The Automorphism Groups Of Connected Graded Frobenius Algebras

Frobenius algebras are a class of finite dimensional algebras with an associative nondegenerate bilinear form.This class of algebras has good duality properties.This thesis mainly studies the structure and automorphism groups of connected graded Frobenius algebras with length 2.The thesis is divided into four chapters.The first chapter is an introduction,which mainly introduces the research background of Frobenius algebra and the main results of this thesis,and gives the preliminary knowledge.In the second chapter,we mainly study the structure of connected graded Frobenius algebras with length 2.We classify connected graded Frobenius algebras over complex field C with three generators of length 2 up to isomorphisms,and obtain some general conclusions of the matrix T corresponding to the twisted superpotential of a connected graded Frobenius algebra with n generators and length 2.In the third chapter,we mainly study the automorphism groups of graded algebras with length 2.First,we prove that the automorphism group of graded algebras with length 2(not necessarily preserving degree)is the semidirect product of a subgroup and the group of degreepreserving automorphisms.In addition,we obtain the degree-preserving automorphism group of graded algebras with length 2.The fourth chapter summarizes the main results of the thesis and prospects the future research work.