1 Introduction↩︎

Given a graph \(E\) and a field \(\mathsf{k}\), Abrams and Aranda Pino in [1], and independently Ara, Moreno, and Pardo in [2], introduced the Leavitt path algebra \(L_\mathsf{k}(E)\), which is a quotient of a path algebra obtained by imposing some additional relations amongst the generators. These Leavitt path algebras generalise the algebras \(L_{\mathsf{k}}(1, 1 +k)\) constructed by Leavitt [3], as universal rings without invariant basis number, and are also the discrete, purely algebraic version of graph \(C^*\)-algebras. We refer the reader to [4] and [5] for a detailed history and overview of Leavitt path algebras.

The classification of Leavitt path algebras has received a lot of attention, as they are closely related to the classification of graph \(C^*\)-algebras and those of symbolic dynamics. However, while graph \(C^*\)-algebras have been classified by means of \(K\)-theoretic invariants (see, e.g., [6]–[8]), an analogous classification for Leavitt path algebras remains conjectural. The Graded Classification Conjecture for Leavitt path algebras states that the graded Grothendieck group \(K^{\operatorname{gr}}_0\), is a complete invariant for these algebras ([5], [9], [4]):

G. Arnone in [10] and L. Vas in [11] independently answered part (1) of Conjecture 1 in the affirmative, using different approaches. In fact Arnone showed that the lifting map can be diagonal preserving graded \(*\)-homomorphism, while Vas established the result for graphs that may contain infinitely many edges. Part (2) of the Graded Classification Conjectures remains open, as neither a proof nor a counterexample is currently known. The graded Grothendieck group \(K_0^{\operatorname{gr}}\) is closely related to Krieger’s dimension group [12], thereby connecting the classification of Leavitt path algebras with the classification of shift of finite types in symbolic dynamics [13]. For a comprehensive account of the graded classification conjectures and their connections to related topics, we refer the reader to [5].

Whereas Leavitt path algebras model Leavitt algebras \(L_\mathsf{k}(1,1+k)\), weighted Leavitt path algebras were introduced by the second author in [14] to cover the more general Leavitt algebras \(L_\mathsf{k}(n,n+k)\) [3], which cannot, in general, be realised as ordinary Leavitt path algebras. We also note that weighted Leavitt path algebras are closely connected to sandpile models (see, e.g., [15]–[17]). Since weighted Leavitt path algebras are also naturally \(\mathbb{Z}\)-graded rings, it is natural to investigate whether the graded Grothendieck group \(K_0^{\operatorname{gr}}\) carries structural information about these algebras analogous to that provided by \(K_0^{\operatorname{gr}}\) in the ordinary Leavitt path algebra setting.

In this article, we show that \(K^{\operatorname{gr}}_0\) detects the presence of weights, thereby distinguishing weighted Leavitt path algebras from their unweighted counterparts. This is not the case when one uses the ordinary \(K_0\)-group. Indeed, for \(k,l, n\in \mathbb{N}\), there are order isomorphisms \[K_0(L_\mathsf{k}(1,1+k))\cong K_0(L_\mathsf{k}(n,n+k))\cong K_0(L_\mathsf{k}(n+l,n+k+l))\cong \mathbb{Z}_{k}.\] Nevertheless, the algebras are pairwise non-isomorphic: the first is an ordinary Leavitt path algebra, while the others arise only as weighted Leavitt path algebras.

For a vertex-weighted graph \((E,w)\), we show that \(K_0^{\operatorname{gr}}\) is a full functor, that means, Conjecture 1(1) holds for vertex-weighted graphs. By specializing to the case where all weights are equal to one, we recover the corresponding results of Arnone and Vas mentioned above. Furthermore, we obtain that every order preserving \(\mathbb{Z}\)-module isomorphism between \(K_0^{\operatorname{gr}}(L_{\mathsf{k}}(E, w))\) and \(K_0^{\operatorname{gr}}(L_{\mathsf{k}}(F, w))\) induces an isomorphism between the semilattices of all vertex-generated ideals of \(L_{\mathsf{k}}(E, w)\) and \(L_{\mathsf{k}}(F, w)\), which provides further evidence in support of Conjecture 1(2) for vertex-weighted graphs.

