Frobenius-Perron Dimension Of Category Of Finite Dimensional Modules

The spectral radius of matrix,also known as Frobenius-Perron dimension,is a useful tool for studying linear category and plays an important role in the classification of categories of algebraic modules.Our main research antithesis of academic dissertation is the Frobenius-Perron dimension of three algebras,they are representation-directed algebra,quotient algebra of canonical algebra of type ADE and a class of algebra of bound quiver containing loops.In Chapter one,we introduce the background and the latest development of the research topic,and summarize the main work of this paper.In Chapter two,We review the Frobenius-Perron dimension of linear category and its endofunctors,and give the relationship of Frobenius-Perron dimension and τ FrobeniusPerron dimension of category of modules between an algebra and its quotient algebra.We proved that the τ Frobenius-Perron dimension of a category of modules is not less than its Frobenius-Perron dimension;The Frobenius-Perron dimension of a module category of an algebra is not less than the Frobenius-Perron dimension of category of any of its quotient algebras;the τ Frobenius-Perron dimension of category of modules is not less than the τ Frobenius-Perron dimension of category of any of its quotient algebras.In Chapter three,we calculate Frobenius-Perron dimension of several types of category of algebraic modules.The main conclusions are as follows:the Frobenius-Perron dimension of category of modules of representation-directed algebra is 0;the Frobenius-Perron dimension of category of modules of quotient canonical algebra is 0 or 1,and the sufficient and necessary condition for 0 is that there exists commutative relation between any two paths with common starting point and end point in the arrow graph;if the quotient algebra of algebra of path containing loops satisfies that all loops are central nilpotent and the quotient algebra obtained by the ideal generated by loop is representation-directed,then the Frobenius-Perron dimension of this algebra is the maximum number of loops at each vertex in the quiver.