EXTREME CYCLES. THE CENTER OF A LEAVITT PATH ALGEBRA

2017

These notes contain all the explanations and references corresponding to the course I delivered during the first CIMPA research school in Panama in October 2015 (see the complete program of the school here: http://cimpa. up.ac.pa/program.html). The purpose of the course is to explain the essentials on Leavitt path algebras and then to determine the center of a Leavitt path algebra whose associated graph is row finite. We start by the very beginning of the theory of Leavitt path algebras and give the tools needed to understand the structure of the center. Finally, we give a taste of the work that has been done concerning the study of the Lie estructure of a Leavitt path algebra.

Forum Mathematicum, 2000

We characterize, in terms of its idempotents, the Leavitt path algebras of an arbitrary graph that satisfies Condition (L) or Condition (NE). In the latter case, we also provide the structure of such algebras. Dual graph techniques are considered and demonstrated to be useful in the approach of the study of Leavitt path algebras of arbitrary graphs. A refining of the so-called Reduction Theorem is achieved and is used to prove that I(P c (E)), the ideal of the vertices which are base of cycles without exits of the graph E, a construction with a clear parallelism to the socle, is a ring isomorphism invariant for arbitrary Leavitt path algebras. We also determine its structure in any case.

Proceedings of the National Academy of Sciences, 2013

We determine the structure of Leavitt path algebras of polynomial growth and discuss their automorphisms and involutions. Gelfand-Kirillov dimension | Toeplitz algebra F ollowing the works in refs. 1-4, Abrams and Aranda Pino (5) and Ara et al. (6) introduced Leavitt path algebras of directed graphs as algebraic analogs of C* algebras of Cuntz and Krieger. This construction provided a rich supply of finitely presented algebras having interesting and extreme properties. Let Γ = ðV ; EÞ be a finite directed graph with the set of vertices V and the set of edges E. For an edge e ∈ E; we let sðeÞ and rðeÞ ∈ V denote its source and range, respectively. A vertex v for which s −1 ðvÞ is empty is called a sink. A path p = e 1. .. e n in a graph Γ is a sequence of edges e 1. .. e n such that rðe i Þ = sðe i+1 Þ; i = 1; 2;. .. ; n − 1. In this case we say that the path p starts at the vertex sðe 1 Þ and ends at the vertex rðe n Þ. If sðe 1 Þ = rðe n Þ then the path is closed. If p = e 1. .. e n is a closed path and the vertices sðe 1 Þ;. .. ; sðe n Þ are distinct, then the subgraph ðfsðe 1 Þ;. .. ; sðe n Þg; fe 1 ;. .. ; e n gÞ of the graph Γ is called a cycle. Let Γ be a finite graph and let F be a field. The Leavitt path F algebra LðΓÞ is the F algebra presented by the set of generators fvjv ∈ V g fe; e p je ∈ Eg and the set of relations (i) v i v j = δ vi;vj v i for all v i ; v j ∈ V ; (ii) sðeÞe = erðeÞ = e; rðeÞe p = e p sðeÞ = e p for all e ∈ E; (iii) e p f = δ e; f rðeÞ for all e; f ∈ E; and (iv) v = P sðeÞ = v ee p for an arbitrary vertex v ∈ V ∖ fsinksg. The mapping that sends v to v,

2012

This paper is devoted to the study of the center of several types of path algebras associated to a graph E over a field K. In a first step we consider the path algebra KE and prove that if the number of vertices is infinite then the center is zero; otherwise, it is K, except when the graph E is a cycle in which case the center is K[x], the polynomial algebra in one indeterminate. Then we compute the centers of prime Cohn and Leavitt path algebras. A lower and an upper bound for the center of a Leavitt path algebra are given by introducing the graded Baer radical for graded algebras.

Journal of Pure and Applied Algebra, 2008

In this paper we characterize the minimal left ideals of a Leavitt path algebra as those ones which are isomorphic to principal left ideals generated by line point vertices, that is, by vertices whose trees do not contain neither bifurcations nor closed paths. Moreover, we show that the socle of a Leavitt path algebra is the two-sided ideal generated by these line point vertices. This characterization allows us to compute the socle of some algebras that arise as the Leavitt path algebra of some row-finite graphs. A complete description of the socle of a Leavitt path algebra is given: it is a locally matricial algebra.

Springer eBooks, 2020

2015

Este libro es un curso sobre algebras de caminos de Leavitt que imparti en la Universidad de Monastir en abril de 2009.

Journal of Algebra, 2008

We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between Zgraded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms we are able to achieve two ends. First, we show that the K 0 groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in K 0 , classify the algebras in these sets up to isomorphism. Second, we show that the isomorphism between matrix rings over the classical Leavitt algebras, established previously using number-theoretic methods, can be reobtained via appropriate isomorphisms between Leavitt path algebras.

Hacettepe Journal of Mathematics and Statistics, 2014

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph-which we name as a truncated tree associated with A-whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant κ(A) for A and count the number of isomorphism classes of Leavitt path algebras with κ(A) = n. Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices.