Superpotentials And Generalized Determinants

Frobenius algebras are a class of algebra with lots of symmetric properties,which are related to different branches of mathematics.In this note,we focus on connected graded Frobenius algebras.This class of Frobenius algebras are related to noncommutative projective geometry.Superpotentials are widely used in the constructions of various algebraic structures such as Calabi-Yau algebras Therefore,we are trying to construct connected graded Frobenius algebras in this wayDeterminant,which is a basic mathematical tool,has important applications in different branches of mathematics.Furthermore,many researches have extend?ed determinants to distinguished types.The introduction and exploration of the generalized determinants such as homological determinant,quantum determinant.are helpful to the proof of many kinds of algebraic theoriesIn this paper,We give a construction of connected graded Frobenius algebras in terms of twist superpotentials.We present a definition of generalized determi-nant,which is derived from a connected graded Frobenius algebra.Then we study the properties of the generalized determinantsIn the second chapter,it is proved that a connected graded Frobenius algebra is determined by a twisted superpotential,and vice versa.In the last section.Some examples of connected graded Frobenius are given with few generatorsIn the third chapter,we introduce the concept of generalized determinants.called σ-determinants.Some properties of σ-determinants are studied.We intro-duce a generalized matrix algebra and study a group-like element of it as well.