A course on Leavitt path algebras

A course on Leavitt path algebras Mercedes Siles Molina Monastir (Tunisia), April 2009 !! Contents Contents 1 1 Finite dimensional Leavitt path algebras 3 1.1 The IBN property and the type of a ring. . . . . . . . . . . . . 3 1.2 Path algebras and Leavitt path algebras . . . . . . . . . . . . 5 1.3 Examples of Leavitt path algebras . . . . . . . . . . . . . . . . 7 1.4 Finite dimensional Leavitt path algebras . . . . . . . . . . . . 8 2 Uniqueness Theorems. Simple Leavitt path algebras 13 2.1 Semiprimeness in path algebras and in Leavitt path algebras . 13 2.2 Uniqueness theorems . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Simple Leavitt path algebras . . . . . . . . . . . . . . . . . . . 18 2.4 Purely infinite Leavitt path algebras . . . . . . . . . . . . . . 25 2.5 The dichotomy principle for simple Leavitt path algebras . . . 29 1 2 Chapter 1 Definitions, examples. Finite dimensional Leavitt path algebras Introduction Leavitt path algebras are a specific type of path K-algebras associated to a graph E, modulo some relations. Its appearance, for row-finite graphs, took place in [2] and [15]. They can be considered, on the one hand, natural generalizations of Leavitt algebras L(1, n) of type (1, n), introduced and investigated by Leavitt in [41] in order to give examples of algebras not satisfying the IBN property. On the other hand, they are the algebraic version of Cuntz-Krieger graph C∗-algebras, a class of algebras intensively investigated by analysts for more than two decades. For a complete explanation of the history of Leavitt path algebras see [20]. In this chapter we will relate Leavitt path algebras to the work by Leavitt and will give some important and interesting examples. Our main concern will be to classify the finite dimensional Leavitt path algebras. 1.1 The IBN property and the type of a ring. Let Rbe a unital ring. We say that Rsatisfies the invariant basis number (IBN) property if any two bases (i.e., linearly independent spanning sets) for a free left R-module have the same number of elements. In words, the IBN property says that if mand nare integers with the property that the free left modules RRmand RRnare isomorphic, then m=n. 3 4Chapter 1. Finite dimensional Leavitt path algebras Noetherian rings and commutative rings are included among the many classes of rings having this property. But the IBN property does not hold for all rings, as the following example shows. Example 1.1.1 For a field K, let V=K(N), which is a countably infinite dimensional vector space over K, and let R= EndK(V). It is not difficult to see that R∼ =RFMN(K), the countable row-finite matrices over K(use the standard basis for V, view the elements of Vas row-vectors, and apply transformations on the right hand side). Then RRm∼ =RRnfor all m, n ∈N: The first step is to show RR1∼ =RR2; such an isomorphism is given by the map which associates X∈Rwith the pair of matrices (X1, X2), where X1 (resp. X2) is built from the odd-numbered (resp. even numbered) columns of X. But then RR1∼ =RR2gives RR1⊕RR1∼ =RR2⊕RR1, so RR2∼ =RR3, and the result follows by continuing in this way. It is easy to determine, algebraically, whether or not for a ring Rwe have RR1∼ =RRnfor some n > 1. Note that such an isomorphism exists if and only if there is a set of 2nelements in Rwhich produce the appropriate isomorphisms as matrix multiplications by an n-row vector and an n-column vector with entries in R. Specifically, it is easy to show that RR1∼ =RRn for some n > 1 if and only if there exist elements x1, ..., xn, y1, ..., yn∈Rfor which xiyj=δij1Rfor all i, j, and Pn i=1 yixi= 1R. Suppose that a unital ring Rdoes not have IBN. Let m∈Nbe minimal with RRm∼ =RRnfor some n>mand find the minimal such nfor m. Then it is said that Rhas module type (m, n). (Warning: some authors call the module type of such a ring (m, n−m).) For example, RFMN(K) has module type (1,2). In his paper, Leavitt proved that for each pair of positive integers