Weighted Leavitt path algebras -- an overview

Abstract

Weighted Leavitt path algebras were introduced in 2013 by Roozbeh Hazrat. These algebras generalise simultaneously the usual Leavitt path algebras and William Leavitt's algebras L(m, n). In this paper we try to give an overview of what is known about the weighted Leavitt path algebras. We also prove some new results (in particular on the graded K-theory of weighted Leavitt path algebras) and mention open problems. Contents 1. Introduction 1 2. Preliminaries 3 3. Unweighted and weighted Leavitt path algebras 4 4. Linear bases and the Gelfand-Kirillov dimension 6 5. Weighted Leavitt path algebras that are isomorphic to unweighted Leavitt path algebras 11 6. Finite-dimensionality, Noetherianess and von Neumann regularity 15 7. Realisation as generalised corner skew Laurent polynomial rings 21 8. Local valuations 26 9. The V-monoid and K 0 28 10. The graded V-monoid and K gr 0 33 11. Representations 40 12. Open problems 50 References 51 1. Introduction Leavitt path algebras are algebras associated to directed graphs. They were introduced by G. Abrams and G. Aranda Pino in 2005 [1] and independently by P. Ara, M. Moreno and E. Pardo in 2007 [7]. The Leavitt path algebras turned out to be a very rich and interesting class of algebras, whose studies so far have comprised over 150 research papers and counting. A comprehensive treatment of the subject can be found in the book [2]. The definition of the Leavitt path algebras was inspired by the algebras L(m, n) studied by W. Leavitt in the 1950's and 60 's [19, 20, 21, 22]. Recall that for positive integers m < n, the Leavitt algebra L(m, n) is universal with the property that L(m, n) m ∼ = L(m, n) n as left L(m, n)-modules.Weighted Leavitt path algebras are algebras associated to weighted graphs, i.e. directed graphs with a weight map associating to each edge a positive integer. These algebras were introduced by R. Hazrat in 2013 [13]. If all the weights are equal to 1, then the weighted Leavitt path algebras reduce to the usual Leavitt path algebras. While their unweighted cousins only include the Leavitt algebras L(1, n) (n > 1) as special cases, the weighted Leavitt path algebras embrace all Leavitt algebras (see [29]).

Leavitt path algebras of weighted and separated graphs

“…A bipartite separated graph is a separated graph (E, C) such that E 0 = E 0,0 ⊔ E 0,1 , and s(e) ∈ E 0,0 , r(e) ∈ E 0,1 for each e ∈ E 1 . Note that our notation concerning ranges and sources of edges is the same as the one from [13], [16], [1] and [6], but it is distinct from the one used in [4], [7] and other sources.…”

“…Recently, an interesting connection between the K-theory of Leavitt path algebras of weighted graphs and the theory of abelian sandpile models has been developed in [2]. We refer the reader to [16] for a nice survey on Leavitt path algebras of weighted graphs.…”

Connections between Abelian sandpile models and the $K$-theory of weighted Leavitt path algebras

“…These are algebras associated to weighted graphs (see §2). We refer the reader to [15,21] for a detailed analysis of these algebras.…”

“…(cf. [20], [21]) Let (E, w) be a row-finite vertex weighted graph. Then there is an isomorphism of monoids…”

“…When n ≥ 2, this universal ring is not realizable as a Leavitt path algebra. With this in mind, the notion of weighted Leavitt path algebras L k (E, w) associated to weighted graphs (E, w) were introduced by the second author in 2011 ( [15]; see [21] for a nice overview of this topic). The weighted Leavitt path algebras L k (E, w) provide a natural context in which all of Leavitt's algebras (corresponding to any pair n, k ∈ N) can be realised as a specific example.…”