Delvaux, L. Yetter-Drinfel’d modules for group-cograded multiplier Hopf algebras. (English) Zbl 1194.16026 Commun. Algebra 36, No. 8, 2872-2882 (2008). Given a pairing \(\langle A,B\rangle\) of regular multiplier Hopf algebras, as in B. Drabant and A. Van Daele [Algebr. Represent. Theory 4, No. 2, 109-132 (2001; Zbl 0993.16024)], with an admissable action \(\pi\) of \(G\) on \(B\), the Drinfel’d double \(D^\pi\) can be constructed following L. Delvaux and A. Van Daele [Algebr. Represent. Theory 10, No. 3, 197-221 (2007; Zbl 1161.16028)]. Group-cograded multiplier Hopf algebras were introduced by A. T. Abd El-Hafez, L. Delvaux and A. Van Daele [Algebr. Represent. Theory 10, No. 1, 77-95 (2007; Zbl 1129.16027)], and this double construction generalizes M. Zunino [J. Algebra 278, No. 1, 43-75 (2004; Zbl 1058.16035)]. In the paper under review, a characterization of left \(D^\pi\)-modules is given purely in terms of modules and comodules over \(B\). This characterization leads to a notion of “\(\pi\)-Yetter-Drinfel’d modules” over \(B\). Under additional restrictions on \(\pi\), the author further proves that the monoidal category of \(\pi\)-Yetter-Drinfel’d modules is \(\pi\)-braided. This analysis is applicable to the “mirror construction” of M. Zunino [J. Pure Appl. Algebra 193, No. 1-3, 313-343 (2004; Zbl 1075.16019)]. Reviewer: Edward S. Letzter (Philadelphia) Cited in 3 Documents MSC: 16T05 Hopf algebras and their applications 16T20 Ring-theoretic aspects of quantum groups 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) Keywords:Drinfeld doubles; group-cograded multiplier Hopf algebras; Hopf group-coalgebras; Yetter-Drinfeld modules; monoidal categories; comodules PDF BibTeX XML Cite \textit{L. Delvaux}, Commun. Algebra 36, No. 8, 2872--2882 (2008; Zbl 1194.16026) Full Text: DOI References: [1] DOI: 10.1007/s10468-006-9043-0 · Zbl 1129.16027 · doi:10.1007/s10468-006-9043-0 [2] DOI: 10.1081/AGB-120016027 · Zbl 1038.16028 · doi:10.1081/AGB-120016027 [3] DOI: 10.1081/AGB-200065373 · Zbl 1081.16040 · doi:10.1081/AGB-200065373 [4] DOI: 10.1016/j.jalgebra.2003.03.003 · Zbl 1044.16028 · doi:10.1016/j.jalgebra.2003.03.003 [5] DOI: 10.1016/j.jpaa.2003.10.031 · Zbl 1056.16027 · doi:10.1016/j.jpaa.2003.10.031 [6] DOI: 10.1007/s10468-006-9042-1 · Zbl 1161.16028 · doi:10.1007/s10468-006-9042-1 [7] Delvaux L., J. Algebra 289 pp 484– · Zbl 1079.16022 · doi:10.1016/j.jalgebra.2005.02.023 [8] DOI: 10.1023/A:1011470032416 · Zbl 0993.16024 · doi:10.1023/A:1011470032416 [9] DOI: 10.1080/00927879908826688 · Zbl 0951.16013 · doi:10.1080/00927879908826688 [10] Kassel C., Quantum Groups (1995) · doi:10.1007/978-1-4612-0783-2 [11] DOI: 10.1007/BF02392485 · Zbl 0833.18005 · doi:10.1007/BF02392485 [12] DOI: 10.1080/00927879108824306 · Zbl 0767.16014 · doi:10.1080/00927879108824306 [13] DOI: 10.2307/2154659 · Zbl 0809.16047 · doi:10.2307/2154659 [14] DOI: 10.1006/aima.1998.1775 · Zbl 0933.16043 · doi:10.1006/aima.1998.1775 [15] Van Daele , A. , Zhang , Y. ( 2000 ).A Survey on Multiplier Hopf Algebras. Proceedings of the conference in Brussels on Hopf algebras and Quantum Groups. In: Caenepeel/Van Oystaeyen, eds. New York : Marcel Dekker , pp. 269 – 309 . · Zbl 1020.16032 [16] DOI: 10.1023/A:1009938708033 · Zbl 0929.16038 · doi:10.1023/A:1009938708033 [17] DOI: 10.1016/j.jalgebra.2004.03.019 · Zbl 1058.16035 · doi:10.1016/j.jalgebra.2004.03.019 [18] DOI: 10.1016/j.jpaa.2004.02.014 · Zbl 1075.16019 · doi:10.1016/j.jpaa.2004.02.014 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.