To do so, we begin by defining the graph monoid \(M_{(E,w)}\) and the talented monoid \(T_{(E,w)}\) associated with a vertex-weighted graph \((E,w)\). We show that \(T_{(E,w)}\) is a cancellative monoid and that its Grothendieck group completion is precisely the graded Grothendieck group \(K_0^{\operatorname{gr}}(L_\mathsf{k}(E,w))\). The cancellativity of \(T_{(E,w)}\) implies that it coincides with the positive cone of \(K_0^{\operatorname{gr}}(L_\mathsf{k}(E,w))\), allowing one to work directly with the monoid rather than with the \(K_0^{\operatorname{gr}}\)-group. Working with the talented monoid has the advantage of being independent of the algebra itself while still providing detailed information about its elements (such as minimal elements, atoms, and related invariants) and, consequently, about the geometry of the underlying graph, including the number of cycles and their lengths. Moreover, we use Bergman machinery [18] and its graded version developed in [19] to establish Conjecture 1(1) for vertex-weighted graphs.

The paper is organised as follows. In Section 2, we prove that the graph monoid associated with an acyclic vertex-weighted graph is cancellative (Theorem 4). Consequently, the talented monoid associated with a vertex-weighted graph is cancellative (Proposition 11(2)). We also give nice connections between the graph monoid \(M_{(E,w)}\) of a vertex-weighted graph \((E, w)\) and the talented monoid \(T_{(E,w)}\) (Proposition 11). Moreover, we show that, in general, the talented monoid associated with a vertex-weighted graph cannot be realized as the talented monoid of an ordinary graph (Example 5). In Section 3, based on Section 2, we demonstrate that the positive cone of the graded Grothendieck group \(K^{\operatorname{gr}}_0(L_{\mathsf{k}}(E, w))\) of the Leavitt path algebra of a vertex-weighted graph \((E, w)\) is exactly the talented monoid of \((E, w)\) (Corollary 14), and show that the graded Grothendieck group \(K^{\operatorname{gr}}_0\) distinguishes the class of vertex-weighted Leavitt path algebras from the unweighted class (Theorem 17). Moreover, we prove that every order preserving \(\mathbb{Z}\)-module isomorphism between \(K_0^{\operatorname{gr}}(L_{\mathsf{k}}(E, w))\) and \(K_0^{\operatorname{gr}}(L_{\mathsf{k}}(F, w))\) provides an isomorphism between the semilattices of all vertex-generated ideals of \(L_{\mathsf{k}}(E, w)\) and \(L_{\mathsf{k}}(F, w)\) (Theorem 18). In Section 4, we prove that Conjecture 1(1) holds for vertex-weighted graphs (Theorems 19 and 23) and show that the graded Grothendieck group \(K_0^{\operatorname{gr}}\) classifies the Leavitt algebras \(L_\mathsf{k}(n,n+k)\) (Theorem 24).

2 Weighted graphs and graph monoids↩︎

The main goal of this section is to establish the graph monoid associated with an acyclic vertex-weighted graph is cancellative (Theorem 4). Also, we completely describe the structure of all order ideals of the graph monoid associated with a vertex-weighted graph (Theorem 8) via the lattice of all saturated hereditary subsets. Moreover, we give several connections between the graph monoid \(M_{(E,w)}\) associated with a vertex-weighted graph \((E, w)\) and the talented monoid \(T_{(E,w)}\) (Proposition 11); in particular, we provide a lattice isomorphism between the lattice of all order ideals of \(M_{(E,w)}\) and the lattice of all \(\mathbb{Z}\)-order ideals of \(T_{(E,w)}\).

We start by setting some terminology and notation about graphs. A (directed) graph is a quadruple \(E=(E^0,E^1,s,r)\), where \(E^0\) and \(E^1\) are sets and \(s,r:E^1\rightarrow E^0\) are maps. Throughout, “graph" will always mean”directed graph". The elements of \(E^0\) are called vertices and the elements of \(E^1\) edges. (We allow the empty set to be viewed as a graph with \(E^0 = E^1 = \varnothing.\)) If \(e\) is an edge, then \(s(e)\) is called its source and \(r(e)\) its range. If \(v\) is a vertex and \(e\) an edge, we say that \(v\) emits \(e\) if \(s(e)=v\), and \(v\) receives \(e\) if \(r(e)=v\). An edge \(e\) is called a loop at \(v\) in case \(s(e) = v = r(e)\). A vertex is called a sink if it emits no edges. A vertex is called regular if it is not a sink and does not emit infinitely many edges. The subset of \(E^0\) consisting of all the regular vertices is denoted by \(E^0_{\operatorname{reg}}\). Similarly, the subset of \(E^0\) consisting of all the sinks is denoted by \(E^0_{\operatorname{sink}}\).

