¦°-comodule Algebra And ¦Ð-i Mold Algebra

Early in the 1950s, the concept of the Hopf algebra was derived from an axiomatization of H.Hopf relating to topogical of Lie groups. We know that it is extensively connected with Lie algebra, differential geometry, algebraic topology and statistical physics. Hopf algebra is an interesting subject for people, and has been widely studied over the past several decades. One got many important results about constructing and classifying Hopf algebra. The notion of Hopfπ-coalgebra, whereπis a multiplicative group, generalizes Hopf algebra. Hopfπ-coalgebra was used by V. G. Turaev to construct Hennings-like and Kuperberg-like invariants of principalπ-bundles over link complements and over 3-manifolds. This paper mainly studied Maschke type theorem and Frobenius properties of Doi-Hopf modules for Hopfπ-algebra. VireLizier also studied some properties of the Hopfπ-coalgebra, see [1].In this paper,let H be a Hopfπ- coalgebra, we mainly discuss theπ- H- comodule algebra andπ- H- comodule subalgebra. Fisrtly,we produce the notion ofπ- tensor ofπ- H- comodules,and we obtain that it is still aπ- H- comodule,so we can get a necessary and sufficient condition for aπ- H- comodule algebra ; Secondly,we discuss the relation between aπ- H- comodule algebra A andπ- H-- module coalgebra A- ; Finally, we introduce the notion ofπ- H-comodule subalgebra, and we obtain a necessary and sufficient condition betweenπ- H-comodule andπ- H-- module coideal.In section 1, we have produced Hopfπ- coalgebra, Hopfπ- algebra,π-module,π-comodule and so on, and some essential explanations.In section 2, we define theπ-tensor ofπ-comodules of Hopfπ- coalgebra H ,and wo prove it still aπ- H- comodule;then we define theπ- tensor ofπ-modules of Hopfπ-algebra H ,and wo prove it still aπ- H-module. In section 3, we discuss the relation betweenπ- H- comodule algebra andπ- H-- module algebra, we obtain a conclusion.Theorem 3.7 Let H be a finite type Hopfπ- coalgebra , A is aπ- H- comodule algebra, then A- is aπ- H-- module coalgebra.In section 4, we discuss the relation betweenπ- H-comodule subalgebra andπ- H-- module coideal,and we obtain a necessary and sufficient condition.Theorem 4.11Let H be a Hopfπ- coalgebra , A is aπ- H- comodule algebra,then D is aπ- H-comodule subalgebra of A if and only if D⊥is aπ- H-- module coideal of A-.In section 5, we discuss a special case of theorem 4.11,and we obtain a deduction.Corollary 5.3 Let H be a Hopfπ- coalgebra, then A is a rightπ- coideal of H if and only if A⊥is a rightπ- ideal H-.