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Leavitt path algebras of finite irreducible

representation type

Pere AraDepartment de Matematiques

Universitat Autonoma de Barcelona08913 Bellaterra (Barcelona), Spain.

E-mail: para@mat.uab.cat

Kulumani M. RangaswamyDepartment of Mathemastics

University of Colorado at Colorado SpringsColorado Springs, Colorado 80918, USA.

E-mail: krangasw@uccs.edu

Abstract

Let E be an arbitrary directed graph with no restrictions on the num-ber of vertices and edges and let K be any field. It is shown that the Leav-itt path algebra LK(E) is of finite irreducible representation type, thatis, it has only finitely many distinct isomorphism classes of simple rightLK(E)-modules if and only if LK(E) is a semi-artinian von Neumann reg-ular ring with at most finitely many ideals. Equivalent conditions on thegraph E are also given. Examples are constructed showing that for each(finite or infinite) cardinal κ there exists a Leavitt path algebra LK(E)having exactly κ distinct isomorphism classes of simple right modules.

1 Introduction and Preliminaries

The notion of Leavitt path algebras was introduced and initially studied in [1],[5] as algebraic analogues of graph C∗-algebras and the study of their variousring-theoretic properties has been the subject of a series of papers in recent years(see, e.g., [1] - [8], [11] - [13], [14]). In [10], Goncalves and Royer indicated amethod of constructing various representations of a Leavitt path algebra LK(E)over a graph E by using the concept of algebraic branching systems. Expandingthis, Chen [9] studied special types of irreducible representations of LK(E)by using the sinks as well as the infinite paths which are not tail-equivalentin the graph E and he noted that these can also be considered as algebraicbranching systems. In this paper we investigate the Leavitt path algebras LK(E)which are of finite irreducible representation type, that is, LK(E) having only

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finitely many distinct isomorphism classes of simple right LK(E)-modules. Foran arbitrary graph E with no restrictions (such as being row-finite or countable),it is shown that LK(E) is of finite irreducible representation type if and onlyif LK(E) is a semi-artinian (von Neuman regular) ring with at most finitelymany two-sided ideals. Equivalent conditions on the graph E are also given. Inparticular, when E is a finite graph, then LK(E) has this property exactly whenLK(E) is an artinian semisimple ring (equivalently, the graph E is acyclic). Wealso construct, for each arbitrary (finite or infinite) cardinal κ, a Leavitt pathalgebra for which the cardinality of the distinct isomorphism classes of simpleright modules is exactly κ.

A (directed) graph E = (E0, E1, r, s) consists of two sets E0 and E1 togetherwith maps r, s : E1 → E0. The elements of E0 are called vertices and theelements of E1 edges. If s−1(v) is a finite set for every v ∈ E0, then the graphis called row-finite. All the graphs E that we consider here are arbitrary in thesense that they need not be row-finite. Also K stands for an arbitrary field.

A vertex v is called a sink if it emits no edges and a vertex v is called aregular vertex if it emits a non-empty finite set of edges. For each e ∈ E1, wecall e∗ a ghost edge. We let r(e∗) denote s(e), and we let s(e∗) denote r(e). Afinite path µ of length n > 0 is a finite sequence of edges µ = e1e2 · · · en withr(ei) = s(ei+1) for all i = 1, · · ·, n − 1. In this case µ∗ = e∗n · · · e∗2e

∗1 is the

corresponding ghost path. The set of all vertices on the path µ is denoted byµ0. Any vertex v is considered a path of length 0.

Given an arbitrary graph E and a field K, the Leavitt path K-algebra LK(E)is defined to be the K-algebra generated by a set {v : v ∈ E0} of pairwiseorthogonal idempotents together with a set of variables {e, e∗ : e ∈ E1} whichsatisfy the following conditions:

(1) s(e)e = e = er(e) for all e ∈ E1.(2) r(e)e∗ = e∗ = e∗s(e) for all e ∈ E1.(3) (The ”CK-1 relations”) For all e, f ∈ E1, e∗e = r(e) and e∗f = 0 if

e 6= f .(4) (The ”CK-2 relations”) For every regular vertex v ∈ E0,

v =∑

e∈E1,s(e)=v

ee∗.

The Leavitt path algebra of a graph E over K is denoted by LK(E). We referthe reader to [1] and [11] for the various properties of a Leavitt path algebra,the general notation, terminology and results. We just review some relatedconcepts.