n > m and any field Kthere exists a K-algebra of module type (m, n). To do this, observe that, as it has been used before, isomorphisms between free modules can be realized as matrix multiplications by matrices having coefficients in R. So we need only construct algebras which contain elements which behave “correctly”. Do this as a quotient of a free associative K-algebra in the appropriate number of variables satisfying the appropriate relations. For example, to get an algebra of type (1,3) we need an algebra containing elements x1, x2, x3, y1, y2, y3for which xiyj=δij1Rfor all i, j, and P3 i=1 yixi= 1R. Consider the polynomial algebra over a field Kin 6 non-commuting variables. Then factor by the ideal generated by the appropriate relations. It is not difficult to show that this quotient is not zero if m≥2, but this is much more difficult to show (directly) if m= 1. 1.2. Path algebras and Leavitt path algebras 5 The quotient algebra described above is denoted LK(m, n), and called the Leavitt K-algebra of type (m, n). 1.2 Path algebras and Leavitt path algebras Definitions 1.2.1 A(directed) graph E= (E0, E1, r, s) consists of two countable sets E0, E1and maps r, s :E1→E0. The elements of E0are called vertices and the elements of E1edges. If s−1(v) is a finite set for every v∈E0, then the graph is called rowfinite. If E0is finite then, by the row-finite hypothesis, E1must necessarily be finite as well; in this case we simply say that Eis finite. A vertex which emits no edges is called a sink. A path µin a graph Eis a sequence of edges µ=e1. . . ensuch that r(ei) = s(ei+1) for i= 1, . . . , n −1. In this case, s(µ) := s(e1) is the source of µ,r(µ) := r(en) is the range of µ, and nis the length of µ, i.e, l(µ) = n. We denote by µ0the set of its vertices, that is: µ0={s(e1), r(ei) : i= 1, . . . , n}. Although much work has been done on arbitrary graphs, we will be concerned only with row-finite graphs. Example 1.2.2 Consider the following graph: •u2e2//•u3e jj •u1 e1<< z z z z z z z z f "" D D D D D D D D •v Then E0={u1, u2, u3, v},E1={e1, e2, ef},r(e1) = u1,s(e1) = u2, etc. The vertex vis a sink. Some paths are, for example, v,f,e1e2,e,ee,e2eee, etc. For µ=e1e2e3,µ0={u1, u2, u3}. Definition 1.2.3 Now let Kbe a field and let KE denote the K-vector space which has as a basis the set of paths. It is possible to define an algebra structure on KE as follows: for any two paths µ=e1. . . em, ν =f1. . . fn, we define µν as zero if r(µ)6=s(ν) and as e1. . . emf1. . . fnotherwise. This K-algebra is called the path algebra of Eover K. Example 1.2.4 Consider a field Kand the following graph E: 6Chapter 1. Finite dimensional Leavitt path algebras •u2e2//•u3 •u1 e1<< z z z z z z z z f "" D D D D D D D D •v Then the path algebra KE has, as a vector space over K, dimension 8, while the path K-algebra asociated to the graph in example 1.2.2 is infinite dimensional. There are several ways of defining Leavitt path algebras. Definition 1.2.5 Given a graph Ewe define the extended graph of Eas the new graph b E= (E0, E1∪(E1)∗, r0, s0), where (E1)∗={e∗ i:ei∈E1}and the functions r0and s0are defined as r0|E1=r, s0|E1=s, r0(e∗ i) = s(ei) and s0(e∗ i) = r(ei). Definition 1.2.6 Let Kbe a field and Ebe a row-finite graph. The Leavitt path algebra of Ewith coefficients in Kis defined as the path algebra over the extended graph b E, with relations: (CK1) e∗ iej=δijr(ej) for every ej∈E1and e∗ i∈(E1)∗. (CK2) vi=P{ej∈E1:s(ej)=vi}eje∗ jfor every vi∈E0which is not a sink. This algebra is denoted by LK(E) (or by L(E) if there is no risk of confusion with the field K). The conditions (CK1) and (CK2) are called the Cuntz-Krieger relations. In particular condition (CK2) is the Cuntz-Krieger relation at vi. If viis a sink, we do not have a (CK2 ) relation at vi. Note that the condition of row-finiteness is needed in order to define the equation (CK2). There exists a natural inclusion of the path algebra KE into the Leavitt path algebra LK(E) sending vertices to vertices and