A graph is called row-finite if any vertex emits a finite number (possibly zero) of edges. The graph \(E\) is called finite if \(E^0\) and \(E^1\) are finite sets. In this paper we assume throughout that all graphs are row-finite, and thus \(E^0_{\operatorname{reg}}\) consists of all vertices which are not sinks.

A path \(p\) in \(E\) is a sequence \(p=e_{1}e_{2}\cdots e_{n}\) of edges in \(E\) such that \(r(e_{i})=s(e_{i+1})\) for \(1\leq i\leq n-1\). We define \(s(p) = s(e_{1})\), and \(r(p) =r(e_{n})\). By definition, the length \(|p|\) of \(p\) is \(n\). We assign the length zero to vertices. A closed path (based at \(v\)) is a path \(p\) such that \(s(p)=r(p)=v\). A cycle (based at \(v\)) is a closed path \(p=e_1 e_2 \cdots e_n\) based at \(v\) such that \(s(e_i)\neq s(e_j)\) for any \(i\neq j\). We say a vertex \(v\in E^0\) has no bifurcation if \(| s^{-1}(u)| \leq 1\) for all \(u\in T(v)\), where \(T(v)\) is the usual tree of the vertex \(v\).

A weighted graph is a pair \((E,w)\), where \(E\) is a graph and \(w:E^1\rightarrow \mathbb{N}^+\) is a map. If \(e\in E^1\), then \(w(e)\) is called the weight of \(e\). For each regular vertex \(v\) in a weighted graph \((E,w)\) we set \(w(v):=\max\{w(e)\mid e\in s^{-1}(v)\}\). This gives a map (called \(w\) again) \(w:E_{\operatorname{reg}}^0\rightarrow \mathbb{N}^+\). A weighted graph is called a vertex-weighted graph if, for each regular vertex \(v\), all edges emitted from \(v\) have the same weight, i.e., if \(w(e) = w(e')\) for all \(e,e' \in s^{-1}(v)\). In this case coincides with the weight of any edge emitting from \(v\). A weighted graph \((E, w)\) is called acyclic if the graph \(E\) has no cycles.

Let \((E,w)\) be a weighted graph. An edge \(e\in E^{1}\) is called unweighted if \(w(e)=1\) and weighted if \(w(e)>1\). The subset of \(E^1\) consisting of all unweighted edges is denoted by \(E_{uw}^1\) and the subset consisting of all weighted edges by \(E_{w}^1\).

The notion of the monoid associated to a weighted graph was introduced by the second author in [14] and later generalised to arbitrary weighted graphs by R. Preusser in [20].

We note that for an unweighted graph \(E\) (that is, when the weight function is constantly equal to \(1\)), the graph monoid (1 ) reduces to the commutative monoid \(M_E\) originally defined in [2]. Moreover, there exist weighted graph monoids that cannot be realised as monoids associated to unweighted graphs (see, for example, [16]).

Let \((E, \omega)\) be a vertex weighted graph. There is an explicit description of the congruence on the free commutative monoid \(\mathbb{F}_E\) given by the defining relations of \(M_{(E, \omega)}\) in Definition 2, as follows. For \(v\in E^0\), define the “\(\operatorname{{\boldsymbol{r}}}\)-transform on \(\mathbb{F}_E\)" by setting \[\label{shtrans} \operatorname{{\boldsymbol{r}}}(\omega(v) v) \;: = \;\sum_{e\in s^{-1}(v)}r(e) \text{ for al } v\in E^0_{\operatorname{reg}}.\tag{2}\] The nonzero elements of \(\mathbb{F}_E\) can be written uniquely up to permutation as \(\sum_{i=1}^{n}k_iv_{i}\), where \(v_{i}\) are distinct vertices and \(k_i\in \mathbb{N}^+\). Define a binary relation \(\rightarrow_{1}\) on \(\mathbb{F}_E\) by \[\label{hfgtrgt655} \sum_{i=1}^{n}k_iv_{i} \;\longrightarrow_{1} \; \Big( \sum_{i\neq j}k_iv_{i} \Big) +(k_j-w(v_j))v_j+ \operatorname{{\boldsymbol{r}}}(w(v_j)v_j),\tag{3}\] whenever \(j\in \{1, \cdots, n\}\) and \(k_j\geq w(v_{j})\). Let \(\rightarrow\) be the transitive and reflexive closure of \(\rightarrow_{1}\) on \(\mathbb{F}_E\). Namely \[\label{hfgtrgt6551} a\rightarrow b \;\; \;\text{ if } a=b, \;or \;a=a_0 \rightarrow_1 a_1 \rightarrow_1 \dots \rightarrow_1 a_k=b.\tag{4}\]