A path µ = e1 . . . en in E is closed if r(en) = s(e1), in which case µ issaid to be based at the vertex s(e1). A closed path µ as above is called simpleprovided it does not pass through its base more than once, i.e., s(ei) 6= s(e1) forall i = 2, ..., n. The closed path µ is called a cycle if it does not pass throughany of its vertices twice, that is, if s(ei) 6= s(ej) for every i 6= j. A graph E issaid to satisfy Condition (K) provided no vertex v ∈ E0 is the base of preciselyone simple closed path, i.e., either no simple closed path is based at v, or at

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least two are based at v. An exit for a path µ = e1 . . . en is an edge e such thats(e) = s(ei) for some i and e 6= ei.

If there is a path from vertex u to a vertex v, we write u ≥ v. A subset Dof vertices is said to be downward directed if for any u, v ∈ D, there exists aw ∈ D such that u ≥ w and v ≥ w. A subset H of E0 is called hereditary if,whenever v ∈ H and w ∈ E0 satisfy v ≥ w, then w ∈ H . A hereditary set issaturated if, for any regular vertex v, r(s−1(v)) ⊆ H implies v ∈ H .

A subset S of E0 is said to have the Countable Separation Property (CSP)with respect to a set C, if C is a countable subset of E0 with the property thatfor each u ∈ S there is a v ∈ C such that u ≥ v.

For any vertex v, the tree of v is T (v) = {w : v ≥ w}. We say there is abifurcation at a vertex v, if v emits more than one edge. In a graph E, a vertexv is called a line point if there is no bifurcation or a cycle based at any vertexin T (v). Thus, if v is a line point, there will be a single finite or infinite linesegment µ starting at v (µ could just be v) and any other path α with s(α) = vwill just be an initial sub-segment of µ. It was shown in [6] that v is a line pointin E if and only if vLK(E) (and likewise LK(E)v) is a simple left (right) ideal.Moreover, the ideal generated by all the line points in E is the socle of LK(E).If v is a line point, then it is clear that any w ∈ T (v) is also a line point.

We shall be using the following concepts and results from [14] in our in-vestigation. A breaking vertex of a hereditary saturated subset H is an in-finite emitter w ∈ E0\H (that is, s−1(w) is an infinite set) with the prop-erty that 1 ≤ |s−1(w) ∩ r−1(E0\H)| < ∞. The set of all breaking ver-tices of H is denoted by BH . For any v ∈ BH , vH denotes the elementv −

∑s(e)=v,r(e)/∈H ee∗. Given a hereditary saturated subset H and a subset

S ⊆ BH , (H,S) is called an admissible pair. The admissible pairs form a par-tially ordered set under the relation (H1, S1) ≤ (H2, S2) if and only if H1 ⊆ H2

and S1 ⊆ H2 ∪ S2. Given an admissible pair (H,S), I(H,S) denotes the idealgenerated by H ∪ {vH : v ∈ S}. It was shown in [14] that the graded ide-als of LK(E) are precisely the ideals of the form I(H,S)for some admissibilepair (H,S). Moreover, LK(E)/I(H,S)

∼= LK(E\(H,S)). Here E\(H,S) is theQuotient graph of E in which (E\(H,S))0 = (E0\H) ∪ {v′ : v ∈ BH\S} and(E\(H,S))1 = {e ∈ E1 : r(e) /∈ H} ∪ {e′ : e ∈ E1, r(e) ∈ BH\S} and r, s areextended to (E\(H,S))0 by setting s(e′) = s(e) and r(e′) = r(e)′. Thus whenS = BH we shall identify the graph E\(H,BH) with the graph E\H .

A useful observation is that every element a of LK(E) can be written as

a =n∑

i=1

kiαiβ∗i , where ki ∈ K, αi, βi are paths in E and n is a suitable integer.

This implies that LK(E) is a ring with local units, that is, to each element a ofLK(E), there is an idempotent u ∈ LK(E) such that ua = a = au. Moreover,LK(E) = ⊕v∈E0vLK(E) = ⊕v∈E0LK(E)v. (see [1]).

Further, there is an isomorphism φ : LK(E) → LK(E)op given by the map-

pingn∑

i=1

kiαiβ∗i →

n∑i=1

kiβiα∗i (induced by the involution ∗) and consequently the

results that we derive for right LK(E)-modules are also valid for left LK(E)-modules.

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A ring R is said to be right semi-artinian, if every non-zero right R-moduleM contains a simple submodule and so its socle Soc(M) is non-zero. Equiva-lently, R is the union of a possibly transfinite ascending chain of ideals 0 = S0 <S1 < · · · < Sα < Sα+1 < · · · , where, for each α, Sα+1/Sα = Soc(R/Sα) and if αis a limit ordinal Sα = ∪γ<αSγ . Because of the involution mentioned above, aLeavitt path algebra is right semi-artinian if and only if it is left semi-artinian.So we simply write LK(E) being semi-artinian.