edges to edges. We will use this monomorphism without any explicit mention to it. Another way of introducing Leavitt path algebras is as follows. Definition 1.2.7 For a field Kand a row-finite graph E, the Leavitt path K-algebra LK(E) is defined as the universal K-algebra generated by a set {v:v∈E0}of pairwise orthogonal idempotents, together with a set of variables {e, e∗:e∈E1}, which satisfy the following relations: 1.3. Examples of Leavitt path algebras 7 (1) s(e)e=er(e) = efor all e∈E1. (2) r(e)e∗=e∗s(e) = e∗for all e∈E1. (3) e∗e0=δe,e0r(e) for all e, e0∈E1. (4) v=P{e∈E1|s(e)=v}ee∗for every v∈E0that emits edges. Universal means that if Ais a K-algebra containing a set of pairwise orthogonal idempotents {av:v∈E0}and a set of elements {be, be∗:e∈E1} satisfying the relations (1)-(4), then there exists an algebra homomorphism Φ : LK(E)→Asatisfying Φ(v) = avfor all v∈E0, Φ(e) = beand Φ(e∗) = be∗for all e∈E1. The uniqueness of the Leavitt path algebra associated to a graph Eand to a field Kfollows from the universal property. Again the identities (3) and (4) are called the Cuntz-Krieger relations. The elements of E1are called (real) edges, while for e∈E1we call e∗a ghost edge. The set {e∗|e∈E1}will be denoted by (E1)∗. We let r(e∗) denote s(e), and we let s(e∗) denote r(e). If µ=e1. . . enis a path, then we denote by µ∗the element e∗ n. . . e∗ 1of LK(E). 1.3 Examples of Leavitt path algebras Many well-known examples of Kalgebras can be seen as Leavitt path Kalgebras over concrete graphs. Example 1.3.1 (Leavitt algebras of type (1, n),n > 1). The Leavitt path K-algebra of the following graph •ve1 hh e2 ss e3  en RR is the Leavitt algebra of type (1, n), for n > 1. Example 1.3.2 (Matrix algebras). Consider the graph •u1e1//•u2•un−1en−1//•un Then Mn+1(K)∼ =LK(E), via the map ui7→ eii, ei7→ ei+1i, and e∗ i7→ eii+1, where eij denotes the matrix unit in Mn(K) with all entries equal zero except that in row iand column j. 14 Chapter 2. Uniqueness Theorems. Simple Leavitt path algebras homogeneous element of KE, then x(KE)x6= 0. Write x=Pn i=1 kiαi, with 06=ki∈Kand α1, . . . , αndifferent paths of the same degree (i.e. of the same length). Denote the source and range of α1by u1and v1, respectively. Then, by (3), α1∗x=k1α1∗α1=k1v1. By the hypothesis, there exists a path α10such that s(α10) = v1and r(α10) = u1. Observe that α10x6= 0; otherwise 0=(α10)∗α10x=u1x, a contradiction since a set of different paths is always linearly independent over K(Lemma 1.4.6) and α1=u1α16= 0. Therefore 06=k1α10x=k1v1α10x=α1∗xα10x∈α1∗x(KE)x.2 Lemma 2.1.2 ([18, Lemma 1.5]). Let Ebe an arbitrary graph. Let vbe a vertex in E0such that there exists a cycle without exits cbased at v. Then: vLK(E)v=(n X i=−m kici|ki∈K;m, n ∈N)∼ =K[x, x−1], where ∼ =denotes a graded isomorphism of K-algebras, and considering (by abuse of notation) c0=wand c−t= (c∗)t, for any t≥1. Proof. First, it is easy to see that if c=e1. . . enis a cycle without exits based at vand u∈T(v), then s(f) = s(g) = u, for f, g ∈E1, implies f=g. Moreover, if r(h) = r(j) = w∈T(v), with h, j ∈E1, and s(h), s(j)∈T(v) then h=j. We have also that if µ∈E∗and s(µ) = u∈T(v) then there exists k∈N,1≤k≤nverifying µ=ekµ0and s(ek) = u. Let x∈vLK(E)vbe given by x=Pp i=1 kiαiβ∗ i+δv, with s(αi) = r(β∗ i) = s(βi) = vand αi, βi∈E∗. Consider A={α∈E∗:s(α) = v}; we prove now that if α∈A,deg(α) = mn +q, m, q ∈Nwith 0 ≤q < n, then α=cme1. . . eq. We proceed by induction on deg(α). If deg(α) = 1 and s(α) = s(e1) then α=e1. Suppose now that the result holds for any β∈A with deg(β)≤sn +tand consider any α∈A, with deg(α) = sn +t+ 1. We can write α=α0fwith α0∈A,f∈E1and deg(α0) = sn +t, so by the induction hypothesis α0=cse1. . . et. Since s(f) = r(et) = s(et+1) implies f=et+1, then α=α0f=cse1. . . et+1. We shall show that the elements αiβ∗ iare in the desired form, i.e., cd with d∈Z. Indeed, if deg(αi) = deg(βi) and αiβ∗ i6= 0, we have αiβ∗ i= cpe1. . . eke∗ k. . . e∗ 1c−p=vby (4). On the other hand deg(αi)> deg(βi) and αiβ∗ i6= 0 imply αiβ∗ i=cd+qe1. . . eke∗ k. . . e∗ 1c−q=cd, d ∈N∗. In a similar way, from deg(αi)< deg(βi) and αiβ∗ i6= 0 it follows that αiβ∗ i= cqe1. . . eke∗ k. . . e∗ 1c−q−d=c−d, d ∈N∗. Define ϕ:K[x, x−1]→LK(E) by ϕ(1) = v, ϕ(x) = cand ϕ(x−1) = c∗. It is a straightforward routine to check that ϕis a graded monomorphism with image vLK(E)v, so that vLK(E)vis graded isomorphic to K[x, x−1] as a graded K-algebra. 