Leavitt Path Algebras Research Articles

On some ideal structure of Leavitt path algebras with coefficients in integral domains

In this paper, we present results concerning the structure of the ideals in the Leavitt path algebra of a (countable) directed graph with coefficients in an integral domain, such as, describing the set of generators for an ideal; the necessary and sufficient conditions for an ideal to be prime; the necessary and sufficient conditions for a Leavitt path algebra to be simple. Besides, some other interesting properties of ideal structure in a Leavitt path algebra are also mentioned.

Strong gradings on Leavitt path algebras, Steinberg algebras and their $$C^*$$-completions

Strong gradings on Leavitt path algebras, Steinberg algebras and their $$C^*$$-completions

Realizing ultragraph Leavitt path algebras as Steinberg algebras

In this article, we realize the finite range ultragraph Leavitt path algebras as Steinberg algebras. This realization allows us to use the groupoid approach to obtain structural results about these algebras. Using the skew product of groupoids, we show that ultragraph Leavitt path algebras are graded von Neumann regular rings. We characterize strongly graded ultragraph Leavitt path algebras and show that every ultragraph Leavitt path algebra is semiprimitive. Moreover, we characterize irreducible representations of ultragraph Leavitt path algebras. We also show that ultragraph Leavitt path algebras can be realized as Cuntz-Pimsner rings.

Graded K-theory and Leavitt path algebras

Let G be a group and \(\ell \) a commutative unital \(*\)-ring with an element \(\lambda \in \ell \) such that \(\lambda + \lambda ^*= 1\). We introduce variants of hermitian bivariant K-theory for \(*\)-algebras equipped with a G-action or a G-grading. For any graph E with finitely many vertices and any weight function \(\omega :E^1 \rightarrow G\), a distinguished triangle for \(L(E)=L_\ell (E)\) in the hermitian G-graded bivariant K-theory category \(kk^h_{G_{{\text {gr}}}}\) is obtained, describing L(E) as a cone of a matrix with coefficients in \(\mathbb {Z}[G]\) associated to the incidence matrix of E and the weight \(\omega \). In the particular case of the standard \(\mathbb {Z}\)-grading, and under mild assumptions on \(\ell \), we show that the isomorphism class of L(E) in \(kk^h_{\mathbb {Z}_{{\text {gr}}}}\) is determined by the graded Bowen–Franks module of E. We also obtain results for the graded and hermitian graded K-theory of \(*\)-algebras in general and Leavitt path algebras in particular which are of independent interest, including hermitian and bivariant versions of Dade’s theorem and of Van den Bergh’s exact sequence relating graded and ungraded K-theory.

Ring theoretic properties of partial skew groupoid rings with applications to Leavitt path algebras

Let [Formula: see text] be a group-type partial action of a connected groupoid [Formula: see text] on a ring [Formula: see text] and [Formula: see text] be the corresponding partial skew groupoid ring. In the first part of this paper, we investigate the relation of several ring theoretic properties between [Formula: see text] and [Formula: see text]. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action [Formula: see text], we characterize when [Formula: see text] is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general partial skew groupoid rings. Several examples that illustrate the results on Leavitt path algebras are presented.

Annihilator ideals of graph algebras

If I is a (two-sided) ideal of a ring R, we let $${\text {ann}}_l(I)=\{r\in R\mid rI=0\},$$ $${\text {ann}}_r(I)=\{r\in R\mid Ir=0\},$$ and $${\text {ann}}(I)={\text {ann}}_l(I)\cap {\text {ann}}_r(I)$$ be the left, the right and the double annihilators. An ideal I is said to be an annihilator ideal if $$I={\text {ann}}(J)$$ for some ideal J (equivalently, $${\text {ann}}({\text {ann}}(I))=I$$ ). We study annihilator ideals of Leavitt path algebras and graph $$C^*$$ -algebras. Let $$L_K(E)$$ be the Leavitt path algebra of a graph E over a field K. If I is an ideal of $$L_K(E),$$ it has recently been shown that $${\text {ann}}(I)$$ is a graded ideal (with respect to the natural grading of $$L_K(E)$$ by $$\mathbb Z$$ ). We note that $${\text {ann}}_l(I)$$ and $${\text {ann}}_r(I)$$ are also graded. If I is graded, we show that $${\text {ann}}_l(I)={\text {ann}}_r(I)={\text {ann}}(I)$$ and describe $${\text {ann}}(I)$$ in terms of the properties of a pair of sets of vertices of E, known as an admissible pair, which naturally corresponds to I. Using such a description, we present properties of E which are equivalent with the requirement that each graded ideal of $$L_K(E)$$ is an annihilator ideal. We show that the same properties of E are also equivalent with each of the following conditions: (1) the lattice of graded ideals of $$L_K(E)$$ is a Boolean algebra; (2) each closed gauge-invariant ideal of $$C^*(E)$$ is an annihilator ideal; (3) the lattice of closed gauge-invariant ideals of $$C^*(E)$$ is a Boolean algebra. In addition, we present properties of E which are equivalent with each of the following conditions: (1) each ideal of $$L_K(E)$$ is an annihilator ideal; (2) the lattice of ideals of $$L_K(E)$$ is a Boolean algebra; (3) each closed ideal of $$C^*(E)$$ is an annihilator ideal; (4) the lattice of closed ideals of $$C^*(E)$$ is a Boolean algebra.

