarrollo Regional.

Project Number

MTM2016-79661

References

• [1] Y. Bespalov, T. Kerler, V. Lyubashenko and V. Turaev, Integrals for braided Hopf Algebras, J. Pure Appl. Algebra 148 (2), 123-164, 2000.
• [2] G. Böhm, Doi-Hopf modules over weak Hopf algebras, Comm. Algebra 28(10), 4687- 4698, 2000.
• [3] G. Böhm and J. Gómez-Torrecillas, On the double crossed product of weak Hopf algebras, Hopf algebras and tensor categories, 153-173, Contemp. Math. 585, Amer. Math. Soc., Providence, RI, 2013.
• [4] G. Böhm, F. Nill and K. Szlachányi, Weak Hopf algebras, I: Integral theory and $C^*$- structure, J. Algebra 221 (2), 385-438, 1999.
• [5] S. Caenepeel and E.J. De Groot, Modules over weak entwining structures, New trends in Hopf algebra theory 31-54, Contemp. Math. 267, Amer. Math. Soc., Providence, RI, 2000.
• [6] S. Caenepeel, D. Wang and Y. Yin, Yetter-Drinfeld modules over weak Hopf algebras, Ann. Univ. Ferrara - Sez. VII - Sc. Mat. 51, 69-98, 2005.
• [7] Y. Doi, Braided bialgebras and quadratic bialgebras, Comm. Algebra 21 (5), 1731- 1749, 1993.
• [8] Y. Doi and M. Takeuchi, Multiplication alteration by two-cocycles. The quantum version, Comm. Algebra 22 (14), 5175-5732, 1994.
• [9] J.M. Fernández Vilaboa, R. González Rodríguez and A.B. Rodríguez Raposo, Preunits and weak crossed products, J. Pure Appl. Algebra 213 (12), 2244-2261, 2009.
• [10] T. Hayashi, Face algebras I. A generalization of quantum group theory, J. Math. Soc. Japan 50 (2), 293-315, 1998.
• [11] T. Hayashi, A brief introduction to face algebras, New trends in Hopf algebra theory, 161-176, Contemp. Math. 267, Amer. Math. Soc., Providence, RI, 2000.
• [12] A. Joyal and R. Street, Braided monoidal categories, Macquarie Univ. Reports 860081, 1986.
• [13] A. Joyal and R. Street, Braided tensor categories, Adv. Math. 102 (1), 20-78, 1993.
• [14] R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras, vol. I. Elementary theory. Reprint of the 1983 original. Graduate Studies in Mathematics 15, American Mathematical Society, Providence, RI, 1997.
• [15] C. Kassel, Quantum Groups, Quantum groups. Graduate Texts in Mathematics 155, Springer-Verlag, New York, 1995.
• [16] S. Mac Lane, Categories for the working mathematician. Second edition. Graduate Texts in Mathematics 5, Springer-Verlag, New York, 1998.
• [17] A. Nenciu, The center construction for weak Hopf algebras, Tsukuba J. Math. 26 (1), 189-204, 2002.
• [18] D. Nikshych and L. Vainerman, Finite quantum groupoids and their applications, New directions in Hopf algebras, 211-262, Math. Sci. Res. Inst. Publ. 43, Cambridge Univ. Press, 2002.
• [19] R. Meyer, Local and Analytic Cyclic Homology, EMS Tracts in Mathematics 3, European Mathematical Society (EMS), Zürich, 2007.
• [20] D.E. Radford, Hopf algebras, Series on Knots and Everything 49, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
• [21] P. Schauenburg, Duals and doubles of quantum groupoids ($\times_{R}$-Hopf algebras), New trends in Hopf algebra theory, 273-299, Contemp. Math. 267, Amer. Math. Soc., Providence, RI, 2000.
• [22] P. Schauenburg, Weak Hopf algebras and quantum groupoids, Noncommutative geometry and quantum groups, 171-188, Banach Center Publ. 61, Polish Acad. Sci. Inst. Math., Warsaw, 2003.
• [23] R. Street, Weak distributive laws, Theory Appl. Categ. 22, 313-320, 2009.
• [24] M. Takeuchi, Groups of algebras over $A\otimes\overline{A}$, J. Math. Soc. Japan 29 (3), 459-492. 1997.
• [25] T. Yamanouchi, Duality for generalized Kac algebras and a characterization of finite groupoid algebras, J. Algebra 163 (1), 9-50, 1994.