2 2.1. Semiprimeness in path algebras and in Leavitt path algebras 15 For a not necessarily associative K-algebra A, and fixed x, y ∈A, the left and right multiplication operators Lx, Ry:A→Aare defined by Lx(y) := xy and Ry(x) := xy. Denoting by EndK(A) the K-algebra of K-linear maps f:A→A, the multiplication algebra of A(denoted M(A)) is the subalgebra of EndK(A) generated by the unit and all left and right multiplication operators La, Ra:A→A. There is a natural action of M(A) on Asuch that A is an M(A)-module whose submodules are just the ideals of A. This is given by M(A)×A−→ A, where f·a:= f(a) for any (f, a)∈ M(A)×A. Given x, y ∈Awe shall say that xis linked to yif there is some f∈ M(A) such that y=f(x). This fact will be denoted by x`y. The result that follows states that any nonzero element in a Leavitt path algebra is linked to either a vertex or to a nonzero polynomial in a cycle with no exits. So it gives a full account of the action of M(LK(E)) on LK(E). This result is very powerful as the main ingredient to show that the socle of a Leavitt path algebra of a row-finite graph is the ideal generated by the line points. Other interesting results are also obtained as a consequence of it. Proposition 2.1.3 ([17, Proposition 3.1]). Let Ebe an arbitrary graph. Then, for every nonzero element x∈LK(E), there exist µ1, . . . , µr, ν1, . . . , νs ∈E0∪E1∪(E1)∗such that: 1. µ1. . . µrxν1. . . νsis a nonzero element in Kv, for some v∈E0, or 2. there exist a vertex w∈E0and a cycle without exits cbased at wsuch that µ1. . . µrxν1. . . νsis a nonzero element in wLK(E)w. Both cases are not mutually exclusive. Proof. Show first that for a nonzero element x∈L(E), there exists a path µ∈L(E) such that xµ is nonzero and in only real edges. Consider a vertex v∈E0such that xv 6= 0. Write xv =Pm i=1 βie∗ i+β, with ei∈E1,ei6=ejfor i6=jand βi, β ∈L(E), βin only real edges and such that this is a minimal representation of xv in ghost edges. If xvei= 0 for every i∈ {1, . . . , m}, then 0 = xvei=βi+βei, hence βi=−βei, and xv =Pm i=1 −βeie∗ i+β=β(Pm i=1 −eie∗ i+v)6= 0. This implies that Pm i=1 −eie∗ i+v6= 0 and since s(ei) = vfor every i, this means that there exists f∈E1,f6=eifor every i, with s(f) = v. In this case, xvf =βf 6= 0 (because βis in only real edges), with βf in only real edges, which would conclude our discussion. If xvei6= 0 for some i, say for i= 1, then 0 6=xve1=β1+βe1, with β1+βe1having strictly less degree in ghost edges than x. Repeating this argument, in a finite number of steps we prove our first statement. 16 Chapter 2. Uniqueness Theorems. Simple Leavitt path algebras Now, assume x=xv for some v∈E0and xin only real edges. Let 06=x=Pr i=1 kiαibe a linear combination of different paths αiwith ki6= 0 for any i. We prove by induction on rthat after multiplication on the left and/or the right we get a vertex or a polynomial in a cycle with no exit. For r= 1 if α1has degree 0 then it is a vertex and we have finished. Otherwise we have x=k1α1=k1f1· · · fnso that k−1 1f∗ n· · · f∗ 1x=vwhere v=r(fn)∈E0. Suppose now that the property is true for any nonzero element which is a sum of less than rpaths in the conditions above. Let 0 6=x=Pr i=1 kiαi such that deg(αi)≤deg(αi+1) for any i. If for some iwe have deg(αi) = deg(αi+1) then, since αi6=αi+1, there is some path µsuch that αi=µfν and αi+1 =µf0ν0where f, f0∈E1are different and ν, ν0are paths. Thus 06=f∗µ∗xand we can apply the induction hypothesis to this element. So we can go on supposing that deg(αi)<deg(αi+1) for each i. We have 0 6=α∗ 1x=k1v+Pikiβi, where v=r(α1) and βi=α∗ 1αi. If some βiis null then