LEAVITT PATH ALGEBRAS OF WEIGHTED AND SEPARATED GRAPHS

AbstractIn this paper, we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra$L(E,\omega )$of a row-finite vertex weighted graph$(E,\omega )$is$*$-isomorphic to the lower Leavitt path algebra of a certain bipartite separated graph$(E(\omega ),C(\omega ))$. For a general locally finite weighted graph$(E, \omega )$, we show that a certain quotient$L_1(E,\omega )$of$L(E,\omega )$is$*$-isomorphic to an upper Leavitt path algebra of another bipartite separated graph$(E(w)_1,C(w)^1)$. We furthermore introduce the algebra${L^{\mathrm {ab}}} (E,w)$, which is a universal tame$*$-algebra generated by a set of partial isometries. We draw some consequences of our results for the structure of ideals of$L(E,\omega )$, and we study in detail two different maximal ideals of the Leavitt algebra$L(m,n)$.

Irreducible representations of Leavitt algebras

For a weighted graph E, we construct representation graphs F, and consequently, LK(E)-modules VF, where LK(E) is the Leavitt path algebra associated to E, with coefficients in a field K. We characterise representation graphs F such that VF are simple LK(E)-modules. We show that the category of representation graphs of E, RG(E), is a disjoint union of subcategories, each of which contains a unique universal object T which gives an indecomposable LK(E)-module VT and a unique irreducible representation graph S, which gives a simple LK(E)-module VS.Specialising to graphs with one vertex and m loops of weight n, we construct irreducible representations for the celebrated Leavitt algebras LK(n,m). On the other hand, specialising to graphs E with weight one, we recover the simple modules of Leavitt path algebras LK(E) constructed by Chen via infinite paths or sinks and give a large class of non-simple indecomposable LK(E)-modules.Our approach gives a completely new way to construct indecomposable and simple modules for Leavitt path algebras of (weighted) graphs. Besides being more visual, this approach allows for carrying calculus on these modules with ease. On the other hand, the approach also allows us, to the best of our knowledge, for the first time, to produce systematically many examples of non-simple indecomposable modules, for these algebras, including Leavitt algebras LK(n,m).

Homology of étale groupoids a graded approach

We introduce a graded homology theory for graded étale groupoids. We prove that the graded homology of a strongly graded ample groupoid is isomorphic to the homology of its ε-th component with ε the identity of the grade group. For Z-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras.

Classifying Leavitt path algebras up to involution preserving homotopy

We prove that the Bowen–Franks group classifies the Leavitt path algebras of purely infinite simple finite graphs over a regular supercoherent commutative ring with involution where 2 is invertible, equipped with their standard involutions, up to matricial stabilization and involution preserving homotopy equivalence. We also consider a twisting of the standard involution on Leavitt path algebras and obtain partial results in the same direction for purely infinite simple graphs. Our tools are K-theoretic, and we prove several results about (Hermitian, bivariant) K-theory of Leavitt path algebras, such as Poincaré duality, which are of independent interest.

Grothendieck groups of purely infinite simple Leavitt path algebras for punctured power graphs of finite groups

In [S. Das, M. K. Sen and S. K. Maity, Leavitt path algebras for power graphs of finite groups, J. Algebra Appl., doi:10.1142/S0219498822502097], we studied several algebraic properties of Leavitt path algebra of the directed power graph [Formula: see text] and also of the directed punctured power graph [Formula: see text] of a finite group [Formula: see text]. In this paper, we compute the Grothendieck group of purely infinite simple Leavitt path algebra of the directed punctured power graph of a finite group.