Finally, let \(\sim\) be the congruence on \(\mathbb{F}_E\) generated by the relation \(\rightarrow\). That is, \(a\sim b\) in case there is a string \(a=a_0, a_1,\dots, a_n=b\) in \(\mathbb{F}_E\) such that \(a_i\rightarrow_1 a_{i+1}\) or \(a_{i+1}\rightarrow_1 a_{i}\) for each \(0\leq i \leq n-1\). Then \[M_{(E,w)}=\mathbb{F}_E/\sim.\] To avoid cumbersome equivalence class notation, is standard (but not technically correct) to denote the elements of \(M_{(E,w)}\) and the elements of \(\mathbb{F}_E\) using the same symbols. For instance, we will sometimes write \(a=b\) in \(M_{(E,w)}\) for elements \(a,b \in \mathbb{F}_E\).

The relation 4 may be interpreted as follows. Let

If \(k_i\geq w(x_i)\), then we may “let the vertex \(x_i\) flow” by replacing the term \(w(x_i)x_i\) with the sum of the vertices adjacent to \(x_i\). More precisely, we obtain that

\[y_1=\left(\sum_{j\neq i}k_jx_j\right)+(k_i-w(x_i))x_i+\sum_{e\in s^{-1}(x_i)}r(e),\] and write \(x\to y_1\). Observe that this operation is permitted only when the summand \(k_i x_i\) appears in the expression for \(x\) and satisfies \(k_i\geq w(x_i)\).

Repeating this procedure and allowing a vertex of \(y_1\) to flow, we obtain another element \(y_2\in \mathbb{F}_E\) such that \(y_1\to y_2\). In general, the relation \(\to_1\) is applied successively to vertices appearing in the decomposition of elements of \(\mathbb{F}_E\). By definition of the relation \(\to\), every element \(y\in \mathbb{F}_E\) satisfying \(x\to y\) is obtained from \(x\) through a finite sequence of such vertex flows.

Next we recall the “confluence" property of the monoid \(M_{(E,w)}\), established by Abrams and the second author in [15]. Informally, this property asserts that whenever two elements \(a,b\in \mathbb{F}_E\) represent the same element in the quotient monoid \[M_{(E,w)}=\mathbb{F}_E/\sim,\] there exists a common element of \(\mathbb{F}_E\) to which both \(a\) and \(b\) flow.

Consequently, we obtain the following result, which follows immediately from [14], where Bergman algebras are used in the proof.

The next part of this section is to show that the weighted graph monoid associated to an acyclic vertex-weighted graph is cancellative. To do so, we first establish several useful results. Recall that for a graph \(E=(E^0,E^1,s,r)\), there exists a preorder “\(\geq\)” on \(E^0\) defined by \(u\ge v\) whenever there exists a path \(p\) in \(E\) such that \(s(p)=u\) and \(r(p)=v\). We write \(u > v\) if \(u\ge v\) and \(u\neq v\). We also note that if \(E\) is acyclic, then the preorder is exactly a partial order.

Let \((E,w)\) be a weighted graph. A weighted subgraph \(X\) of \((E, w_E)\) is the weighted graph \[X = (X^0, X^1, r_X, s_X, w_X)\] such that \(X^0\subseteq E^0\), \(X^1\subseteq E^1\), and \(r_X, s_X\) are respectively the restrictions of \(s_E, r_E\) on \(X^1\) and

A weighted subgraph \(X\) is called complete if \[s^{-1}_X(v) = s^{-1}_E(v)\] for all \(v\in X^0_{\operatorname{reg}}\). By [14], the weighted graph \(E\) can be expressed as the direct limit \[(E,w)=\varinjlim (E_i,w_i),\] where \(\{(E_i,w_i)\}\) is a directed system of finite complete weighted subgraphs of \((E,w)\).

We are now in a position to state the first main result of this section.

In the following examples, we present a vertex-weighted graph whose associated graph monoid is cancellative, and another vertex-weighted graph whose graph monoid is not cancellative.

