On graded irreducible representations of Leavitt path algebras

Using the E-algebraic branching systems, various graded irreducible representations of a Leavitt path K-algebra L of a directed graph E are constructed. The concept of a Laurent vertex is introduced and it is shown that the minimal graded left ideals of L are generated by the Laurent vertices or the line points leading to a detailed description of the graded socle of L  . Following this, a complete characterization is obtained of the Leavitt path algebras over which every graded irreducible representation is finitely presented. A useful result is that the irreducible representation V[p]V[p] induced by infinite paths tail-equivalent to an infinite path p (we call this a Chen simple module) is graded if and only if p is an irrational path. We also show that every one-sided ideal of L is graded if and only if the graph E contains no cycles. Supplementing the theorem of one of the co-authors that every Leavitt path algebra L is graded von Neumann regular, we show that L is graded self-injective if and only if L is a graded semi-simple algebra, made up of matrix rings of arbitrary size over the field K   or the graded field K[xn,x−n]K[xn,x−n] where n∈Nn∈N.