Magnitudes Of Finite Dimension Algebras

In topology and geometry,as a classical invariant of Euler characteristic and dimension,magnitude has been widely known.In 2016,J.Chuang,A.King,and J.Leinster int,roduced and studied the magnitudes of a finite dimensional algebra,and it is same as t,he classical invariant of above all,they propose that the magnitudes of finite dimensional algebras also is important in algebras,specifically,in the repre-sentation theory of an associative.In this thesis,we use the method of homology to calculate the magnitude of two kinds of finite dimensional algebras and the method of the one point extension algebras and the tensor product algebra.This thesis is divided into two parts.The first part introduces the background of the project,necessary foundations and main results of this thesis.The second part consists of five chapters,and main results are as follows.In the first chapter,we introduce the basic concepts and notation of magnitudes of the finite dimensional algebras.Firstly,we introduce the basic concepts and no?tations of the magnitudes of matrixs,and propose the calculation of the magnitudes of finite dimensional algebras is transforemed into the calculation of the magnitudes of the matrix.In the second cha.pter,we caculate the magnitudes of a class of Nakayama algebras with acyclic quiver.We let A=kQ/I is a class of Nakayama algebras with acyclic quiver,Q is the acyclic quiver with n points,(?),we study the homological properties of Nakayama algebras,on the basis of these lemmas,we use the method of homology to get magnitude of A,the content of the theorem is that if x=0,|IP(A)|=a,ifx?0,|IP(A)|=a+1.In the third chapter,we caculate the magnitudes of the canonical algebras.By computing homology groups between the single modules,we calculate the magni-tudes of the canonical algebras is 1.In the forth chapter,we caculate the one point extension algebras.By comparing the relation between the Cartan matrix of one point extension algebra of A and the Cartan matrix of the original algebra,the magnitude of a one point extension algebra is computed.As a corollary,we calculate the magnitude of the tubular algebras.In the fifth chapter,we caculate the magnitude of the tensor product algebra.Let A,B is a finite dimension algebra of finite global dimension,by calculating the relation of the Cartan matrix of the tensor product algebra A(?)B and the Cartan matrix of the A,B,then |IP(A(?)B)| = |IP(A)||IP(B)|.Finally,a brief conclusion and a prospect are made in the last chapter.