apply the induction hypothesis to α∗ 1xand we are done. Otherwise if some βidoes not start (or finish) in vwe apply the induction hypothesis to vα∗ 1x6= 0 (or α∗ 1xv 6= 0). Thus we have 06=z:= α∗ 1x=k1v+ r X i=1 kiβi, where 0 <deg(β1)<· · · <deg(βr) and all the paths βistart and finish in v. Now, if there is a path τsuch that τ∗βi= 0 for some βibut not for all of them, then we apply our inductive hypothesis to 0 6=τ∗zτ. Otherwise for any path τsuch that τ∗βj= 0 for some βj, we have τ∗βi= 0 for all βi. Thus βi+1 =βirifor some path riand zcan be written as z=k1v+k2γ1+k3γ1γ2+· · · +krγ1· · · γr−1, where each path γistarts and finishes in v. If the paths γiare not identical we have γ16=γifor some i, then 0 6=γ∗ izγi=k1vproving our thesis. If the paths are identical then zis a polynomial in the cycle c=γ1with independent term k1v, that is, an element in vL(E)v. If the cycle has an exit, it can be proved that there is a path ηsuch that η∗c= 0, in the following way: Suppose that there is a vertex w∈T(v), and two edges e, f, with e6=f,s(e) = s(f) = w, and such that c=aweb =aeb, for aand bpaths in L(E). Then η=af gives η∗c=f∗a∗aeb =f∗eb = 0. Therefore, η∗zη is a nonzero scalar multiple of a vertex. Moreover, if cis a cycle without exits, by Lemma 2.1.2, vL(E)v=(n X i=−m lici,with li∈Kand m, n ∈N), 2.2. Uniqueness theorems 17 where we understand c−m= (c∗)mfor m∈Nand c0=v. Finally, consider the graph Econsisting of one vertex and one loop based at the vertex to see that both cases can happen at the same time. This completes the proof. 2 Corollary 2.1.4 For any nonzero x∈ L we have x`vfor some v∈E0or x`p(c, c∗)where cis a cycle with no exits and pa nonzero polynomial in c and c∗. Proof. Use Lemma 2.1.2 together with Proposition 2.1.3. 2 Proposition 2.1.5 Let Ebe an arbitrary graph. Then LK(E)is semiprime. Proof. Take a nonzero ideal Isuch that I2= 0. If Icontains a vertex we are done. On the contrary there is a nonzero element p(c, c∗)∈Iby Corollary 2.1.4. If we consider the (nonzero) coefficient of maximum degree in cand write p(c, c∗)2= 0 we immediately see that this scalar must be zero, a contradiction. 2 2.2 Uniqueness theorems An edge eis an exit for a path µ=e1. . . enif there exists isuch that s(e) = s(ei) and e6=ei. A graph is said to satisfy Condition (L) if every cycle in the graph has an exit. For any K-algebra Athe M(A)-submodules of Aare just the ideals of Aand the cyclic M(A)-submodules of Aare the ideals generated by one element (principal ideals in the sequel), so Corollary 2.1.4 states that the nonzero principal ideals of any Leavitt path algebra contain either vertices or nonzero elements of the form p(c, c∗). Therefore, for graphs in which every cycle has an exit, each nonzero ideal contains a vertex. Definition 2.2.1 Let A=Pn∈ZAnbe a Z-graded algebra. An ideal Iof A is said to be a graded ideal if I∩An⊆I. Definition 2.2.2 We say that a graph Esatisfies Condition (L) if every cycle has an exit. The following result is a consequence of Proposition 2.1.3. Corollary 2.2.3 Let Ebe an arbitrary graph. (i) Every Z-graded nonzero ideal of LK(E)contains a vertex. 18 Chapter 2. Uniqueness Theorems. Simple Leavitt path algebras (ii) Suppose that Esatisfies Condition (L). Then every nonzero ideal of LK(E)contains a vertex. Proof. The second assertion has been proved above. So assume that Iis a graded ideal of L which contains no vertices. Let 0 6=x∈Iand use Corollary 2.1.4 to find elements y, z ∈LK(E) such that yxz =Pn i=−mkici6= 0. But Ibeing a graded ideal implies that every summand is in I. In particular, for t∈ {−m, . . . , n}such that ktct6= 0 we have 0 6= (kt)−1c−tktct=w∈I, which is absurd. 