In giving a method to construct simple modules over LK(E), Chen [9] definesan equivalence relation among infinite paths by using the following notation. Ifp = e1e2 · · ·en · ·· is an infinite path where the ei are edges, then for any positiveinteger n, let τ≤n(p) = e1e2 · · ·en and τ>n(p) = en+1en+2 · · · . Two infinite pathsp and q are said to be tail equivalent, in symbols, p ∼ q, if there exist positiveintegers m and n such that τ>n(p) = τ>m(q).

Given an equivalence class of infinite paths [p], let V[p] denote the K-vectorspace having the set {q : q ∈ [p]} as a basis. Then Chen [9] defines an LK(E)action on V[p] making V[p] a right LK(E)-module by first defining a homomor-phism f : LK(E) −→ End(V[p]) and showing that, via this map f , V[p] indeedbecomes a simple right LK(E)-module. Moreover, Theorem 3.2 of [9] shows thatthe two simple LK(E)-modules V[p], V[q] are isomorphic if and only [p] = [q]. Ina similar fashion, given a sink v in E, Chen considers the K-vector space Nv

having as a basis the set all the paths that end in v and defines an LK(E) actionon Nv making Nv a simple right LK(E)-module ([9], Theorem 3.5).

In this connection, we mention the following alternate method of construct-ing primitive ideals and simple modules using sinks in an arbitrary graph.

Lemma 1.1 Let E be an arbitrary graph and K be any field. Suppose v is asink in E and Hv = {u ∈ E0 : u � v}. Then H is a hereditary saturated subsetof E0 and the ideal Iv = I(Hv ,BHv )

is a primitive ideal of LK(E). Moreover,if u, v are distinct sinks, then the corresponding primitive ideals Iu and Iv aredistinct, thus giving rise to non-isomorphic simple modules over LK(E).

Proof. It is straightforward to see that Hv is a hereditary and saturated subsetof E0. Now Mv = E0\Hv = {u ∈ E0 : u ≥ v} is a downward directed set,which trivially satisfies the countable separation property (with respect to v)and in which every cycle has an exit. Hence, by Theorem 4.3 (iii) of [11], theideal Iv = I(Hv ,BHv )

is a primitive ideal of LK(E).If u 6= v are distinct sinks in E, then Iu 6= Iv since u ∈ Hv = Iv ∩ E0 but

u /∈ Hu = Iu ∩ E0.

2 A characterization

In this section, we give the necessary and sufficient conditions for a Leavittpath algebra LK(E) to be of finite irreducible representation type, where E isan arbitrary graph with no restrictions either on the number of vertices or onthe number of edges emitted by a single vertex. A description of LK(E) with

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finitely many irreducible representations is also given when E is a finite graphand when LK(E) is semi-artinian.

We begin with the first main theorem of this section.

Theorem 2.1 Let E be an arbitrary graph and K be any field. If LK(E) isof finite irreducible representation type, then the graph E has no cycles, has atmost finitely many sinks and finitely many inequivalent infinite paths. Also Ehas at most finitely many hereditary saturated subsets H and for each such Hthe corresponding set BH of breaking vertices is finite. In this case, LK(E) isa von Neumann regular ring with only a finite number of two-sided ideals all ofwhich are principal (graded) ideals.

Proof. Since there are only finitely many non-isomorphic simple right modules,LK(E) has only finitely many distinct primitive ideals. We wish to show thatthere are no non-graded prime ideals in LK(E). Now any non-graded primeideal of LK(E) is primitive (see Theorem 4.3, [11]). From Theorem 3.12 of [11],the non-graded prime ideals of LK(E) are precisely the ideals P with P∩E0 = Hand S = {v ∈ BH : vH ∈ P} and generated by I(H,S)∪{f(c)}, where c is a cyclewithout exits in E\H and f is an irreducible polynomial in K[x, x−1]. So, ifLK(E) has one such non-graded prime ideal P , then corresponding to each ofthe infinitely many irreducible polynomials g in K[x, x−1], LK(E) will have anon-graded prime (and hence a primitive) ideal Pg generated by I(H,S)∪{g(c)}.This contradicts that LK(E) has only finite number of primitive ideals. Thusevery prime ideal of LK(E) must be graded and we appeal to Corollary 3.13 of[11] to conclude that the graph E satisfies Condition (K). From Theorem 6.16of [14], we then conclude that every ideal of LK(E) is graded.

Thus if I is an ideal of LK(E) then, being a graded ideal, I = I(H,S), whereH = I∩E0 and S = {v ∈ BH : vH ∈ I} and, by [14], LK(E)/I ∼= LK(E\(H,S)).As the Jacobson radical of the Leavitt path algebra LK(E\(H,S)) is zero (see[8]), we conclude that I is the intersection of all the primitive ideals containingI. As there are only finitely many primitive ideals in LK(E), we then concludethat LK(E) contains only finitely many distinct ideals which are all of the formI(H,S) for some admissible pair (H,S). It is then clear that there are onlyfinitely many hereditary saturated subsets H of E0 and that the correspondingset BH of breaking vertices is finite. As these finitely many ideals triviallysatisfy the ascending chain condition, they are all finitely generated and hence,by (Corollary 8 [12]), are principal ideals.

