Representations Of Skew Group Algebras Induced From Isomorphically Invariant Modules Over Path Algebras

The present thesis concerns the representation of kQ#kG in terms of kQ with G an automorphism group of Q. Generally, we give invariant properties of representations of algebras under cleft extensions over a semisimple Hopf algebras. We prove that, when H is semisimple and cosemisimple Hopf algebra over algebraically closed field, the representation type of an algebra is an invariant property under a finite dimensional H-cleft extension.Let k be an algebraically closed field whose characteristic doesn't divide the order of G, the dual quiver QG of kQ with respect to G, which is the Ext-quiver of the basic algebra Morita equivalent to kQ#kG, is constructed by constructing isomorphism class of simple(projective) kQ#kG-modules from all simple(projective) kQ-modules.Trivially, for G=???a cyclic group, we investigate the relationship between in-decomposable kQ-modules and indecomposable kQ#k???-modules. We prove that any indecomposable kQ#k???-module is an indecomposable???-equivalent kQ-module. We construct all non-isomorphic indecomposable kQ#k???-modules from indecomposable???-equivalent kQ-modules and give the number of all non-isomorphic indecomposable kQ#k(a)-modules which are obtained from the same indecomposable???-equivalent kQ-module.Finally, we concern the Clebsch-Gordan problem for kQ#???, i.e., the problem of decomposing the tensor product of any two modules of kQ#k???, which is totally depended on the construction of indecomposable kQ#k???-modules from indecom-posable kQ-modules. This is then applied to give the Clebsch-Gordan formulas for kQ#k(a) for Q of type An,Dn and Kronecker quiver. The obtained Clebsch-Gordan formulas are used to study the set of complete orthogonal idempotents of representa-tion ring R(Q,?) induced from R(Q).