2 Theorem 2.2.4 Let Ebe an arbitrary graph, and let LK(E)be the associated Leavitt path algebra. (1) Graded Uniqueness Theorem. If Ais a Z-graded ring and π:LK(E)→Ais a graded ring homomorphism with π(v)6= 0 for every vertex v∈E0, then πis injective. (2) Cuntz-Krieger Uniqueness Theorem. Suppose that Esatisfies Condition (L). If π:LK(E)→Ais a ring homomorphism with π(v)6= 0, for every vertex v∈E0, then πis injective. Proof. In both cases, the kernel of the ring homomorphism πis an algebra ideal (a graded ideal in the first one). By Corollary 2.2.3, Ker(π) must be zero because otherwise it would contain a vertex (apply (i) in the corollary to (1) and (ii) to the other case), which is not possible by the hypotheses. 2 2.3 Simple Leavitt path algebras In this section we use Proposition 2.1.3 to proof a characterization of simple Leavitt path algebras (see [2, Theorem 3.11] and [18, Corollary 3.8]). Recall that an algebra Ais said to be simple if A26= 0 and it has no nonzero proper ideals. If the algebra is graded by a group G, write A= Pσ∈GAσ, it is called graded simple if A26= 0 and it has no nonzero proper graded ideals (an ideal Iof Ais graded if whenever y=Pσ(yσ), every yσ∈I). In general, and in the particular case of Leavitt path algebras, simplicity and graded-simplicity are not equivalent, as we shall see. For n≥2 we write Ento denote the set of paths of length n, and E∗=Sn≥0Enthe set of all paths. We define a relation ≥on E0by setting v≥wif there is a path µ∈E∗with s(µ) = vand r(µ) = w. A subset H 2.3. Simple Leavitt path algebras 19 of E0is called hereditary if v≥wand v∈Himply w∈H. A hereditary set is saturated if every vertex which feeds into Hand only into His again in H, that is, if s−1(v)6=∅and r(s−1(v)) ⊆Himply v∈H. Denote by H (or by HEwhen it is necessary to emphasize the dependence on E) the set of hereditary saturated subsets of E0. For a graph E, the empty set and E0 are hereditary and saturated subsets of E0. The set T(v) = {w∈E0|v≥w}is the tree of v, and it is the smallest hereditary subset of E0containing v. We extend this definition for an arbitrary set X⊆E0by T(X) = Sx∈XT(x). The hereditary saturated closure of a set Xis defined as the smallest hereditary and saturated subset of E0 containing X. It is shown in [15] that the hereditary saturated closure of a set Xis X=S∞ n=0 Λn(X), where Λ0(X) = T(X), and Λn(X) = {y∈E0|s−1(y)6=∅and r(s−1(y)) ⊆Λn−1(X)} ∪ Λn−1(X), for n≥1. One way of constructing graded ideals in a Leavitt path algebra is the following: Lemma 2.3.1 Let Hbe a hereditary subset of E0, for a graph E. Then I(H) = nXkαβ∗,with k∈K, α, β paths such that r(α) = r(β)∈Ho. In fact, I(H)is a graded ideal. Proof. Denote by Jthe following set: J=nXkαβ∗|k∈K, α, β are paths and r(α) = r(β)∈Ho. The containment J⊆I(H) is clear. For the converse, consider µ, ν, α, β paths in L(E), and u∈Hsuch that µν∗uαβ∗6= 0. By [53, Lemma 3.1], µν∗uαβ∗is µα0β∗if α=να0or µν0∗β∗if ν=αν0. Note that α=να0, u=s(α) and Hhereditary imply r(α0)∈H, hence µα0β∗∈J. In the second case, ν=αν0,u=s(α) and Hhereditary imply s(ν0∗) = r(ν0)∈H, hence µr(ν0)ν0∗β∗∈J, therefore I(H)⊆J. The last statement follows immediately by the form the elements of I(H) have. 2 Moreover, it was proved in [2, Lemma 3.9] 20 Chapter 2. Uniqueness Theorems. Simple Leavitt path algebras Lemma 2.3.2 For every ideal Iof a Leavitt path algebra LK(E),I∩E0is a hereditary and saturated subset of E0. In fact, all graded ideals of a Leavitt path algebra come from hereditary and saturated subsets of vertices. Remark 2.3.3 An ideal Jof L(E) is graded if and only if it is generated by idempotents; in fact, J=I(H), where H=J∩E0∈ HE. (See the proofs of [15, Proposition 4.2 and Theorem 4.3].) Now, the question that arises is if every graded ideal of a Leavitt path algebra is again a Leavitt path algebra (note that this question has sense only for graded ideals as every Leavitt path algebra is graded). Other natural question is if the quotient of a Leavitt path algebra by an ideal is a Leavitt path algebra too. In both cases, as we shall see, the answer is yes. For a graph Eand a hereditary subset Hof E0, we denote by E/H the quotient graph (E0\H, {e∈E1|r(e)6∈ H}, r|(E/H)1, s|(E/H)1), and by EHthe restriction graph (H, {e∈E1|s(e)∈H}, r|(EH)1, s|(EH)1). Observe that while L(EH) can be seen as a subalgebra of L(E), the same cannot be said about L(E/H). Lemma 2.3.4 ([19, Lemma 2.3]) Let Ebe a graph and consider a proper H∈ HE. Define Ψ : L(E)→L(E/H)by setting Ψ(v) = χ(E/H)0(v)v,Ψ(e) = χ(E/H)1(e)eand Ψ(e∗) = χ((E/H)1)∗(e∗)e∗for every vertex vand every edge e, where χ(E/H)0:E0→Kand χ(E/H)1:E1→Kdenote the characteristic functions. Then: 1. The map Ψextends to a K-algebra epimorphism of Z-graded algebras with Ker(Ψ) = I(H)and therefore L(E)/I(H)∼ =L(E/H). 