Abstract
The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions A H ↪ A, where A H is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over A H . Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group \(H_{\phi _{Z(A_H )} }^2 \) (H,Z(A H )), where Z(A H ) is the center of A H .
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This work was supported by the project of Ministerio de Ciencia e Innovación (No.MTM2010-15634) and Fondo Europeo de Desarrollo Regional.
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Álvarez, J.N.A., Vilaboa, J.M.F. & Rodríguez, R.G. Crossed products over weak Hopf algebras related to cleft extensions and cohomology. Chin. Ann. Math. Ser. B 35, 161–190 (2014). https://doi.org/10.1007/s11401-014-0828-x
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DOI: https://doi.org/10.1007/s11401-014-0828-x
Keywords
- Monoidal category
- Weak Hopf algebra
- Cleft extension
- Weak crossed product
- Sweedler cohomology for weak Hopf algebras