Next we wish to show that there are no cycles in E. Suppose, on the contrary,E contains a cycle g based at a vertex v. Since E satisfies Condition (K), therewill be another simple closed path h 6= g based at v. Consider the infinite pathγ = ghgh · · · gh · ·· which, for convenience, we rename as γ = α1α2 · ·· whereαi = g or h. Let P be the set of all prime integers and, for each p ∈ P, letNp = {p, p2, · · ·, pn, · · ·}. Then ℜ = {Np : p ∈ P} is an infinite family of pair-wise disjoint infinite subsets of the set of natural numbers N. Let τ = (g, h)be the transposition on the set {g, h}. For each q ∈ P, define an infinite pathσq = q1q2 · · · qn · ·· where qi = ατ

i or αi according as i ∈ Nq or not. Clearly σq is

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not equivalent to σq′ if q 6= q′ ∈ P. Thus there are infinitely many equivalenceclasses [σq]. By Theorem 3.2 of Chen ([9]), these classes give rise to infinitelymany distinct non-isomorphic simple right LK(E)-modules. This contradictsthe hypothesis that LK(E) is of finite irreducible representation type. Hencethe graph E must be acyclic. By [7], LK(E) is then a von Neumann regularring. This completes the proof.

It is known (see [4]) that if E is a finite acyclic graph E, the Leavitt pathalgebra LK(E) is a direct sum of matrix rings. From this we get the followingcorollary.

Corollary 2.2 Let E be a finite graph. Then LK(E) is of finite irreduciblerepresentation type if and only if LK(E) is isomorphic to a direct sum of finitelymany matrix rings over K, equivalently, the graph E is acyclic.

In order to prove the next corollary, we need the following Lemma.

Lemma 2.3 Suppose R is a von Neumann regular ring and A ⊂ B are twonon-zero proper ideals of R. Then every simple right R-module is isomorphicto a simple right module over R/B,B/A or A. Conversely, every simple rightmodule over R/B, or B/A or A is isomorphic to some simple right R-module.

Proof. Let R/M be a simple right R-module, where M is a maximal right idealof R. Then it is easy to see that R/M is isomorphic to a simple right moduleover R/B,B/A or A according as (i) M ⊃ B, or (ii) M ⊃ A but M + B or (iii)M + A.

To prove the converse, we only consider the case of a simple right B/A-module S, as the other two cases are similar or easy. Write S = (B/A)/(N/A) ∼=B/N for some (maximal) right R-submodule N satisfying A ⊂ N ⊂ B. Letb ∈ B\N . Since R is von Neumann regular, there is an idempotent e suchthat bR = eR ⊂ B. Now M = N ∩ eR is a maximal R-submodule of eR. Let(1 − e)R denote the set {r − er : r ∈ R}. As B = eR + N , B/N ∼= eR/M ∼=[eR⊕ (1 − e)R]/[M ⊕ (1 − e)R] = R/[M ⊕ (1 − e)R]. Thus S is isomorphic tothe simple right R-module R/(M ⊕ (1− e)R).

Corollary 2.4 Suppose LK(E) is semi-artinian ring. Then LK(E) is of finiteirreducible representation type if and only if LK(E) has finitely many ideals,equivalently, there are only finitely many hereditary saturated subsets H of E0

and the corresponding sets BH of breaking vertices are finite.

Proof. In view Theorem 2.1, we need only to prove the sufficiency. Suppose thesemi-artinian ring R = LK(E) has only finitely many two-sided ideals. Thenwe can build a finite ascending chain of two-sided ideals 0 < S1 < · · · < Sn = Rwhere, for each i = 1, · · ·, n − 1, Si+1/Si is a simple ring which, being alsosemi-artinian, is a direct sum of isomorphic simple right ideals. Consequently,each Si+1/Si has a single isomorphism class of simple right Si+1/Si-modules.Also by [8], LK(E) is von Neumann regular. By Lemma 2.3, we then concludethat R has exactly n distinct isomorphism classes of simple right R-modules.

The next proposition is key to proving our main theorem.

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Proposition 2.5 Suppose R = LK(F ) is a simple von Neumann regular ring.Then R has either exactly one or uncountably many isomorphism classes ofsimple modules (according as R = Soc(R) or not).