We note that for every (unweighted) graph \(E\), the graph monoid \(M_E\) is cancellative if and only if no cycle in \(E\) has exits (see, e.g., [21]). However, Example 1(1) shows that this characterization does not, in general, extend to vertex-weighted graphs. In light of this observation, it is natural to ask the following question.

In the remainder of this section, we describe the structure of weighted graph monoids \(M_{(E, \omega)}\) in terms of saturated hereditary subsets of \(E\). To do so, we need to recall some useful notions.

We denote by \(\mathcal{H}_{(E,\omega)}\) the set of all saturated hereditary subsets of \(E^0\).

Let \((E, \omega)\) be a vertex-weighted graph and \(H\in \mathcal{H}_{(E,\omega)}\). We denote by \((E_H, \omega_r)\) the restriction weighted graph

and the source, range and weight function in \(E_H\) are exactly the source, range and weight function in \(E\) restricted to \(H\).

The following proposition provides that every nonempty saturated hereditary subset \(H\) of a vertex-weighted graph \(E\) induces a weighted graph submonoid \(M_{(E_H, \omega_r)}\) of \(M_{(E, \omega)}\).

Recall (see [4]) that an order-ideal of a commutative monoid \(M\) is a submonoid \(I\) of \(M\) such that, for any \(x, y\in M\), if \(x+y \in I\) then \(x, y\in I\). An order-ideal may also be described as a submonoid \(I\) of \(M\) which is hereditary with respect to the canonical preorder \(\le\) on \(M\): \(x\le y\) and \(y\in I\) imply \(x\in I\), where the preorder \(\le\) on \(M\) is defined by setting \(x\le y\) if \(y = x + m\) for some \(m\in M\). For each \(X\subseteq M\), the order-ideal of \(M\) generated by \(X\) is the set \[\langle X \rangle :=\big\{m\in M\mid m\le \sum^n_{i=1}x_i,\;n\in \mathbb{N}^+,\;x_i\in X\big\}.\] The set \(\mathcal{L}(M)\) of order-ideals of \(M\) forms a complete lattice such that the join of two elements \(I, J\) is exactly the order-ideal of \(M\) generated by \(I +J\).

We are now in a position to establish the second main result of this section describing order-ideals of the graph monoid \(M_{(E, \omega)}\) of a vertex-weighted graph \((E, \omega)\), which is an analogue of the corresponding result for sandpile monoids established in [17].

For clarification, we consider the following example.

Next we define the “talented” version of the weighted graph monoids (1 ).

For clarification, we illustrate Definition 9 by presenting the following examples. We first recall the notion of refinement for monoids. A commutative monoid \(M\) is called a refinement monoid if for any equality \[a_1 + \cdots + a_m \;=\; b_1 + \cdots + b_n,\] with \(a_i, b_j \in M\), there exist elements \(x_{ij} \in M\) such that \[a_i = \sum_{j=1}^n x_{ij} \quad \text{for all } i=1,\dots,m,\] and \[b_j = \sum_{i=1}^m x_{ij} \quad \text{for all } j=1,\dots,n.\]

We shall show that only the talented monoids of unweighted Leavitt path algebras are refinement (Proposition 16).

In the remainder of this section, we heavily use the concept of the covering graph of a weighted graph.

For clarification, we illustrate Definition 10 by presenting the following examples.

Let \(M\) be a \(\Gamma\)-monoid. A \(\Gamma\)-ordered ideal of \(M\) is a submonoid \(I\) of \(M\) which is closed under the action of \(\Gamma\) and and it is hereditary in the sense that \(x \le y\) and \(y \in I\) imply \(x\in I\).

We end this section by giving connections between the graph monoid \(M_{(E,w)}\) of a vertex-weighted graph \((E, w)\) and the talented monoid \(T_{(E,w)}\).

3 Weighted Leavitt path algebras and the graded Grothendieck group \(K_0^{\mathrm{gr}}\)↩︎

The aim of this section is to compute the graded Grothendieck group of the Leavitt path algebra of a vertex-weighted graph in terms of its talented monoid (Corollary 14), and show that the graded Grothendieck group \(K^{\operatorname{gr}}_0\) distinguishes the class of vertex-weighted Leavitt path algebras from the unweighted class (Theorem 17), as well as prove that every order preserving \(\mathbb{Z}\)-module isomorphism between \(K_0^{\operatorname{gr}}(L_{\mathsf{k}}(E, w))\) and \(K_0^{\operatorname{gr}}(L_{\mathsf{k}}(F, w))\) provides an isomorphism between the semilattices of all vertex-generated ideals of \(L_{\mathsf{k}}(E, w)\) and \(L_{\mathsf{k}}(F, w)\) (Theorem 18).

