Q-Cartan Matrices For Self-injective Nakayama Algebras

Cartan matrix is an important invariant used to study the global dimension of algebras in the representation theory of associative algebras,which plays an extremely important role in the representation theory of algebras.Cartan matrix,as the derived invariant of algebras,has been studied extensively recently.Gabriel theorem established that any finite dimensional algebra can be realized by a quiver with an admissible ideal.Dynkin type(An,Dn,E6,E7,E8)quivers and Euclid type((?)n,Dn,E6,E7,E8)quivers are important types of quivers in the representation theory of algebras.This thesis mainly studies the q-Cartan matrix corresponding to a special kind of quiver of Euclid type((?)n-type quiver).By defining two different kinds of admissible ideals on An-type quiver,the global dimensions of the algebras corresponding to the quiver are finite.The determinant and some properties of q-Cartan matrix corresponding to quiver are given according to the global dimension.More specifically,the main content of this thesis includes two parts.First,this thesis calculate the q-Cartan determinant of Nakayama algebras through the properties of circulant matrix;Secondly,we introduce the admissible ideal of the effective intersection of the k(?)n,and prove that the conjecture of Cartan determinant is also true for the q-Cartan matrix.