Proof. If Soc(R) 6= 0, then clearly the simple ring R = Soc(R) is a direct sum ofisomorphic simple left/right ideals. In this case R has exactly one isomorphismclass of simple R-modules. Suppose Soc(R) = 0. By [6], the graph F then hasno line points and thus no sinks. Moreover, since R is von Neumann regular,the graph E has no cycles (see [7]). Hence, for every v ∈ E0, the tree T (v)has bifurcations and every vertex belongs to an infinite path. Now R is, inparticular, a prime ring and so E0 is downward directed (see [3]).

We begin with constructing some specific infinite paths. Start with a bifur-cation vertex v1 with edges e1, f1 such that v1 = s(e1) = s(f1). Since E0 isdownward directed, there is a vertex w1 in E, a path from r(e1) to w1 and apath from r(f1) to w1. Let v2 be a bifurcation vertex in T (w1) so that there areedges e2, f2 satisfying s(e2) = s(f2) = v2. Fix a path from w1 to v2. Proceedinglike this we get the following diagram

INSERT Figure.jpeg hereAt each step i, by choosing ei or fi we can construct a family ℑ of 2ℵ0

infinite paths. Indeed, there is an obvious bijection ℑ ⇆ 2N. Given p ∈ ℑ,corresponding to (α1, α2, · · ·) ∈ 2N, the paths in ℑ which are equivalent to pare the ones corresponding to (β1, · · ·, βn, αn+1, αn+2, · · ·) ∈ 2N with n ∈ N.Therefore there are only a countable number of paths in ℑ which are equivalentto p. Since ℑ has uncountably many paths, it is clear that there are uncountablymany paths in E which are not tail-equivalent. By Theorem 3.2 of Chen [9], Rhas uncountably many non-isomorphic simple right R-modules.

From Theorem 2.1 and Proposition 2.5, we then obtain the following maintheorem.

Theorem 2.6 Let E be an arbitrary graph and K be any field. Then the fol-lowing are equivalent:

(i) The Leavitt path algebra LK(E) is of finite irreducible representationtype.

(ii) LK(E) is a semi-artinian von Neumann regular ring with finitely manytwo-sided ideals.

(iii) LK(E) is a semi-artinian ring with finitely many two-sided ideals.

Proof. Assume (i). From Theorem 2.1, we know that LK(E) is von Neumannregular and has at most finitely many distinct ideals all of which are graded.So we can build a finite ascending chain of ideals {0} = S0 ⊂ S1 ⊂ ... ⊂Sm = LK(E) where, for each i, Si+1/Si is a simple von Neumann regular ringwhich also, by Lemma 2.3, has only a finite number of distinct isomorphismclasses of simple right modules. Since the graph E is acyclic (and hence satisfiesCondition (K)), each Si+1 is a graded ideal and so is isomorphic to a Leavittpath algebra LK(G) of a suitable graph G (see [13], Theorem 6.1). Moreover,Si+1/Si

∼= LK(F ) where F is an appropriate quotient graph of G (see [14]).

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We then appeal to Proposition 2.5 to conclude that, for each i = 0, ...,m − 1,the simple ring Si+1/Si coincides with its socle and hence is a direct sum ofisomorphic simple modules. It is then immediate that the Leavitt path algebraLK(E) is a semi-artinian ring having at most finitely many ideals, thus proving(ii).

(ii) =⇒ (iii). Obvious.Assume (iii). If a Leavitt path algebra LK(E) is a semi-artinian ring, then

by [8] it is already von Neumann regular. If in addition LK(E) contains at mostfinitely many ideals, then an appeal to Corollary 2.4 implies that LK(E) is offinite irreducible representation type. This proves (i).

REMARK: Actually a conclusion stronger than the statement of Theorem2.6 (ii) holds, namely, if R = LK(E) has at most finitely many irreduciblerepresentations, then R is a semi-artinian von Neumann regular ring of finiteLoewy length m and the only ideals of R are those appearing in the chain ofsuccessive socles {0} = S0 ⊂ S1 ⊂ ... ⊂ Sm = R. Moreover this chain is alsoa compositions series and consists of principal ideals. Note that, even thoughR = LK(E) may be a ring without identity, R is the two-sided ideal generatedby a single element.

3 Equivalent Graphical Conditions

In this section we describe the graphical properties of E under which the Leavittpath algebra LK(E) is of finite irreducible representation type. We begin with asimple lemma describing when two line points generate isomorphic simple rightideals.

Lemma 3.1 Given two line points u, v, uLK(E) ∼= vLK(E) if and only ifT (u) ∩ T (v) is not empty.

Proof. Suppose θ : u LK(E) → v LK(E) is an isomorphism. Then θ is given bythe left multiplication by the non-zero element θ(u) = vsu for some s ∈ LK(E).

