Leavitt path algebras with coefficients in a commutative ring

Given a directed graph E we describe a method for constructing a Leavitt path algebra L_R(E) whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness Theorem and Cuntz-Krieger Uniqueness Theorem for these Leavitt path algebras, giving proofs that both generalize and simplify the classical results for Leavitt path algebras over fields. We also analyze the ideal structure of L_R(E), and we prove that if K is a field, then L_K(E)≅K⊗_ZL_Z(E).     

*-Regular Leavitt path algebras of arbitrary graphs

If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L _K (E). We show that the involution on L _K (E) is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L _K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L _K (E) to be *-regular (i.e., regular with proper involution). This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone. As a corollary, we show that Handelman’s conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.     

Finite-dimensional Leavitt path algebras

We classify the directed graphs E for which the Leavitt path algebra L(E) is finite dimensional. In our main results we provide two distinct classes of connected graphs from which, modulo the one-dimensional ideals, all finite-dimensional Leavitt path algebras arise.     

Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring

For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L _R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L _R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L _R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L _R (E) coincide.     

The graded structure of Leavitt path algebras

A Leavitt path algebra associates to a directed graph a ?-graded algebra and in its simplest form it recovers the Leavitt algebra L(1, k). In this note, we first study this ?-grading and characterize the (?-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, C _n -comet, multi-headed graphs and a mixture of these graphs (i.e., polycephaly graphs). The last two types are examples of graphs whose Leavitt path algebras are strongly graded. We give a criterion when a Leavitt path algebra is strongly graded and in particular characterize unital Leavitt path algebras which are strongly graded completely, along the way obtaining classes of algebras which are group rings or crossed-products. In an attempt to generalize the grading, we introduce weighted Leavitt path algebras associated to directed weighted graphs which have natural ⊕?-grading and in their simplest form recover the Leavitt algebras L(n, k). We then show that the basic properties of Leavitt path algebras can be naturally carried over to weighted Leavitt path algebras.     

Applications of Normal Forms for Weighted Leavitt Path Algebras: Simple Rings and Domains

Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras L _K (n, n + k) constructed by Leavitt. Using Bergman’s diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular.     

The socle series of a Leavitt path algebra

We investigate the ascending Loewy socle series of Leavitt path algebras L _K (E) for an arbitrary graph E and field K. We classify those graphs E for which L _K (E) = S _λ for some element S _λ of the Loewy socle series. We then show that for any ordinal λ there exists a graph E so that the Loewy length of L _K (E) is λ. Moreover, λ ≤ ω ₁ (the first uncountable ordinal) if E is a row-finite graph.     

The Dixmier-Moeglin Equivalence for Leavitt Path Algebras

Let K be a field, let E be a finite directed graph, and let L _K (E) be the Leavitt path algebra of E over K. We show that for a prime ideal P in L _K (E), the following are equivalent:

Regularity Conditions for Arbitrary Leavitt Path Algebras

We show that if E is an arbitrary acyclic graph then the Leavitt path algebra L _K (E) is locally K-matricial; that is, L _K (E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graph-theoretic conditions (e.g. row-finiteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E: (1) L _K (E) is von Neumann regular. (2) L _K (E) is π-regular. (3) E is acyclic. (4) L _K (E) is locally K-matricial. (5) L _K (E) is strongly π-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.     

The matrix type of purely infinite simple Leavitt path algebras

Let R denote the purely infinite simple unital Leavitt path algebra L(E). We completely determine the pairs of positive integers (c, d) for which there is an isomorphism of matrix rings M _c (R) ≌ M _d (R), in terms of the order of [1 _R ] in the Grothendieck group K ₀(R).     

Ideals of identities of representations of nilpotent lie algebras

Let L be a Lie algebra, nilpotent of class 2, over an infinite field K, and suppose that the centre C of L is one dimensional; such Lie algebras are called Heisenberg algebras. Let ρ:L→hom _KV be a finite dimensional representation of the Heisenberg algebra L such that ρ(C) contains non-singular linear transformations of V, and denote l(ρ) the ideal of identities for the representation ρ. We prove that the ideals of identities of representations containing I(ρ) and generated by multilinear polynomials satisfy the ACC. Let sl ₂(L) be the Lie algebra of the traceless 2×2 matrices over K, and suppose the characteristic of K equals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation of sl ₂(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras (over an infinite field of characteristic 2) that contain the identities of the regular representation of sl ₂(K).     

Leavitt Path Algebras of Edge-Colored Graphs

Given any edge-colored graph G and any commutative unital ring R, we construct a generalized Leavitt path algebra L _R (G).We show that L _R (G) is a certain free product of L _R (G _i ), where G _i s are 1-colored subgraphs of G. We also show that L _R (G) may be written as a free product of simpler algebras. In the end, we define a natural ${\mathbb{Z}}$ -grading for L _R (G) and give four necessary conditions for simplicity of L _R (G).     

Nonstable <Emphasis Type="Italic">K</Emphasis>-theory for Graph Algebras

We compute the monoid V(L _K (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L _K (E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L _K (E) and the lattice of order-ideals of V(L _K (E)). When K is the field of complex numbers, the algebra is a dense subalgebra of the graph C ^*-algebra C ^*(E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property. The first author was partially supported by the DGI and European Regional Development Fund, jointly, through Project BFM2002-01390, the second and the third by the DGI and European Regional Development Fund, jointly, through Project MTM2004-00149 and by PAI III grant FQM-298 of the Junta de Andalucía. Also, the first and third authors are partially supported by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.     

On the ideal of functions with compact support in pointfree function rings

As usual, let RL\mathcal{R}L denote the ring of real-valued continuous functions on a completely regular frame L. We consider the ideals R_s(L)\mathcal{R}_{s}(L) and R_K(L)\mathcal{R}_{K}(L) consisting, respectively, of functions with small cozero elements and functions with compact support. We show that, as in the classical case of C(X), the first ideal is the intersection of all free maximal ideals, and the second is the intersection of pure parts of all free maximal ideals. A corollary of this latter result is that, in fact, R_K(L)\mathcal{R}_{K}(L) is the intersection of all free ideals. Each of these ideals is pure, free, essential or zero iff the other has the same feature. We observe that these ideals are free iff L is a continuous frame, and essential iff L is almost continuous (meaning that below every nonzero element there is a nonzero element the pseudocomplement of which induces a compact closed quotient). We also show that these ideals are zero iff L is nowhere compact (meaning that non-dense elements induce non-compact closed quotients).     

Diagonal-preserving ring ⁎-isomorphisms of Leavitt path algebras

The graph groupoids of directed graphs E and F are topologically isomorphic if and only if there is a diagonal-preserving ring ?-isomorphism between the Leavitt path algebras of E and F.     

Simpleness of Leavitt path algebras with coefficients in a commutative semiring

In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, establishing some fundamental properties of those algebras. We provide a complete characterization of ideal-simple Leavitt path algebras with coefficients in a commutative semiring S, extending the well-known characterizations when S is a field or a commutative ring. We also present a complete characterization of congruence-simple Leavitt path algebras over row-finite graphs with coefficients in a commutative semiring S.     

The graded Grothendieck group and the classification of Leavitt path algebras

This paper is an attempt to show that, parallel to Elliott’s classification of AF C*-algebras by means of K-theory, the graded K ₀-group classifies Leavitt path algebras completely. In this direction, we prove this claim at two extremes, namely, for the class of acyclic graphs (graphs with no cycles) and multi-headed comets or rose graphs (graphs in which each head is connected to a cycle or to a collection of loops), or a mixture of these graphs (i.e., polycephaly graphs).