Locally Finite Derivations And LFED Conjecture Of Some Associative Algebras

Affine algebraic geometry mainly studies affine varieties and their polynomial maps.From the perspective of algebra,it is the study of polynomial algebras and their homomorphisms.Understanding polynomial automorphism groups is an important task in algebra and affine algebraic geometry.A fundamental problem is how to determine whether a polynomial map is a polynomial automorphism.The Jacobian conjecture asserts that if the Jacobian determinant of a polynomial map F is a non-zero constant,then F is a polynomial automorphism.The Jacobian conjecture has attracted the attention of many famous mathematicians.Fields Prize winner Smale listed the Jacobian conjecture as one of the eighteen mathematical problems for the 21st century.The Jacobian conjecture holds for polynomial maps of degree 1 and 2.Bass and others reduced the Jacobian conjecture to the following case of polynomial maps F=X+H,where each component of H is a homogeneous polynomial of degree 3.Furthermore,de Bondt and others reduced H to a gradient map of a homogeneous polynomial of degree 4 with the nilpotent Hessian matrix.Zhao gave equivalent conditions for the Hessian matrix of f to be nilpotent,and for the polynomial map X+▽f to be invertible,respectively.This prompted him to propose the theory of Mathieu subspaces.Subsequently,many problems in affine algebraic geometry became proving that a subspace,especially the image or kernel of some linear map(such as a derivation),is a Mathieu subspace.Van den Essen,Wright and Zhao proved that if D is a locally finite derivation of K[x,y],then Im D is a Mathieu subspace.Then,Zhao formulated LFED conjecture:Images of locally finite derivations or ε-derivations of K-algebras of characteristic zero are Mathieu subspaces.Van den Essen and Zhao found that the LFED conjecture implies Duistermaat-van der Kallen theorem the case of Mathieu’s conjecture about Abelian Lie groups.This reflects the theoretical significance of the LFED conjecture from another perspective.Although the LFED conjecture has been studied by many scholars,it is far from being resolved.At present,the LFED conjecture has only been proved for derivations and ε-derivations on some special algebras.Zhao discovered the important role of idempotents for determining Mathieu subspaces,so he proposed a weaker conjecture than the LFED conjecture-idempotent conjecture:If D is a locally finite derivation or ε-derivation of K-algebras of characteristic zero,then(e)?Im D,for any idempotent e ∈ Im D.In Chapter 3,we prove that images of locally finite ε-derivations on K[x,y]are Mathieu subspaces.This,combining with a conclusion on derivations due to Van den Essen,Wright and Zhao,ensures that the LFED conjecture holds for K[x,y].Next,we give a sufficient condition for monomial spaces in polynomial algebras to be Mathieu subspaces.In Chapter 4,we characterize idempotents of incidence algebras on partial order sets.Then we prove the idempotent conjecture for derivations of incidence algebras,and the LFED conjecture for derivations on algebraic incidence algebras.Finally,we prove the LFED conjecture for locally finite algebras.Derivations,especially locally nilpotent derivations and locally finite derivations,are also important research topics and tools in algebra and affine algebraic geometry.A basic question is whether the kernel of a derivation on polynomial algebra K[X]is a finitelly generated algebra.Specifically,is the kernel of a locally nilpotent derivation finite generated?These problems are special cases of Hilbert’s fourteenth problem and are closely related to Hilbert’s fourteenth problem.Zariski proved that the Hilbert’s fourteenth problem is affirmative if n<2.From this result,it can be concluded that the finite generation problem of kernel of derivation is affirmative if n<3.On the one hand,most of the known counterexamples of Hilbert’s fourteenth problem can be constructed through kernels of locally nilpotent derivations.On the other hand,the first counterexample of the finitelly generation problem for kernels of locally nilpotent derivation was constructed using Nagata’s first counterexample of Hilbert’s fourteenth problem.Locally nilpotent derivations contained locally finite derivations.According to Jordan-Chevalley decomposition,a locally finite derivation can be decomposed into a sum of a semisimple derivation and a locally nilpotent derivation.Locally finite derivations are closely related to the Jacobian conjecture and structure of polynomial automorphism groups.At present,much have not been known about locally finite derivations.Rentschler used Jung’s theorem on automorphism group of K[x,y]to obtain the classification of locally nilpotent derivations of K[x,y].Bass et al.provided a classification of locally finite derivations on K[x,y].However,when u>2,the classification of locally nilpotent derivations and locally finite derivations of the polynomial algebra in n variables remains unresolved.There is also the finitelly generation problem of kernels for derivations of free algebra.Drensky and others characterized kernels of locally nilpotent derivations on free algebras in two variables and proved that the kernels are free algebras.Moreover,they obtained the classification of locally nilpotent derivations.In Chapter 5,we present the classification of locally finite derivations of K<x,y>.As applications,we show that semi-simple derivations can be diagonalized,and kernels of locally finite derivations are free algebras,when K is an algebraically closed field.Finally,we prove that the LFED conjecture for diagonal derivations on K<X>.