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Yetter-Drinfel’d modules for group-cograded multiplier Hopf algebras. (English) Zbl 1194.16026
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)].

MSC:
16T05 Hopf algebras and their applications
16T20 Ring-theoretic aspects of quantum groups
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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