Lie Derivations Of Incidence Algebras

Let A be an associative algebra.We define the Lie operation as the usual commutator[x,y]:=xy-yx for all x,y?A and then(A,[,])becomes a Lie algebra.This paper is intended to explore the relationship between the associative structure and the Lie structure of A.In 1961,Herstein proposed a series of conjectures concerning the form of Jordan and Lie maps on associative simple and prime rings.Since then,studying the Herstein's conjectures and related problems becomes a hot point in the field of algebra,which is called the Herstein's program.The connection between the Herstein's program and operator algebras has been widely studying in the field of analysis.However,some works about such connection,in which the algebraic methods took the dominant position,just occurred more than ten years ago.The proof of Xiao in[32]suggested that it's possible to study the Herstein's program in incidence algebras by combinatorial methods.Motivated by the idea,in this dissertation we study the operator theory in an algebraic combinatorial flavor,providing a new way to study the related problems.Let(X,?)be a finite pre-ordered set,R a 2-torsion free commutative ring with the identity.We prove that every Lie derivation L of the incidence algebra I(X,R)is proper,in other words,L = D + F,where D is a derivation,F is a central-valued map and annihilates all commutators.If(X,?)is a locally finite pre-ordered and connected set,we also show that every Lie derivation of the incidence algebra I(X,R is proper.