Weak Hopf Group-Coalgebras

Througout the paper,we let be a discrete group(vvith neutral element l)and k be a field.All k-algebras over k are supposed to be associtive and unitary.If U and V are k-spaceswill denote the flip defined by The main contents of this paper is Weak Hopf π-Coalgebras.We mainly consider the Weak π-comodule over it and we prove the foundermental structure theorem of Weak Hopf π-Comodule; We discuss the relations between Weak integrals of H1 and the semislmplicity of H(a Weak Hopf π-Coalgebra).We study π-integrals and the antipode of a Weak Hopf π-Coalgebra.We study it in mainly four parts form three respects and we get some special characterizations.In chapter one,we list the contents and the properties to be used in the following parts.In the chapter two ,we construct the Weak Hopf π- Coalgebras. The Weak π-counit subalgebras are studied thoroughly and we proved all respects needed in the following and our main results are as the followings: is a k-algebra.Lemma 1Corollorv 1 ] be a Weak Hopf π- Coalgebras.then:Lemma 2 be the counit of H = and be the antipode of H1 then:Proposition 1 Let we can claim that are all subalgebras of Ha and they are commutative. Lemma 3 Suppose a, . then we have:Theorem 1 Provided that H = be a Weak Hopf π-Coalgebra;then we have:In the third section,we mainly study the π-Comudules:the Weak Hopf π-Comodules;and we get the main results as: Theorem 2 Suppose C is a π-Coalgebra ,then we say:(a) There is a one-to-one correspondence between (isomorphic classses of ) right π - Comoduleover C and (isomorphic classes of )rational π-graded left C*-modules.(b) Every graded submodule of a rational π-graded left C*-module is rational.(c) Any π-graded left C*- module.L = has a unique maximal rational graded submodule,noted Lrayt, which is equal to the sum of all rational graded submodules of L.Moreover ,if is defined as in previous ,thenLemma 4 (M,p) is a right π-Comodule over C if and only if is aπ-graded left C*-module.Lemma 5 Let (M, p) be a right Weak Hopf π- Comoduleover H = a Weak Hopf π-coalgebra,then for all (1) is a right submodule over Ha of M.Theorem 3 If H is a Weak Hopf π- coalgebra and M = {Ma}a is a right Weak Hopf π- Comodule over H;we claim that:In the fourth section ,we mainly study the Weak integrals of H\ and the semisimplicity of H.We discuss the relation between the π-integral and the antipiode of H,We get the following main results:Lemma 6 Let H = be a Weak Hopf π- Coalgebra ,then the following statements for an element are equivalent(a) 1 a integral of H1(b) (c) (d) (e) (f ) is a right integral of H\Theorem 4 Let H = be a Weak Hopf π- Coalgebra.thenthe following conditions on H1 are equivalent(a) H1 is semisimple.(b) There exixts a normalized left integral 1 of H1(c) H1 is a separable k-algebra.Lemma 7 H = be a Weak Hopf π- Coalgebra and is a normalized left integral of H1; then for all Theorem 5 Let H = be a finite - dimension weak Hopf π-Coalgebra ,then H is semisimple if and only if H1 is semisimple.Definition 4,3 Weak Hopf π-integral:Let be a Weak Hopf π-Coalgebra ;a left π-integral for H is a family of k-linear forms such that,for all And the right π-integral is defined siminlarly such as: Theorem 6 Let H be a Weak Hopf π-Coalgebra ,right Weak π-integral, thenSuppose is & dual integral of (that is then At last we discuss the correspondence respects of the crossed Weak Hopf Group-Coalgebras.We proved Theorm 7 and theorm 8 .At the same time we gave the definition of Weak π-integrals and \ve used it in theorm 8.