Diagonal Crossed Products Over Weak Hopf Algebras

Let H be a finite-dimensional weak Hopf algebra and A an H-bimodule algebra. As a generalization of smash products over weak Hopf algebras, In this thesis we mainly introduce the notion of diagonal crossed products over weak Hopf algebras H; We prove the Maschke theorem of the diagonal crossed products over weak Hopf algebras; We construct the Morita context<AH, A (?) H,A,Q,τ,μ> connecting the diagonal crossed product A (?) H and the invariant subalgebra AH.This thesis is divided into four parts:In Part1, we recall the definitions of weak bialgebras and weak Hopf algebras over a field k and list some properties of weak Hopf algebras; We introduce the notions of left and right H-module algebras and give some examples; Moreover we recall the notion of the integrals in weak Hopf algbras.In Part2, we introduce the notion of the diagonal crossed products over weak Hopf alge-bras and discuss its properties. We indicate that a diagonal crossed product over a weak Hopf algebra is an associative algebra with a unit and can be seen as a generalization of a smash product over a weak Hopf algebra.In Part3, we prove the Maschke theorem of the diagonal crossed products over semisimple weak Hopf algebras which generalizes the corresponding result given by Cohen and Fishman.In Part4, we show that a diagonal crossed product over weak Hopf algebras is an A-ring with a grouplike character and we construct the Morita context<AH, A (?) H,A,Q,τ,μ connecting the diagonal crossed product A (?) H and the invariant subalgebra AH over weak Hopf algebras, which generalizes Theorem10in [13].