We can clearly assume that s = vsu. Write s =

m∑

i=1

kiαiβ∗i where ki ∈ K and

α, β are finite paths in E. If a term kiαiβ∗i = kivαiβ

∗i u 6= 0, then v = s(αi),

u = s(βi) and r(αi) = r(βi) = w, so w ∈ T (u) ∩ T (v). Conversely, supposew ∈ T (u) ∩ T (v). Since u is a line point, so is w and there is a unique path µfrom u to w. Then ua 7−→ wµ∗ua is an isomorphism from the simple moduleuLK(E) to wLK(E) with the map wb 7−→ uµwb being the inverse isomorphism.By a similar argument, vLK(E) ∼= wLK(E). Consequently, uLK(E) ∼= vLK(E).

The next Theorem describes the graphical conditions on E under whichLK(E) is of finite irreducible representation type.

Theorem 3.2 Let E be an arbitrary graph and let K be any field. Then theLeavitt path algebra LK(E) is of finite irreducible representation type if and onlyif

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(i) E is acyclic;(ii) E0 has only finitely many distinct hereditary saturated subsets H and

for each such H the corresponding set BH of breaking vertices is finite;(iii) In the poset of admissible pairs in E, (E0, φ) is the supremum of a finite

ascending chain

(H0 = φ, φ) < (H1, φ) < (H1, BH1) < (H2, BH1

∩BH2) < (H2, BH2

) <

(H3, BH2∩BH3

) < (H3, BH3) < · · ·

where, φ denotes the empty set and, for each j ≥ 0, Hj+1 is a hereditary satu-rated subset of E0 and Hj+1\Hj is the hereditary saturated closure of the set ofline points in E\(Hj , BHj

).

Proof. If LK(E) has finite irreducible representation type, then in the proof ofTheorem 2.1 it was shown that E is acyclic and that LK(E) has at most finitelymany two-sided ideals (which are all graded). The latter property is equivalentto Condition (ii) by ([14], Theorem 5.7). To prove condition (iii), we shall usethe fact, established in Theorem 2.6, that LK(E) (and each of its homomorphicimages) is semi-artinian. We wish to construct a chain of ideals of the formI(H,S). Let J1 = Soc(LK(E)). By [6], J1 is the ideal generated by the hereditarysaturated closure H1 of the set T1 of all the line points in E. Now J1 = I(H1,φ),where φ denotes the empty set and, by [14], J1 is the kernel of an epimorphismf : LK(E) −→ LK(E\(H1, φ)) where (E\(H1, φ))

0 = E0\H1 ∪ {v′ : v ∈ BH1}

and that f(vH1) = v′ for all v ∈ BH1. Let J2 = I(H1,BH1

) =< H1, {vH1 :

v ∈ BH1} >. so that J2/J1 ∼=< {v′ : v ∈ BH1

} >⊂ LK(E\(H1, φ)). By [14],LK(E)/J2 ∼= LK(E\(H1, BH1

)) ∼= LK(E\H1) since (E\(H1, BH1))0 = E0\H1

and (E\(H1, BH1))1 = {e ∈ E1 : r(e) /∈ H1}. Now LK(E)/J2 has a non-zero

socle (being semi-artinian) and let H2 be a hereditary saturated subset of E0

containing H1 such that H2\H1 is the hereditary saturated closure of the set T2

of all the line points in E\(H1, BH1). Define

J3 =< J2,H2 >=< H2, {vH1 : v ∈ BH1

} >

so that

J3/J2 ∼=< H2\H1 >= Soc(E\(H1, BH1)) ⊂ LK(E\(H1, BH1

)) ∼= LK(E\H1).

If v ∈ BH1\BH2

, then r(s−1(v)) ⊂ H2 ⊂ J3 and since vH1 ∈ J3, we concludethat v ∈ J3. In the isomorphism LK(E)/J2 ∼= LK(E\H1)), v gets mapped to anelement in (J3/J2) ∩ (E0\H1) = H2\H1 and so v ∈ H2. Thus BH1

\BH2⊂ H2

and so J3 =< H2, {vH2 : v ∈ BH1

∩BH2} >= I(H2,BH1

∩BH2). Note that, in the

isomorphism LK(E)/J1 ∼= LK(E\(H1, φ)), J3/J1 maps to Soc(LK(E\(H1, φ))).Define J4 = I(H2,BH2

). Proceeding like this, we obtain a chain of ideals

{0} = I(H0=φ,φ) ⊂ I(H1,φ) ⊂ I(H1,BH1) ⊂ I(H2,BH1

∩BH2) ⊂

I(H2,BH2) ⊂ I(H3,BH2

∩BH3) ⊂ I(H3,BH3

) ⊂ · · · (∗)