We begin this section by briefly recalling the notion of weighted Leavitt path algebras, which are algebras associated with weighted graphs. We refer the reader to [14], [23] for a detailed analysis of these algebras.

Note that if the weight of each edge is \(1\), then \(L_\mathsf{k}(E,w)\) reduces to the usual Leavitt path algebra \(L_\mathsf{k}(E)\) of the graph \(E\). Also, if \(E\) is the empty graph then \(L_\mathsf{k}(E,w)\) is defined to be the zero ring.

The motivation for defining weighted Leavitt path algebras comes from the fact that ordinary Leavitt path algebras do not encompass Leavitt algebras of the form \(L_\mathsf{k}(n,n+k)\). However, starting from the weighted graphs in (14 ), the associated weighted Leavitt path algebras are precisely \(L_\mathsf{k}(n,n+k)\).

For a vertex-weighted graph \((E,w)\), one can consider a \(\mathbb{Z}\)-grading on \(L_\mathsf{k}(E,w)\) by assigning \[\deg(v)=0,\qquad \deg(e_i)=1,\qquad \deg(e_i^*)=-1,\] for \(v\in E^0\), \(e\in E^1\), and \(1\le i\le w(e)\). If all edges have weight \(1\), then this grading coincides with the standard grading on an unweighted Leavitt path algebra. For an approach to endowing weighted Leavitt path algebras with a grading by an arbitrary group, we refer the reader to [23].

Let \(R\) be a not necessarily unital ring. Recall that a left \(R\)-module \(M\) is called unital if \(RM = M\). We denote by \(R\)-\(\mathrm{Mod}\) the category of unital left \(R\)-modules. Furthermore, we denote by \(R\)-\(\mathrm{Mod}_{\mathrm{proj}}\) the full subcategory of \(R\)-\(\mathrm{Mod}\) whose objects are the projective objects of \(R\)-\(\mathrm{Mod}\) that are finitely generated as a left \(R\)-module. If \(R\) has local units, we define \[\mathcal{V}(R) = \{ [P] ~| ~ P \in R\text{-}\mathrm{Mod}_{\mathrm{proj}} \}\] where \([P]\) denotes the isomorphism class of \(P\) as a left \(R\)-module. \(\mathcal{V}(R)\) becomes an abelian monoid by defining \([P]+[Q] = [P \oplus Q]\). It is well-known that \(\mathcal{V}\) is a functor that commutes with direct limits. There is a different, but equivalent definition of \(\mathcal{V}\) using idempotent matrices over \(R\), cf. [24].

The following result was introduced by the second author in [14] and later generalised to arbitrary weighted graphs by R. Preusser in [20].

Let \(R\) now be a \(\Gamma\)-graded ring. Recall that a \(R\)-module \(M\) is called \(\Gamma\)-graded if there is a decomposition \(M = \oplus_{\gamma \in \Gamma} M_{\gamma}\) such that \(R_{\alpha} M_{\gamma} \subseteq M_{\alpha \gamma}\) for any \(\alpha \gamma \in \Gamma\). We denote by \(R\text{-} \mathrm{Gr}\) the category of \(\Gamma\)-graded unital left \(R\)-modules with morphisms the \(R\)-module homomorphisms that preserve grading. Moreover, we denote by \(R\text{-}\mathrm{Gr}_{\mathrm{proj}}\) the full subcategory of \(R\)-\(\mathrm{Gr}\) whose objects are the projective objects of \(R\text{-}\mathrm{Gr}\) that are finitely generated as a left \(R\)-module. If \(R\) has graded local units, we define \[\mathcal{V}^{\mathrm{gr}} (R) = \{ [P] ~|~ P \in R\text{-}\mathrm{Gr}_{\mathrm{proj}} \}\] where \([P]\) denotes the isomorphism class of \(P\) as a graded left \(R\)-module. \(\mathcal{V}^{\mathrm{gr}} (R)\) becomes a \(\Gamma\)-monoid by defining \([P]+[Q] = [P \oplus Q]\) and \({^{\gamma}}{[P]} = [P(\gamma)]\). The graded Grothendieck group \(K_0^{\mathrm{gr}} (R)\) is the group completion of \(\mathcal{V}^{\mathrm{gr}} (R)\) [24].

