A graded pullback structure of Leavitt path algebras of trimmable graphs
Piotr M. Hajac Atabey Kaygun Mariusz Tobolski

15pt
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  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. 
  
Theory of combustion in disordered media
    Boris Z. Spivak
    December 30, 2023
========================================






§ INTRODUCTION


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 <cit.> and Leavitt path
  algebras <cit.>.  Then we define a trimmable
  graph, and prove our main result: There is a ℤ-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 <cit.>, where it was used to analyze the
generators of K-groups of quantum complex projective spaces.





§ LEAVITT PATH ALGEBRAS



  A graph Q is a quadruple (Q_0,Q_1,s,t) consisting of the
  set of vertices Q_0, the set of edges Q_1, and the source and
  target maps s,t Q_1→ Q_0 assigning to each edge its
  source and target vertex respectively.

We say that a graph Q'=(Q'_0,Q'_1,s',t') is a sub-graph of a
graph Q=(Q_0,Q_1,s,t) iff Q'_0⊆ Q_0,
Q'_1⊆ Q_1, and the source and target maps s' and t'
are respective restrictions-corestrictions of the source and target
maps s and t. 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 k be a field and Q a graph. The path algebra kQ is
  the k-algebra whose underlying vector space has as its basis the
  set of all finite-length paths Path(Q). 
  
  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 kQ is unital if and only if the
set of vertices Q_0 is finite. Then the unit is the sum of all
vertices.  It is also straightforward to verify that kQ is
ℕ-graded by the path length.

To define a Leavitt path algebra, we need ghost edges.  For any
graph Q=(Q_0,Q_1,s,t), we create a new set
Q_1^*:={x^* | x∈ Q_1} and call its elements  ghost
  edges. Now, the source and the target maps for the extended graph
Q:=(Q_0,Q_1∐ Q_1^*, ŝ,t̂ )
are defined as follows:

    ŝ(x):=s(x),  ŝ(x^*):=t(x),  t̂(x):=t(x),  t̂(x^*):=s(x).


Let k be a field and Q a graph. The Leavitt path algebra L_k(Q) of a graph Q is the path algebra of the extended graph
Q divided by the ideal generated by the relations:




  * For all edges x_i,x_j∈ Q_1, we have x_i^*x_j=δ_ijt(x_i).

  * For every vertex v∈ Q_0 whose preimage  s^-1(v)  is not empty and finite, we have

    ∑_x∈ s^-1(v)xx^*=v.



In other words, the Leavitt path algebra L_k(Q) of a graph Q is the universal k-algebra generated by
the elements v∈ Q_0,  x∈ Q_1, x^*∈ Q_1^*, subject to relations:




  (L1) v_iv_j=δ_ijv_i for all v_i,v_j∈ Q_0,

  (L2) s(x)x=xt(x)=x for all x∈ Q_1,

  (L3) t(x)x^*=x^*s(x)=x^* for all x^*∈ Q_1^*,

  (L4) x_i^*x_j=δ_ijt(x_i) for all x_i,x_j∈ Q_1, and 

  (L5) ∑_x∈ s^-1(v)xx^*=v  for all v∈ Q_0 such that  s^-1(v)  is finite and nonempty.


Furthermore, note that the ℕ-grading of the path algebra kQ
induces a ℤ-grading of the Leavitt path algebra L_k(Q) by
counting the length of any ghost edge as -1 (see <cit.>).
Let us recall now the Graded Uniqueness Theorem <cit.> that shows the importance of
this grading.  We will need it in the next section.

  Let Q be an arbitrary graph and k be any field. If A is a
  ℤ-graded ring, and f:L_k(E)→ A is a graded ring
  homomorphism with f(v)≠ 0 for every vertex v∈ Q_0, then f
  is injective.
 

Recall that a vertex v∈ Q_0 is called a sink if s^-1(v)=∅. Next, let x=x_1x_2… x_n
be a path in Q. If the length of x is at least 1, and if s(x)=v=t(x)∈ Q_0,
we say that x is a closed path based at v.
If in addition s(x_i)≠ s(x_j) for every i≠ j, then x is called a cycle based at v.