2. If Xis hereditary in E, then Ψ(X)∩(E/H)0is hereditary in E/H. 3. For X⊇H,X∈ HEif and only if Ψ(X)∩(E/H)0∈ H(E/H). 4. For every X⊇H,Ψ(X)∩(E/H)0= Ψ(X)∩(E/H)0. 2.3. Simple Leavitt path algebras 21 Proof. (1) It was shown in [2, Proof of Theorem 3.11] that Ψ extends to aK-algebra morphism. By definition, Ψ is Z-graded and onto. Moreover, I(H)⊆Ker(Ψ). Since Ψ is a graded morphism, Ker(Ψ) ∈ Lgr(L(E)). By [15, Theorem 4.3], there exists X∈ HEsuch that Ker(Ψ) = I(X). By Lemma ??,H= I(H)∩E0⊆I(X)∩E0=X. Hence, I(H)6= Ker(Ψ) if and only if there exists v∈X\H. But then Ψ(v) = v6= 0 and v∈Ker(Ψ), which is impossible. (2) It is clear by the definition of Ψ. (3) Since Ψ is a graded epimorphism, there is a bijection between graded ideals of L(E/H) and graded ideals of L(E) containing I(H). Thus, the result holds by [15, Theorem 4.3]. (4) It is immediate by part (3). 2 Lemma 2.3.5 Let Ebe a graph. For every hereditary and saturated subset Hof E, the ideal I(H)is isomorphic to L(HE). Corollary 2.3.6 Every graded ideal of a Leavitt path algebra is again a Leavitt path algebra. One interesting property for Leavitt path algebras is that cycles without exists behaves in a similar way to sinks, so, roughly speaking, for a graph having no cycles with exits, and such that every vertex connects to the cycle (the so called Cn-comet), the corresponding Leavitt path algebra is a direct sum of matrices over something appropriate. The notion of Cn-comet was introduced in [7] to describe the locally finite Leavitt path algebras. The role of the cycle Cnwithin a Cn-comet is similar to that played by sinks in more general graphs. If a graph Eis a Cn-comet, then its associated Leavitt path algebra is isomorphic to Mn(K[x, x−1]). Since Cncomets have a finite number of vertices, it is natural to generalize this concept to the case of an infinite (numerable) set of vertices. Definition 2.3.7 We say that a graph Eis a comet if it has exactly one cycle c,T(v)∩c06=∅for every vertex v∈E0, and every infinite path ends in the cycle c. Remark 2.3.8 The following is not an example of a Cn-comet: •v1e1// "" E E E E E E E E•v2e2//  •v3e3// ||y y y y y y y y•v4 vvllllllllllllllll • c ZZ 22 Chapter 2. Uniqueness Theorems. Simple Leavitt path algebras and the reason is that the infinite path γ=e1e2e3. . . does not end either in a sink or in a cycle. Proposition 2.3.9 Let Ebe a graph which is a comet. Then the Leavitt path algebra L(E)is isomorphic to Mn(K[x, x−1]), where n∈Nif Eis finite, or n=∞otherwise. Proof. We can adapt [7, Theorem 3.3] to our situation. Concretely, let cbe the cycle in E,va vertex at which the cycle cis based and consider {pi}the (perhaps infinite) family of all paths in Ewhich end in vbut do not contain the cycle c. Let n∈N∪ {∞} be the number of all such paths. Denote by Nthe set {1, . . . , n}when nis finite and N=Nwhen n=∞. Consider the family B:= {pickp∗ j}i,j∈N,k∈Nof monomials in L(E) where we understand c0=vand cn= (c∗)−nfor negative n. As in [7, Theorem 3.3], we can show that Bis a linearly independent set. We will prove that Bgenerates L(E) as a K-vector space. First, note that since Eis a comet, then T(v) is a finite set for every v∈c0. Not only is this true for any vertex on the cycle cbut also for any vertex in Eas follows: Suppose on the contrary that there exists w∈Ewith |T(w)|=∞. In particular, wdoes not lie on the cycle. As Eis row-finite, we are able to find and edge e1in Ewith s(e1) = wand v1:= r(e1) such that |T(v1)|=∞. Again v1does not lie on the cycle. Repeating this process, we find an infinite path such that none of its vertices