In this paper we study Hopf algebra by using representation theory of asso-ciative algebras in several ways.LetΛbe an artin ring. Let P be an indecomposable projective leftΛ-module. If we viewΛas a leftΛ-module, then the multiplicity of P in the directsum decomposition ofΛequals to the dimension of corresponding simple moduleS as a vector space over a division ring. As an example of application of ourmain result, we simplify a proof of decomposition of restricted quantum groupof U_q(sl_2).The structure of weak Hopf algebra wsl_q(2) is studied in this paper in orderto give a complete description of representation of wsl_q(2). The algebra struc-ture of wsl_q(2) is decomposed into a direct sum of U_q(sl_2) and the algebra ofpolynomials of two indeterminates. The coatgebra structure of wsl_q(2) is provedto be indecomposable. Another weak Hopf algebra vsl_q(2) is decomposed intoa direct sum of U_q(sl_2) and the trivial algebra k. Then we study all possibleweak Hopf algebras corresponding to U_q(sl_2). There are 9 possible non isomor-phic weak Hopf algebras corresponding to U_q(sl_2). We also give the direct sumdecomposition of weak Hopf algebras corresponding to U_q(sl_n).We study the actions of Hopf algebras on algebras and coalgebras. As aspecial case of pointed Hopf algebras, the actions of group Hopf algebras areequivalent to that of groups. We consider graded actions of groups on pathalgebras.The structure constants for coalgebras and Hopf algebras are similar to thoseof algebras. We introduce higher dimensional matrices and use them to describethe structure constants for Hopf algebras. We determine the conditions for apre-coalgebra to be coalgebra and Hopf algebra in form of higher dimensionalmatrices.