A Survey of Some of the Recent Developments in Leavitt Path Algebras

Abstract

AbstractIn this survey article, we describe some of recent ring-theoretic and module-theoretic investigations of a Leavitt path algebra L of an arbitrary directed graph E over a field K. It is shown how a single graph-theoretical property of E often gives rise to several independent ring properties of L, thus making Leavitt path algebras as effective tools in constructing examples of rings with various desired properties. Leavitt path algebras satisfying a polynomial identity are completely described. It is shown how using special vertices, infinite paths or cycles in the graph E, various types of simple modules over L can be constructed. A complete description is given of a Leavitt path algebra L whose simple modules possess various specific properties such as being, flat, injective, graded or finitely presented. In the first three cases, L becomes von Neumann regular while in the last case, when the graph E is finite, L possesses finite GK-dimension. Leavitt path algebras having only finitely many isomorphism classes of simple modules turn out to be semi-artinian von Neumann regular rings in which the ideals form a finite chain under inclusion. The sum and the intersection of any two principal one-sided ideals of L are shown to be again principal one-sided ideals and this leads to the existence of the left/right gcd and the left/right lcm of any two non-zero elements in L.KeywordsLeavitt path algebrasGraph algebrasSimple modulesFinitely presented modulesClassification2010 Mathematics Subject Classification 16D5016D70