lie on c, which contradicts the fact that every infinite path in Eends in the cycle c. Take an arbitrary element Pikiαiβ∗ iof L(E), where αi, βiare paths in Eand ki∈K. Consider the set {r(αi)}. Some of these vertices could lie on the cycle c, in which case we leave the corresponding monomial as is. For those monomials αkβ∗ kwhose {r(αk)}is not on c, we proceed as in [6, Proof of Proposition 3.5] by using relation (4) to expand it as αkβ∗ k=X {e∈E1:s(e)=r(αk)} αkee∗β∗ k=X {e∈E1:s(e)=r(αk)} (αke)(βke)∗. As we have just proved that the tree of any vertex is finite, so will be this process of expanding these monomials until reaching vertices of c. Consider now a monomial αkβ∗ kwith r(αk)∈c0. Let tbe the subpath of cwith s(t) = r(αk) and r(t) = v. Since cdoes not have exits then αkβ∗ k=αktt∗β∗ k= (αkt)(βkt)∗=αβ∗, where αand βare paths in Ethat end in v. Finally, since Eis a comet, we can always factor some powers of cout of αand βso that there exist integers m, n such that α=picmand β=pjcn for some paths pi, pjwhich do not contain the path c. Hence, we obtain that αkβ∗ k=picm−np∗ j∈ B. This proves that Bis a K-generator of L(E). 2.3. Simple Leavitt path algebras 23 Now, by defining φ:L(E)→Mn(K[x, x−1]) on the basis by setting φ(pickp∗ j) = xkeij for eij the (i, j)-matrix unit, then again one easily checks that φis a K-algebra isomorphism. 2 For a graph E, denote by Pc(E) the set of vertices in the cycles without exits of E. Proposition 2.3.10 Let Ebe a graph. Then: (i) I(Pc(E)) = Lj∈ΥI(Pcj(E)), where Υis a countable set and {cj}j∈Υ is the set of all different cycles without exits of E(and by abuse of notation we identify two cycles that have the same vertices). (ii) Pc(E)is hereditary and if Hdenotes the saturated closure of Pc(E), we have that I(Pc(E)) = I(H)∼ =L(HE)∼ =M i∈Υ1 Mni(K[x, x−1])⊕M j∈Υ2 Mmj(K[x, x−1]), where Υ1and Υ2are countable sets, ni∈Nand mj=∞. Proof. We will use Lemma 2.3.1 implicitly. This can be done because Pc(E) is, clearly, a hereditary set. (i). To shorten the notation, write: J=I(Pc(E)) and Jj=I(Pcj(E)). Consider monomials γδ∗with r(δ)∈(cj)0and στ∗∈ J . Since the cycles cjhave no exits, they are disjoint and then, similar arguments to that of the previous paragraph show that γδ∗στ∗, στ∗γδ∗∈ Jcj. Moreover, these arguments also yield that if στ∗∈ Jckwith j6=k, then γδ∗στ∗=στ∗γδ∗= 0. Thus, {Jcj}is indeed a family of orthogonal ideals of J. To show that J=PjJjapply Lemma 2.3.1 to H=∪jc0 j, which is a hereditary set since the considered cycles have no exits. (ii). I(Pc(E)) = I(H) follows by [19, Lemma 2.1] and I(H)∼ =L(HE) by [16, Lemma 1.2] The same results applied to cjinstead of cimply I(Pcj(E)) = I(Hj)∼ =L(HjE), for Hjthe saturated closure of Pcj. By the definition of Hj, and since cjhas no exits, every vertex in Hjconnects to cj. The same can be said about HjE, where cjcan be seen as its only cycle. Now suppose that γis an infinite path in HjE. Again, by the way this graph is constructed, there must exist a finite path pand an infinite path βsuch that γ=pβ, with βbeing completely contained in EHj. Suppose that βdoes not end in the cycle cj. This, together with the fact that cjdoes not have exits, yield that β0∩c0 j=∅. On the other hand, because β0⊆Hjwe can consider mto be the minimum nsuch that Λn(c0 j)∩β06=∅. Now, β0∩c0 j=∅implies that m > 0 so that there exists w∈ {v∈(EHj)0| ∅ 6=r(s−1(v)) ⊆Λm−1(c0 j)}∩β0. 30 Chapter 2. Uniqueness Theorems. Simple Leavitt path algebras vertices are not in H, as desired. Now consider w∈H. By the hypothesis, there exists z∈γsuch that w≥z, and by hereditariness of Hwe get z∈H, contradicting the definition of γ. Conversely, suppose that H={∅, E0}. Take v∈E0and γ∈E≤∞, with v6∈ γ0(the case v∈γ0is obvious). By hypothesis the hereditary saturated subset generated by vis E0, i.e., E0=Sn≥0Λn(v). Consider m, the minimum nsuch that Λn(v)∩γ06=∅, and let w∈Λm(v)∩γ0. If m > 0, then by minimality of mit must be s−1(w)6=∅and r(s−1(w)) ⊆Λm−1(v). 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