We next discuss the smash product of weighted Leavitt path algebras. For a \(\Gamma\)-graded ring \(A\) (possibly without unit), the smash product ring \(A\#\Gamma\) is defined as the set of all formal sums \(\sum_{\gamma \in \Gamma} r^{(\gamma)} P_\gamma\), where \(r^{(\gamma)}\in A\) and \(P_\gamma\) are symbols. Addition is defined component-wise and multiplication is defined by linear extension of the rule \[(rP_{\alpha})(sP_{\beta})=rs_{\alpha\beta^{-1}}P_{\beta},\] where \(r,s\in A\) and \(\alpha,\beta\in\Gamma\). It is routine to check that \(A\#\Gamma\) is a ring. We emphasise that the symbols \(P_{\gamma}\) do not belong to \(A\#\Gamma\); however if the ring \(A\) has unit, then we regard the \(P_\gamma\) as elements of \(A\#\Gamma\) by identifying \(1_A P_\gamma\) with \(P_\gamma\). Each \(P_{\gamma}\) is then an idempotent element of \(A\#\Gamma\).

It is known that (see [24] or [25]) there is an isomorphism of categories of modules \[\label{vast5} \psi: A\text{-}\mathrm{Gr}\longrightarrow A\#\Gamma\text{-}\mathrm{Mod}.\tag{17}\]

For a \(\Gamma\)-graded ample groupoid \(\mathcal{G}\), in [24] it was shown that there is an isomorphism between the Steinberg algebras \(A_\mathsf{k}(\mathcal{G} \times \Gamma) \cong A_\mathsf{k}(\mathcal{G})\# \Gamma\), where \(\mathcal{G} \times \Gamma\) is the crossed-product groupoid. Specialising to the case of graph groupoids \(\mathcal{G}_E\), since \(A_\mathsf{k}(\mathcal{G}_E)\cong L_\mathsf{k}(E)\), we obtain that \[L_\mathsf{k}(\overline{E}) \cong A_\mathsf{k}(\mathcal{G}_{\overline{E}}) \cong A_\mathsf{k}(\mathcal{G}_E \times \mathbb{Z}) \cong A_\mathsf{k}(\mathcal{G}_E)\# \mathbb{Z}\cong L_\mathsf{k}(E) \# \mathbb{Z}.\] Combined with the category isomorphism of 17 , this explains why graded \(K\)-theory is both tractable and powerful in the context of Leavitt path algebras. Although the computations become considerably simpler, the theory still captures substantial information about the algebra itself, as in effect, one is working with the \(K\)-theory of algebras associated to acyclic graphs, namely the covering graphs.

There is no groupoid model for weighted Leavitt path algebras. However, the following lemma once again highlights the effectiveness of graded \(K\)-theory in the weighted graph setting. Compare this with [24] for the case of Leavitt path algebras and [25] for Leavitt path algebras of hyper graphs.

Consequently, we obtain the following useful fact.

We recall Preusser’s conditions on a weighted Leavitt path algebras to be isomorphic to an unweighted one [23], [27].

In [27] it was proved that a weighted Leavitt path algebra \(L_\mathsf{k}(E,w)\) is isomorphic to an unweighted Leavitt path algebras if and only if Preusser’s conditions hold.

We illustrate, by means of an example, how Preusser’s conditions allow one to construct an unweighted graph whose associated Leavitt path algebra is (graded) isomorphic to that of a weighted graph. We emphasis that to preserve the grading, however, the canonical grading on edges must be modified, as the example below demonstrates. Consider the weighted graph \((E,w)\): \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/fgorlpzx.png}\label{zjikrncs}\end{figure}\tag{20}\]

Then following [27], we first obtain the intermediate weighted graph \((\widetilde{E},w)\) \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/gsmvtloy.png}\label{mlhtbfau}\end{figure}\tag{21}\]

