A graded pullback structure of
Leavitt path algebras of trimmable graphs
Motivated by recent results in graph C*-algebras concerning an equivariant pushout structure of the Vaksman-Soibelman quantum odd spheres, we introduce a class of graphs called trimmable. Then we show that the Leavitt path algebra of a trimmable graph is graded-isomorphic to a pullback algebra of simpler Leavitt path algebras and their tensor products.
Key words and phrases:
The goal of this paper is to introduce and apply the concept of a trimmable graph. We begin by recalling the fundamental concepts of path algebras [ass06] and Leavitt path algebras [L62, aa05, A15, aasm17]. Then we define a trimmable graph, and prove our main result: There is a -graded algebra isomorphism from the Leavitt path algebra of a trimmable graph to an appropriate pullback algebra. The graph C*-algebraic version of this result is proven in [dht], where it was used to analyze the generators of K-groups of quantum complex projective spaces.
1. Leavitt path algebras
A graph is a quadruple consisting of the set of vertices , the set of edges , and the source and target maps assigning to each edge its source and target vertex respectively.
We say that a graph is a sub-graph of a graph iff , , and the source and target maps and are respective restrictions-corestrictions of the source and target maps and . Furthermore, we say that two edges are composable if the end of one of them is the beginning of the other. Now we can define a path in a graph as a sequence of composable edges. The length of a path is the number of edges it consists of, infinity included. We treat vertices as zero-length paths that begin and end in themselves.
Let be a field and a graph. The path algebra is the -algebra whose underlying vector space has as its basis the set of all finite-length paths . The product is given by the composition of paths when the end of one path matches the beginning of the other path. The product is defined to be zero otherwise.
One can check that the path algebra is unital if and only if the set of vertices is finite. Then the unit is the sum of all vertices. It is also straightforward to verify that is -graded by the path length.
To define a Leavitt path algebra, we need ghost edges. For any graph , we create a new set and call its elements ghost edges. Now, the source and the target maps for the extended graph are defined as follows:
Let be a field and a graph. The Leavitt path algebra of a graph is the path algebra of the extended graph divided by the ideal generated by the relations:
For all edges , we have .
For every vertex whose preimage is not empty and finite, we have
In other words, the Leavitt path algebra of a graph is the universal -algebra generated by the elements , , , subject to relations:
for all ,
for all ,
for all ,
for all , and
for all such that is finite and nonempty.
Furthermore, note that the -grading of the path algebra induces a -grading of the Leavitt path algebra by counting the length of any ghost edge as (see [aa05, Lemma 1.7]). Let us recall now the Graded Uniqueness Theorem [t-m07, Theorem 4.8] that shows the importance of this grading. We will need it in the next section.
Theorem 1.4 ([t-m07]).
Let be an arbitrary graph and be any field. If is a -graded ring, and is a graded ring homomorphism with for every vertex , then is injective.
Recall that a vertex is called a sink if . Next, let be a path in . If the length of is at least , and if , we say that is a closed path based at . If in addition for every , then is called a cycle based at .
Next, if is either a sink or a base of a cycle of length 1 (a loop), then a singleton set is a basic example of a hereditary subset of [aasm17, Definition 2.0.5 (i)], and it follows from Corollary 2.4.13 (i) in [aasm17] (cf. [t-m07, Theorem 5.7 (2)]) that
Here the graph is obtained by removing from the vertex and every edge that ends in . In other words,
By Lemma 2.4.1 in [aasm17] (cf. [t-m07, Lemma 5.6]), we also know that
2. Trimmable graphs
We are now ready for the main definition of the paper. Merely to focus attention, we assume henceforth that graphs are finite, i.e. that the set of vertices and the set of edges are both finite.
Let be a finite graph consisting of a sub-graph emitting at least one edge to an external vertex whose only outgoing edge is a loop. We call such a graph -trimmable iff all edges from to begin in a vertex emitting an edge that ends inside .
In symbols, a trimmable graph is described as follows:
The condition for a trimmable graph guarantees the fact that when we remove the distinguised vertex , the resulting graph does not have new sinks. One can imagine a -trimmable graph like this:
The following graph is a simple example of a trimmable graph: