The Maschke-type theorems of diagonal crossed products over Hopf group coalgebras

In this paper, a diagonal crossed product over a Hopf group coalgebra is constructed, and the sufficient and necessary conditions for it being a Hopf group coalgebra are given. Then we prove that the representation category of a diagonal crossed product over a Hopf group coalgebra is isomorphic to the category of Yetter-Drinfeld π-modules and extend the classic Maschke-type theorem of Hopf algebra to diagonal crossed products over Hopf group coalgebras.