The Diagram Shows The Two Areas In The Monomial Quasi-hereditary Algebra

Two kinds of subcategories of the representation of quivers and their applications to monomial quasi- hereditary algebraLin WeiqiangAbstract:The representation theory of quivers ,which was initiated by Gabriel in order to reduce various problems of algebra and geometry to problems in linear algebra, is one of the central substances of the representation theory of algebra, In[G2] Gabriel outlined the use of quivers in the representation theory of algebra, and he gave in [G3]the famous Gabriel theory: Any basic finite dimensional k-algebra is of the form kQ/I for some uniquely determined finite quiver Q and some ideal I with ((kQ)⁾ c I c , for some integer n 2 .For a given quiver with relations, a representation of it consists of a class of vector spaces and some linear maps with certain properties. All representations of a quiver with relations construct the representation category of the quiver, which is equivalent to a finitely generated modular category of the algebra kQ/J. This category is a basic object studied by the representation theory of algebra. In [Gil Gabriel completely portrayed the essence of the representation category of path algebra which has only finite number of indeeomposable object. In this paper, we introduce two special classes of representations of quivers, and study many good properties of thecategories rep(Q,I) ,re(Q,I)eⁿstructed by these representations in 2 and in 3 respectively.Quasi-hereditaiy algebra were introduced by E . Cline, B Parshall and L. Scott in order to deal wLh highest weight categories as they arise in the representation theory of semi-simple complex Lie algebra and algebraic groups. Since then, many algebra arising naturally have been shown to be quasi-hereditary. V.Dlab and C.M. Ringel studied quasi-hereditary algebra from the aspect of ring theory the earliest. The central role in the study of quasi-hereditary algebr x are played by the important concept of a standard module, a costandard module, the good module category, the cogood module category and the characteristic module.In 4,We use two kinds of subcategories of the representation of quivers, which are defined in 2 and in 3 respectively, to discuss some properties of the standard module, the costandard module, the good module category, the cogood module category, and the characteristic module of monomial quasi-hereditary, The main result is that we give the character of A rep^Q,I) and F( A) rep^Q.I and their dual result.