Frobenius Algebra And Twisted Superpotential

Frobenius algebra is an important class of associative algebras in the field of algebraic research.The dual properties of Frobenius algebra are associated with a number of different branches of mathematics and physics,such as representation theory,quantum group and Hopf algebra theory,and so on.Classification and construction of Frobenius algebras have always been the concern of mathematicians.There is a special kind of Frobenius algebra called graded Frobenius algebra,with a simple and beautiful structure,among the numerous Frobenius algebras.Over the years,mathematicians have presented several results for graded Frobenius algebras.However,there are still a number of problems that have not been solved in the construction and classification of the graded Frobenius algebras.In this doctoral thesis,we study the construction and classification of the graded Frobenius algebras.In this article,we construct and classify the graded Frobenius algebras in terms of the element called twisted superpotential.The graded Frobenius algebras are divided in to connected and non-connected,hence we discuss these two cases separately:we classify the connected graded Frobenius algebras and its twisted algebra by twisted superpotentials;we give the definition of twisted superpotentials in nonconnected graded Frobenius algebra and use it to construct non-connected graded Frobenius algebra;the twisted superpotentials of the skew group algebras about the connected graded Frobenius algebra can be given as well.The paper is divided into five chapters:In chapter 1,we introduce the research background and main content of the topic.In chapter 2,we recall the basic concepts and results applied in this paper.In chapter 3,according to the correspondence between the twisted superpotential and the connected graded Frobenius algebra,we focus on the isomorphisms between the connected graded Frobenius algebras corresponding to different twisted superpotentials.In order to distinguish the connected graded Frobenius algebras determined by different twisted superpotentials,we introduce the non-degeneracy of twisted superpotential.We give the sufficient and necessary condition such that the connected graded Frobenius algebras determined by two nondegenerate twisted superpotential are isomorphic.As an application,we classify connected Z-graded Frobenius algebra of length 3,whose dimension of the degree 1 is 2.In chapter 4,we study the twisted algebras of the connected graded Frobenius algebra by twisting system.We construct the element wφ,induced by a twisted superpotential w and a graded coalgebra automorphism φ,and proof that wφ is also a twisted superpotential whose corresponding Frobenius algebra is the twisted algebra by(φ*)-1.Then we explore the condition such that the twisted algebras determined by distinct graded algebra automorphisms are isomorphic.In particular,we prove that any twisted algebra of the exterior algebra can be seen as a twisted algebra determined by a graded algebra automorphism.Consequently,we work out all the twisted algebras of the exterior algebra of length 3,whose dimension of the degree 1 is 3.In chapter 5,we discuss the connection between the twisted superpotentials and the non-connected graded Frobenius algebras.We extend the defmition of the twisted superpotentials in the cotensor coalgebra over the field K to the cotensor coalgebra over the coalgebra C.It is proved that we can construct a non-connected graded Frobenius algebra by a twisted superpotentials.The skew group algebra A#G of a connected graded Frobenius algebra A with a finite subgroup G of the graded algebra automorphism group AutGr(A)can be seen as a non-connected graded Frobenius algebra.And the twisted superpotential of A#G is explicitly constructed.