The Leavitt path algebra of a graph

Abstract

For any row-finite graph E and any field K we construct the Leavitt path algebra L(E) having coefficients in K. When K is the field of complex numbers, then L(E) is the algebraic analog of the Cuntz-Krieger algebra C * (E) described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., 2005]. The matrix rings M n (K) and the Leavitt algebras L(1, n) appear as algebras of the form L(E) for various graphs E. In our main result, we give necessary and sufficient conditions on E which imply that L(E) is simple.  2005 Elsevier Inc. All rights reserved.

K-theory for Leavitt path algebras: Computation and classification

“…Furthermore, if E is also a finite graph, then K top 0 (C * (E)) is a finitely generated abelian group and K top 1 (C * (E)) is isomorphic to the free part of K top 0 (C * (E)), so that C * (E) is classified up to Morita equivalence by the single group K top 0 (C * (E)). On the algebraic side, as shown in [2], [5], and [3] one may perform a similar construction to produce algebras from graphs. If E is a graph and k is any field, then one may mimic the graph C * -algebra construction to produce a k-algebra L k (E), which is called the Leavitt path algebra of E over k. For a given graph E, the algebra L k (E) (for any field k) has many properties in common with C * (E).…”

Regularity Conditions for Arbitrary Leavitt Path Algebras

“…In particular, L K (E) is a simple algebra, while L K (E F ) is not. (See [2] for a more complete discussion. )…”

FP-injective semirings, semigroup rings and Leavitt path algebras

“…Let I be the ideal of C R (E) generated by all elements of the form v − e∈s −1 (v) ee * , where v is a non-sink vertex. The R-algebra C R (E)/I is easily seen to be the Leavitt path algebra of E with coefficients in R (see [1], [19] for further reference). Since E is finite, both C R (E) and L R (E) are unital rings, each having identity 1 = v∈E 0 v (see [2,Subsection 4.2] for example).…”