Research On Some Structural Problems About Leavitt Path Algebras

Since Abrams and Pino introduced Leavitt path algebras, it produced fruitful results on Leavitt path algebras. In this thesis, we study the structure of Leavitt path algebras from several aspects, including the bi-derivations, commuting maps and 2-local derivations on Leavitt path algebras associated with some directed graphs. Further, we study diameters of commuting graphs of Leavitt path algebras associated with some directed graphs.This thesis is divided into two parts. The first part introduces the background of the project, necessary foundations and main results of this thesis. The second part consists of four chapters, and main results are as follows.In the first chapter, we introduce the basic concepts, properties and notations.In the second chapter, we study the anti-symmetric bi-derivations, symmetric bi-derivations and commuting maps of Leavitt path algebras. We give the explicit expressions of anti-symmetric bi-derivations, symmetric bi-derivations and commut-ing maps on Leavitt path algebras, whose underlying graphs arc acyclic graphs, the graphs with a single simple closed path (for simplicity shorted as SSCP) and one-pointed extensions of the graphs with a SSCP.In the third chapter, we determined the 2-local derivations on Leavitt path algebras associated with acyclic graphs and the graphs with SSCP.In the forth chapter, we define commuting graphs and the diameters of the commuting graphs on Leavitt path algebras. We prove that diameters of commuting graphs of Leavitt path algebras with acyclic graphs is four. Further, we also show that diameters of commuting graphs of Leavitt path algebras with the graphs of a SSCP and one-pointed extensions of a SSCP are not less than four.Finally, a brief conclusion and a prospect are made in the last chapter.