Abstract
Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beattie, M., Dǎscǎlescu, S., Grünenfelder, L., Nǎstǎsescu, C.: Finiteness conditions, co-Frobenius Hopf algebras and quantum groups. J. Algebra 200, 312–333 (1998)
Delvaux, L.: Semi-direct products of multiplier Hopf algebras: smash products. Comm. Algebra 30, 5961–5977 (2002)
Delvaux, L.: Twisted tensor product of multiplier Hopf (∗-)algebras. J. Algebra 269, 285–316 (2003)
Delvaux, L.: Twisted tensor coproduct of multiplier Hopf algebras. J. Algebra 274, 751–771(2004)
Delvaux, L., Van Daele, A., Wang, S.H.: Quasitriangular (G-cograded) multiplier Hopf algebras. J. Algebra 289, 484–514 (2005)
Drabant, B., Van Daele, A., Zhang, Y.: Actions of multiplier Hopf algebras. Comm. Algebra 27, 4117–4127 (1999)
Majid, S.: Doubles of quasitriangular Hopf algebras. Comm. Algebra 19, 3061–3073 (1991)
Majid, S.: Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group. Comm. Math. Phys. 156, 607–638 (1993)
Majid, S.: Algebras and Hopf algebras in braided categories. In: Advances in Hopf Algebras, pp. 55–105. Marcel Dekker, New York (1994)
Radford, D.E.: The structure of Hopf algebras with a projection. J. Algebra 92, 322–347 (1985)
Van Daele, A.: Multiplier Hopf algebras. Trans. Amer. Math. Soc. 342, 917–932 (1994)
Van Daele, A.: An algebraic framework for group duality. Adv. Math. 140, 323–366 (1998)
Van Daele, A., Zhang, Y.: A survey on multiplier Hopf algebras. In: Hopf algebras and Quantum Groups, pp. 259–309. Marcel Dekker, New York (1998)
Van Daele, A., Zhang, Y.: Galois Theory for multiplier Hopf algebras with integrals. Algebr. Represent. Theory 2, 83–106 (1999)
Zhang, Y.: The quantum double of a co-Frobenius Hopf algebra. Comm. Algebra 27, 1413–1427 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Delvaux, L. Multiplier Hopf Algebras in Categories and the Biproduct Construction. Algebr Represent Theor 10, 533–554 (2007). https://doi.org/10.1007/s10468-007-9053-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-007-9053-6