The assignment that sends vertices of \(E\) to the corresponding ones in \(\widetilde{E}\), \(e_i\mapsto e_i\), \(f_i\mapsto f_i^*\), and ghost edges accordingly, induces an isomorphism \(\phi: L_\mathsf{k}(E,w)\cong L_\mathsf{k}(\widetilde{E},w)\). Thus if we keep the canonical grading for \(E\), and in \(\widetilde{E}\) we assign \(\deg(e_i)=1\) and \(\deg(f_i)=-1\), then the map \(\phi\) is indeed a graded isomorphism. Next we derive a unweighted graph \(F\) from \(\widetilde{E}\) as follows: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/gcvhfkti.png}\label{ekhgjsco}\end{figure}\tag{22}\] The assignment \(u\mapsto u\), \(w\mapsto w\) and \(v\mapsto v_1+v_2\) on vertices and \(e_1\mapsto e_1\), \(e_2\mapsto e_2^*\), \(f_i\mapsto f_i^1+f_i^2\) on edges induce an isomorphism \(L_\mathsf{k}(\widetilde{E},w)\cong L_\mathsf{k}(F)\). Thus if in the graph \(F\) we assign \(\deg(e_1)=1\), \(\deg(e_2)=-1\) and \(\deg(f^1_i)=\deg(f^2_i)=-1\), then above map is indeed a graded isomorphism. Putting these together we have \[L_\mathsf{k}(E,w)\cong_{\operatorname{gr}} L_\mathsf{k}(\widetilde{E},w) \cong_{\operatorname{gr}} L_\mathsf{k}(F).\]

We are in a position to prove one of the main results of the note.

As a corollary of Proposition 16 and [23]), we obtain the following interesting result.

We end this section by the following interesting result.

4 \(K_0^{\mathrm{gr}}\) is a full functor↩︎

The main aim of this section is to prove the following lifting theorem, which gives that Conjecture 1(1) holds for vertex-weighted graphs (Theorem 19), and show that the graded Grothendieck group \(K_0^{\operatorname{gr}}\) classifies the Leavitt algebras \(L_\mathsf{k}(n,n+k)\) (Theorem 24).

In order to prove this theorem, we need to recall the graded version of Bergman machinery developed in [19]. In that work, Bergman’s universal constructions were extended to the setting of graded rings. It was shown that any conical \(\Gamma\)-monoid can be realised as a graded non-stable \(K\)-theory of a \(\Gamma\)-graded ring, which satisfies a “weak" universal property. We collect here the results from [19] that will be needed in the subsequent sections.

For graded finitely generated projective \(R\)-modules \(P\) and \(Q\), we can adjoin a universal graded isomorphism between \(\overline{P}\) and \(\overline{Q}\) by first freely adjoining a graded map \(h: \overline{P}\xrightarrow[]{}\overline{Q}\), and \(\overline{h}: \overline{Q}\xrightarrow[]{}\overline{P}\) using Theorem 20 and then using Theorem 21 to force that \(1-h\overline{h}=0\) and \(1-\overline{h}h=0\). We denote the resulting \(\Gamma\)-graded \(R\text{-ring}_{\mathsf{k}}\) by \[S:= R\langle h, h^{-1} \mid \overline{P}\cong_{\operatorname{gr}} \overline{Q}\rangle.\]

When the field \(\mathsf{k}\) is concentrated in degree zero, we can compute the non-stable graded \(K\)-theory of the graded ring \(S\) ([19]).

We are now in a position to prove Theorem 19. In fact, by using the graded Bergman construction, we obtain a more general result.

We are now in a position to establish the fullness of the graded Grothendieck group functor on the category of all weighted Leavitt path algebras.

We finish the paper by showing that the \(K_0^{\operatorname{gr}}\)-group is a complete invariant for Leavitt algebras. Note that in comparison, there is an order isomorphism \(K_0(L_\mathsf{k}(n,n+k))\cong K_0(L_\mathsf{k}(n+l,n+k+l))\), for \(n,k,l\in \mathbb{N}\), although \(L_\mathsf{k}(n,n+k)\not \cong L_\mathsf{k}(n+l,n+k+l)\).

Roozbeh Hazrat acknowledges Australian Research Council Discovery Project DP230103184. He would like to thank the Vietnam Institute for Advanced Study in Mathematics (VIASM) and Professor T.G. Nam for hosting him in Hanoi. R.J. Damalerio acknowledges the support provided by the DOST–SEI Accelerated Science and Technology Human Resource Development Program (ASTHRDP) and the Centre International de Mathématiques Pures et Appliquées (CIMPA), whose grant supported his research visit to Hanoi. T. G. Nam was partially supported by the Vietnam Academy of Science and Technology under grant CBCLCA.01/26-28.

References↩︎