zbMATH — the first resource for mathematics

Multiplier Hopf algebras in categories and the biproduct construction. (English) Zbl 1163.16026
Summary: 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.

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S40 Smash products of general Hopf actions
Full Text: DOI
[1] Beattie, M., Dǎscǎlescu, S., Grünenfelder, L., Nǎstǎsescu, C.: Finiteness conditions, co-Frobenius Hopf algebras and quantum groups. J. Algebra 200, 312–333 (1998) · Zbl 0902.16028 · doi:10.1006/jabr.1997.7238
[2] Delvaux, L.: Semi-direct products of multiplier Hopf algebras: smash products. Comm. Algebra 30, 5961–5977 (2002) · Zbl 1038.16027 · doi:10.1081/AGB-120016026
[3] Delvaux, L.: Twisted tensor product of multiplier Hopf ()algebras. J. Algebra 269, 285–316 (2003) · Zbl 1036.16030 · doi:10.1016/S0021-8693(03)00467-8
[4] Delvaux, L.: Twisted tensor coproduct of multiplier Hopf algebras. J. Algebra 274, 751–771(2004) · Zbl 1071.16032 · doi:10.1016/j.jalgebra.2003.09.006
[5] Delvaux, L., Van Daele, A., Wang, S.H.: Quasitriangular (G-cograded) multiplier Hopf algebras. J. Algebra 289, 484–514 (2005) · Zbl 1079.16022 · doi:10.1016/j.jalgebra.2005.02.023
[6] Drabant, B., Van Daele, A., Zhang, Y.: Actions of multiplier Hopf algebras. Comm. Algebra 27, 4117–4127 (1999) · Zbl 0951.16013 · doi:10.1080/00927879908826688
[7] Majid, S.: Doubles of quasitriangular Hopf algebras. Comm. Algebra 19, 3061–3073 (1991) · Zbl 0767.16014 · doi:10.1080/00927879108824306
[8] Majid, S.: Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group. Comm. Math. Phys. 156, 607–638 (1993) · Zbl 0804.17013 · doi:10.1007/BF02096865
[9] Majid, S.: Algebras and Hopf algebras in braided categories. In: Advances in Hopf Algebras, pp. 55–105. Marcel Dekker, New York (1994) · Zbl 0812.18004
[10] Radford, D.E.: The structure of Hopf algebras with a projection. J. Algebra 92, 322–347 (1985) · Zbl 0549.16003 · doi:10.1016/0021-8693(85)90124-3
[11] Van Daele, A.: Multiplier Hopf algebras. Trans. Amer. Math. Soc. 342, 917–932 (1994) · Zbl 0809.16047 · doi:10.2307/2154659
[12] Van Daele, A.: An algebraic framework for group duality. Adv. Math. 140, 323–366 (1998) · Zbl 0933.16043 · doi:10.1006/aima.1998.1775
[13] 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) · Zbl 1020.16032
[14] Van Daele, A., Zhang, Y.: Galois Theory for multiplier Hopf algebras with integrals. Algebr. Represent. Theory 2, 83–106 (1999) · Zbl 0929.16038 · doi:10.1023/A:1009938708033
[15] Zhang, Y.: The quantum double of a co-Frobenius Hopf algebra. Comm. Algebra 27, 1413–1427 (1999) · Zbl 0921.16027 · doi:10.1080/00927879908826503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.