Van Daele, A.; Wang, Shuanhong New braided crossed categories and Drinfel’d quantum double for weak Hopf group coalgebras. (English) Zbl 1154.16031 Commun. Algebra 36, No. 6, 2341-2386 (2008). The paper studies properties of weak Hopf group-coalgebras, previously introduced by the authors. A fundamental theorem for weak Hopf group-comodules is proved, and the concept of a Yetter-Drinfeld module over weak crossed structures is defined. The method uses the approach of S. Caenepeel and M. De Lombaerde [Commun. Algebra 34, No. 7, 2631-2657 (2006; Zbl 1103.16024)] to Turaev’s categorical theory. An analog of a Drinfeld quantum double is defined over a weak crossed Hopf group-coalgebra, and it is shown that the category of modules over such a Drinfeld quantum double is isomorphic to the category of Yetter-Drinfeld modules. In this way new classes of braided crossed categories in the sense of Turaev are constructed. Reviewer: Sorin Dascalescu (Bucureşti) Cited in 2 ReviewsCited in 16 Documents MSC: 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 57R56 Topological quantum field theories (aspects of differential topology) Keywords:braided crossed categories; Drinfeld quantum doubles; Turaev categories; weak Hopf algebras; weak Hopf group-coalgebras; weak Hopf group-comodules; Yetter-Drinfeld modules Citations:Zbl 1103.16024 PDFBibTeX XMLCite \textit{A. Van Daele} and \textit{S. Wang}, Commun. Algebra 36, No. 6, 2341--2386 (2008; Zbl 1154.16031) Full Text: DOI References: [1] DOI: 10.1007/s10468-006-9043-0 · Zbl 1129.16027 · doi:10.1007/s10468-006-9043-0 [2] DOI: 10.1016/j.jalgebra.2005.09.012 · Zbl 1117.16024 · doi:10.1016/j.jalgebra.2005.09.012 [3] DOI: 10.1080/00927870008827113 · Zbl 0965.16024 · doi:10.1080/00927870008827113 [4] DOI: 10.1006/jabr.1999.7984 · Zbl 0949.16037 · doi:10.1006/jabr.1999.7984 [5] DOI: 10.1080/00927870600651430 · Zbl 1103.16024 · doi:10.1080/00927870600651430 [6] Caenepeel S., Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations (2002) · Zbl 1008.16036 [7] Caenepeel S., Ann. Univ. Ferrara-Sez. VII-Sc. Mat. 51 pp 69– (2005) [8] DOI: 10.1007/s10468-006-9042-1 · Zbl 1161.16028 · doi:10.1007/s10468-006-9042-1 [9] DOI: 10.1016/0001-8708(89)90018-2 · Zbl 0679.57003 · doi:10.1016/0001-8708(89)90018-2 [10] DOI: 10.1006/aima.1993.1055 · Zbl 0817.18007 · doi:10.1006/aima.1993.1055 [11] Kassel , C. ( 1995 ). Quantum Groups . Graduate Texts in Mathematics, Vol. 155 . New York : Springer . · Zbl 0808.17003 [12] Maclane S., Categories for the Working Mathematician (1971) · Zbl 0705.18001 [13] DOI: 10.1016/S0166-8641(02)00055-X · Zbl 1021.16026 · doi:10.1016/S0166-8641(02)00055-X [14] Sweedler M., Hopf Algebras (1969) [15] Turaev V. G., Quantum Invariants of Knots and 3-Manifolds. de Gruyter Stud. Math. 18 (1994) [16] DOI: 10.2307/2154659 · Zbl 0809.16047 · doi:10.2307/2154659 [17] DOI: 10.1006/aima.1998.1775 · Zbl 0933.16043 · doi:10.1006/aima.1998.1775 [18] Van Daele A., A Von Neumann Algebra Approach (2006) [19] DOI: 10.1016/S0022-4049(01)00125-6 · Zbl 1011.16023 · doi:10.1016/S0022-4049(01)00125-6 [20] DOI: 10.1081/AGB-120039402 · Zbl 1073.16034 · doi:10.1081/AGB-120039402 [21] DOI: 10.1016/j.jpaa.2004.02.014 · Zbl 1075.16019 · doi:10.1016/j.jpaa.2004.02.014 [22] DOI: 10.1016/j.jalgebra.2004.03.019 · Zbl 1058.16035 · doi:10.1016/j.jalgebra.2004.03.019 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.