9

whose union is LK(E). Here, for each j ≥ 0, Hj+1 is a hereditary saturatedsubset of E0 and Hj+1\Hj is the hereditary saturated closure of the set Tj+1

of all the line points in E\(Hj , BHj). Note that, for each j ≥ 0, there are only

finitely many equivalence classes of line points in E\(Hj , BHj), due to Condition

(ii) . Our construction shows that, in the poset of admissible pairs, (E0, φ) isthen the supremum of a finite ascending chain

(φ, φ) < (H1, φ) < (H1, BH1) < (H2, BH1

∩BH2) <

(H2, BH2) < (H3, BH2

∩BH3) < (H3, BH3

) < · · · (∗∗)

where the sets Hj are as described above.Conversely, Condition (i) implies, by [7], that LK(E) is von Neumann regular

and Condition (ii) implies that it has only finitely many two-sided ideals (whichare all graded ideals). Consider chain (∗∗) indicated in Condition (iii) and thecorresponding chain of ideals (∗) constructed above. Denote I(H1,φ) by S1 and,for each i ≥ 2, denote I(Hi,BHi−1

∩BHi) by Si. Then we get the ascending finite

chain{0} ⊂ S1 ⊂ · · · ⊂ St = LK(E)

where S1 = Soc(LK(E)) and for each i, Si+1/Si = Soc(LK(E)/Si). HenceLK(E) is a semi-artinian ring. By Theorem 2.6, LK(E) is of finite irreduciblerepresentation type.

4 Examples

In this section we show that, for each (finite or infinite) cardinal κ, there existsa Leavitt path algebra LK(E) with exactly κ distinct isomorphism classes ofsimple right LK(E)-modules.

We consider the ”Pyramid” graphs introduced in [8].Let P1 be the graph

v11• −→

v12• −→

v13• −→ · · ·

consisting of a single infinite path. Now all the vertices v1i are line points inP1 and so v1iLK(P1) is a simple module for all i (see [6]). Also LK(P1) =⊕iv1iLK(P1) = Soc(LK(P1)). From Lemma 3.1 it is clear that, for all i < j,v1iLK(P1) ∼= v1jLK(P1). Since LK(P1) is a direct sum of simple modules, everysimple right LK(P1)-module is isomorphic to the simple right ideal v1iLK(P1)and we conclude that all the simple right LK(P1)-modules are isomorphic.

Let P2 be the graph

v11• −→

v12• −→

v13• −→ • · · ·

↑ տտտ ∞v21• −→

v22• −→

v23• −→ • · · ·

where տտտ ∞ denotes that each of the infinitely many vertices v2n (n ≥ 2) inthe second infinite path is connected to the vertex v11 by an edge. Now the line

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points in the graph P2 are the vertices in the first row, namely, v11, v12, v13, · · ·and they generate the socle S of P2 which, from the explanation in describ-ing LK(P1) above, is a direct sum of isomorphic faithful simple right LK(P2)-modules. Also, by [14], P2/S ∼= LK(F ) where F ∼= P1 and so P2/S = Soc(P2/S)is a direct sum of isomorphic simple modules annihilated by the ideal S. ThusLK(P2) has exactly two distinct isomorphism classes of simple LK(P2)-modules.

Let P3 be the graph

v11• −→

v12• −→

v13• −→ • · · ·

↑ տտտ ∞v21• −→

v22• −→

v23• −→ • · · ·

↑ տտտ ∞v31• −→

v32• −→

v33• −→ • · · ·

As before, v21 and each of the vertices v2n (n ≥ 2) in the second infinite path isconnected to the same vertex v11 by an edge and, v31 and each of the verticesv3n (n ≥ 2) in the third infinite path is connected to the same vertex v21by an edge. These are indicated by the symbols տտտ ∞. As above, thevertices v11, v12, v13, · · · generate the socle S of LK(P3) which is a direct sum ofisomorphic faithful simple LK(P3)-modules and LK(P3)/S ∼= LK(P2). Clearly,by Lemma 2.3 and the description of LK(P2) above, S2/S ∼= Soc(LK(P2)) isa direct sum of isomorphic simple LK(P3)-modules annihilated by the ideal Sand that LK(P3)/S2

∼= LK(P2)/Soc(LK(P2)) ∼= Soc[LK(P2)/Soc(LK(P2)] is adirect sum of isomorphic simple LK(P3)-modules annihilated by the ideal S2.Thus we conclude that LK(P3) has exactly three distinct isomorphism classesof simple right LK(P3)-modules.

Proceeding like this, we conclude, by simple induction, that for any positiveinteger n, the Leavitt path algebra LK(Pn) of the Pyramid graph Pn with n”layers” has exactly n distinct isomorphism classes of simple right LK(Pn)-modules.

