A class of corners of a Leavitt path algebra

 Tóm tắt — Let E be a directed graph, K a field and L K ( E ) the Leavitt path algebra of E over K . The goal of this paper is to describe the structure of a class of corners of Leavitt path algebras L K ( E ). The motivation of this work comes from the paper “Corners of Graph Algebras” of Tyrone Crisp in which such corners of graph C *-algebras were investigated completely. Using the same ideas of Tyrone Crisp, we will show that for any finite subset X of vertices in a directed graph E such that the hereditary subset H E ( X ) generated by X is finite, the corner ( ) ( )( )     K v X v X v L E v is isomorphic to the Leavitt path algebra L K ( E X ) of some graph E X . We also provide a way how to construct this graph E X . Từ khóa — Abstract — Cho E là một đồ thị có hướng, K là và L K ( là


INTRODUCTION
eavitt path algebras for graphs were developed independently by two groups of mathematicians. The first group, which consists of Ara, Goodearl and Pardo, was motivated by the K-theory of graph algebras. They introduced Leavitt path algebras [3] in order to answer analogous K-theoretic questions about the algebraic Cuntz-Krieger algebras. On the other hand, Abrams and Aranda Pino introduced Leavitt path algebras LK(E) in [2] to generalise Leavitt's algebras, specifically the algebras LK (1,n).
The goal of this paper is to describe the structure of a class of corners of Leavitt path algebras LK(E). The motivation of this work comes from [4] in which such corners of graph C * -algebras were investigated completely. Using the same ideas from [4], we will show that for any finite subset X of vertices in a directed graph E such that the hereditary subset HE(X) generated Author Trinh Thanh Deo -University of Science, VNUHCM (email: ttdeo@hcmus.edu.vn) by X is finite, the corner ( ) ( )( ) isomorphic to the Leavitt path algebra LK(EX) of some graph EX. We also provide a way how to construct this graph EX.
The graph C * -algebra of an arbitrary directed graph E plays an important role in the theory of C * -algebras. In 2005, G. Abrams and G. Aranda-Pino [2] defined the algebra LK(E) of a directed graph E over a field K which was the universal Kalgebra, named Leavitt path algebra, generated by elements satisfying relations similar to the ones of the generators in the graph C * -algebra of E and was considered as a generalization of Leavitt algebras L(1,n). Historically, G. Abrams and G. Aranda-Pino found his inspiration from results on graph C * -algebras to define Leavitt path algebras, so that one of first topics in Leavitt path algebras was to find some analogues for Leavitt path algebras of graph C * -algebras such as in [1,5]. In [4], the class of corners PXC * (E)PX were investigated completely when X was a finite subset of E 0 with HE(X) was finite. In the present note, we consider the similar problem for Leavitt path algebra LK(E). In the next section, we recall briefly the notation and results on the graph theory. In Section 3, we present the way to find a graph EX and an isomorphism of The ideas and arguments we use in Section 3 is almost similar to [4] but there are two important things here: arguments in [4] will be rewritten according to the language of Leavitt path algebras and, secondary, we will modify a little bit these arguments to pass difficulties of hypothesis between graph C *algebras and Leavitt path algebras.

PRELIMINARIES ON GRAPH THEORY
A directed graph E = (E 0 , E 1 , r, s) consists of two countable sets E 0 , E 1 and maps r,s: E 0  E 1 .   Let F be a subgraph of E, that is, F is a graph whose vertices and edges form subsets of the vertices and edges of E respectively. For vertices u,vE 0 we write uF v if there is a path F * such that s() = u and r() = v. We say that a subset X  E 0 is hereditary if vX and uE 0 such that vF u, then u  X. For any subset Y  E 0 we shall denote by HE(Y) the smallest hereditary subset of E 0 containing Y. The set HE(Y)\Y is referred to as the hereditary complement of Y in E. The subgraph T=(T 0 ,T 1 ,r,s) is called a directed forest in E if it satisfies the two following conditions: (1) T is acyclic, that is, for every path e1…en If T is a directed forest of E, then T r denotes the subset of T 0 consisting of those vertices v with |T 1 r 1 (v)| = 0, and T l denotes the subset of T 0 consisting of those vertices v with |T 1 s 1 (v)| = 0. The sets T r and T l are called the roots and the the leaves of T.
The following lemmas are from [4]. The key result of building a new graph EX in this paper is the existence of the directed forest with given roots. In general, a forest with given roots [4, Lemma 3.6] may not exists, but in some special cases, we can find such forest.
Proof. This lemma is just a corollary of [4, Lemma 3.6]. 

RESULTS
We have mentioned graph C * -algebras in the Introduction, but this paper focus only on Leavitt path algebras. In this section, before going to the main goal of paper, we briefly recall just the definition of the Leavitt path algebra of a graph. For a definition of these algebras with remarks one can see in [2].
Given a graph E = (E 0 ,E 1 ,r,s), we denote the new set of edges (E 1 ) * , which is a copy of E 1 but with the direction of each edge reversed; that is, if e  E 1 runs from u to v, then e *  (E 1 ) * runs from v to u. We refer to E 1 as the set of real edges and (E 1 ) * as the set of ghost edges.
The path p = e1 ... en made up of only real edges is called the real path, and we denote the ghost path en * ... e1 * by p * .
Let K be a field and E a directed graph. The Leavitt path K-algebra LK(E) of E over K is the (universal) K-algebra generated by a set {v| vE 0 } of pairwise orthogonal idempotents, together with CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 4, 2018 a set of variables {e, e * | eE 1 } which satisfy the following relations: (1) s(e)e = er(e) =e for all eE 1 .
The rest of the proof is from the second part of Let E be a directed graph, and assume that X is a finite subset of E 0 such that HE(X) is finite. By Lemma 2, there exists a row-finite, path-finite directed forest T in E with T r =X and T 0 = HE(X). .  (2) gives ** , , ( .