Crossed products over weak Hopf algebras related to cleft extensions and cohomology

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.

Keywords

Monoidal category / Weak Hopf algebra / Cleft extension / Weak crossed product / Sweedler cohomology for weak Hopf algebras

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