Let Pω =⋃

n∈N

Pn be the ”Pyramid” graph of length ω constructed inductively

and represented pictorially as follows.

v11• −→

v12• −→

v13• −→ • · · ·

↑ տտտ ∞v21• −→

v22• −→

v23• −→ • · · ·

↑ տտտ ∞v31• −→

v32• −→

v33• −→ • · · ·

↑•

տտտ ∞−→ • −→ • −→ •

· · ·

···

···

···

···

Again, by induction, it follows that LK(Pω) has exactly ω distinct isomorphismclasses of simple LK(Pω)-modules.

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The graph Pω+1 is obtained from the graph Pω by adding a single vertex vω+1

and connecting it by an edge to each of the vertices vj1 for j < ω in the graphPω. Specifically, (Pω+1)

0 = (Pω)0 ∪ {vω+1}, (Pω+1)

1 = (Pω)1 ∪ {eω+1,j : j < ω}

where, for each j, s(eω+1,j) = vω+1 and r(eω+1,j) = vj1. If Sω denotes theω-socle being the ideal generated by all the vertices in Pω , then LK(Pω+1)/Sω

is a simple LK(Pω+1)-module whose annihilator ideal is Sω and it is isomorphic

to the Leavitt path algebra of a graph {vω+1

• } consisting of a single vertex andno edges. Clearly LK(Pω+1) has ω + 1 distinct isomorphism classes of simpleLK(Pω+1)-modules.

Proceeding this way, as was shown in [8], we can construct, by transfiniteinduction, a Pyramid graph Pλ for each ordinal λ. The Leavitt path algebraLK(Pλ) is a semi-artinian von Neumann ring of Loewy length λ such that, foreach α < λ, the quotient Sα+1/Sα of successive socles is isomorphic to theLeavitt path algebra of an infinite line segment (like the graph P1) or a graph

{v•} consisting of a single vertex and no edges, according as α is a successor

or a limit ordinal. Thus Sα+1/Sα has exactly one isomorphism class of simplemodules (annihilated by Sα). By transfinite induction, one can then show thatthe Leavitt path algebra LK(Pλ) has exactly |λ| isomorphism classes of simpleLK(Pλ)-modules.

Acknowledgement: The first-named author was supported by DGI MICIIN-FEDER MTM2011-28992-C02-01, and by the Comissionat per Universitats iRecerca de la Generalitat de Catalunya.

Part of this work was done when the second-named author visited the Univer-sitat Autonoma de Barcelona during May 2013 and he gratefully acknowledgesthe support and the hospitality of the faculty of the Department of Mathematics.

References

[1] G. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph, J. Al-gebra 293 (2005) 319 - 334.

[2] G. Abrams, G. Aranda Pino, M. Siles Molina, Finite dimensional Leavittpath algebras, J. Pure Appl. Algebra 209 (2007) 753 - 762.

[3] G. Abrams, J. Bell, K.M. Rangaswamy, On prime non-primitive von Neu-mann regular algebras, Trans. Amer. Math. Soc. (to appear).

[4] G. Abrams, P. Ara, M. Siles Molina, Leavitt path algebras, A primer andhandbook, Springer (To appear).

[5] P. Ara, M.A. Moreno, E. Pardo, Non-stable K-theory for graph algebras,Algebra Represent. Theory, 10 (2007) 157-178.

[6] G. Aranda Pino, D. Martın Barquero, C. Martın Gonzalez, and M. SilesMolina, Socle theory for Leavitt path algebras of arbitrary graphs, Rev.Mat. Iberoamericana, 26 (2010) 611 - 638.

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[7] G. Abrams, K. M. Rangaswamy, Regularity conditions for Leavitt pathalgebras of arbitrary graphs, Algebra Represent. Theory, 13 (2010) 313 -334.

[8] G. Abrams, K. M. Rangaswamy, M. Siles Molina, The socle series of aLeavitt path algebra, Israel J. Math. 184 (2011) 413 - 435.

[9] X. W. Chen, Irreducible representations of Leavitt path algebras, ForumMath (2012) 20; DOI 10,1515.

[10] D. Goncalves, D. Royer, On the representations of Leavitt path algebras,J. Algebra 333 (2011) 258 - 272.

[11] K. M. Rangaswamy, The theory of prime ideals of Leavitt path algebrasover arbitrary graphs, J. Algebra 375 (2013) 73 - 96.

[12] K. M. Rangaswamy, On generators of two-sided ideals of Leavitt path al-gebras over arbitrary graphs, Commun. Algebra (to appear).

[13] E. Reuf, M. Tomforde, Ideals in graph algebras, arXiv:1205.1247, 25 August2012.

[14] M. Tomforde, Uniqueness theorems and ideal structure of Leavitt pathalgebras, J. Algebra 318 (2007) 270 -299.

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