Realization for tensor products of Leavitt path algebras

Realization for tensor products of Leavitt path algebras

Invariant ideals in Leavitt path algebras

Invariant ideals in Leavitt path algebras

A note on the Dixmier-Moeglin equivalence in Leavitt path algebras of arbitrary graphs over a field

The Dixmier-Moeglin Equivalence (for short, the DM-equivalence) is the equivalence of three distinguishing properties of prime ideals in a non-commutative algebra A. These properties are of (i) being primitive, (ii) being rational and (iii) being locally closed in the Zariski topology of Spec(A). The DM-equivalence holds in many interesting algebras over a field. Recently, it was shown that the prime ideals of a Leavitt path algebra of a finite graph satisfy the DM-equivalence. In this note, we investigate the occurrence of the DM-equivalence in a Leavitt path algebra L of an arbitrary directed graph E. Our analysis shows that locally closed prime ideals of L satisfy interesting equivalent properties such as being strongly primitive and completely irreducible. This leads to new characterizations of primitive ideals in a Leavitt path algebra of a finite graph. Examples illustrate the results and the constraints.

Lifting graph C∗$C^*$‐algebra maps to Leavitt path algebra maps

Let ξ : C ∗ ( E ) → C ∗ ( F ) $\xi :C^*(E)\rightarrow C^*(F)$ be a unital ∗ $*$ -homomorphism between simple purely infinite Cuntz–Krieger algebras of finite graphs. We prove that there exists a unital ∗ $*$ -homomorphism ϕ : L ( E ) → L ( F ) $\phi :L(E)\rightarrow L(F)$ between the corresponding Leavitt path-algebras such that ξ $\xi$ is homotopic to the map ϕ ̂ : C ∗ ( E ) → C ∗ ( F ) $\hat{\phi }:C^*(E)\rightarrow C^*(F)$ induced by completion. We show moreover that ϕ ̂ $\hat{\phi }$ is a homotopy equivalence in the C ∗ $C^*$ -algebraic sense if and only if ϕ $\phi$ is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz–Krieger algebras is homotopic to the completion of a unital algebraic homotopy equivalence.

GRADED VERSION OF EXEL’S EFFROS-HAHN CONJECTURE FOR LEAVITT PATH ALGEBRAS

In this paper, we prove a graded version of Exel’s Effros-Hahn conjecture for Leavitt path algebras. More concretely, we show that any graded primitive ideal of the Leavitt path algebra is the annihilator of a module induced from a graded simple module over an isotropy group algebra. A graded version of Steinberg’s results towards Exel’s conjecture in (B. Steinberg, Ideals of etale ´groupoid algebras and Exel’s Effros-Hahn conjecture, J. Noncommut. Geom., Vol. 15, 2021, pp. 829-839) is also obtained for graded ample groupoid algebras.

A talented monoid view on Lie bracket algebras over Leavitt path algebras

In this paper, we study properties such as simplicity, solvability and nilpotency for Lie bracket algebras arising from Leavitt path algebras, based on the talented monoid of the underlying graph. We show that graded simplicity and simplicity of the Leavitt path algebra can be connected via the Lie bracket algebra. Moreover, we use the Gelfand–Kirillov dimension for the Leavitt path algebra for a classification of nilpotency and solvability.

Non-existence of graded unital homomorphisms between Leavitt algebras and their Cuntz splices

Let [Formula: see text], let [Formula: see text] be the graph consisting of one vertex and [Formula: see text] loops and let [Formula: see text] be its Cuntz splice. Let [Formula: see text] and [Formula: see text] be the Leavitt path algebras over a unital ring [Formula: see text]. Let [Formula: see text] be the cyclic group on [Formula: see text] elements. Equip [Formula: see text] and [Formula: see text] with their natural [Formula: see text]-gradings. We show that under mild conditions on [Formula: see text], which are satisfied, for example, when [Formula: see text] is a field or a principal ideal domain, there are no unital [Formula: see text]-graded ring homomorphisms [Formula: see text] nor in the opposite direction.

Ring theoretical properties of epsilon-strongly graded rings and Leavitt path algebras

We extend Dade’s theorem to the realm of nearly epsilon-strongly graded rings, and present new characterizations of strongly and epsilon-strongly graded rings. Further, we give conditions for an epsilon-strongly graded ring to be written as a direct sum of strongly graded rings and a ring with trivial gradation. Our results are applied to characterize strongly graded Leavitt path algebras endowed with the canonical gradation. Finally we show that for any there is a Leavitt path algebra which is a sum of n − 1 strongly graded rings and a ring with trivial gradation.

The socle of Leavitt path algebras over a semiprime ring

The Reduction Theorem in Leavitt path algebra over a commutative unital ring is very important to prove that the Leavitt path algebra is semiprime if and only if the ring is also semiprime. Any minimal ideal in the semiprime ring and line point will construct a left minimal ideal in the Leavitt path algebra. Vice versa, any left minimal ideal in the semiprime Leavitt path algebra can be found both minimal ideal in the semiprime ring and line point that generate it. The socle of semiprime Leavitt path algebra is constructed by minimal ideals of the semiprime ring and the set of all line points.