Next, if v∈ Q_0 is either a sink or a base of a cycle of length 1 (a loop), then 
a singleton set {v}⊆ Q_0   is a basic example of a hereditary
  subset of Q_0 <cit.>, and
it follows from Corollary 2.4.13 (i) in <cit.> (cf. <cit.>) that

    L_k(Q) I(v)≅ L_k(Q∖{v}).

 Here the graph Q∖{v} is obtained by removing from Q the
  vertex v and every edge that ends in v.  In other  words, 

    (Q∖{v})_0=Q_0∖{v}   and   
        (Q∖{v})_1={x∈ Q_1:t(x)≠ v}.

By Lemma 2.4.1 in <cit.> (cf. <cit.>), we also know that

    I(v)={∑_i=1^nk_ix_iy^*_i  |   n≥ 1,  x_i ,y_i∈ Path(Q), t̂(x_i)=ŝ(y^*_i)=v}.





§ 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 Q be a finite graph consisting of a sub-graph Q' emitting at least one
  edge to an external vertex v_0 whose only outgoing edge x_0
  is a loop.  We call such a graph (Q',v_0)-trimmable iff 
all  edges from Q' to v_0 begin in a vertex emitting an edge that ends
  inside  Q'. 

In symbols, a trimmable graph is described as follows:

    Q_0=Q'_0∪{v_0},   v_0∉Q'_0 ,   Q_1=Q'_1∪ t^-1(v_0),
    
    
    s^-1(v_0)={x_0},   t(x_0)=v_0 ,    t^-1(v_0)∖{x_0}≠∅,
    
    ∀  v∈ s(t^-1(v_0)∖{x_0})    s^-1(v)∖ t^-1(v_0)≠∅.


The condition for a trimmable graph guarantees the fact
that when we remove the distinguised vertex v_0, the resulting
graph does not have new sinks.  One can imagine a
(Q',v_0)-trimmable graph like this:



    < g r a p h i c s >
.



  The following graph is a simple example of a trimmable graph:
  
    v_2
    
          v_1 [u] [r]     v_0 @(u,r)
        _   .

 Note that  the direction of the edge joining vertices v_1 and v_2 is important as the following graph is no longer trimmable:
  
    v_2 [d] 
    
          v_1 [r]     v_0 @(u,r)
       _   .


We need the trimmability condition to guarantee the existence of  maps given in the  lemma  below. This lemma is an algebraic incarnation
of a graph-C*-algebraic lemma proved in <cit.>. 
The only difference between their proofs  is that instead of
using the Gauge Uniqueness Theorem <cit.> we use 
the Graded Uniqueness Theorem <cit.> (Theorem <ref>).

Let Q be a (Q'_0,v_0)-trimmable graph. Denote by Q” the sub-graph of Q obtained  by removing the edge x_0.
The following formulas define homomorphisms of algebras:





  * π_1 L_k(Q)→ L_k(Q'),

    π_1(α) = 
       α    if α∈ Q'_0 ∪ Q'_1∪ Q'^*_1,
    
         0     otherwise.


  * π_2 L_k(Q”)→ L_k(Q'),

    π_2(α) = 
       α    if α∈ Q'_0 ∪ Q'_1∪ Q'^*_1,
    
         0     otherwise.


  * f L_k(Q)→ L_k(Q”)⊗ k[u,u^-1],

    f(α) = 
    α⊗ 1     if α∈ Q_0,
    
      v_0⊗ u     if α = x_0,
    
      v_0⊗ u^-1    if α = x_0^*,
    α⊗ u     if α∈ Q_1∖{x_0},
    α⊗ u^-1    if α∈ Q_1^*∖{x_0^*}.


  * δ L_k(Q') → L_k(Q')⊗ k[u,u^-1],

    δ(α) = 
    α⊗ 1     if α∈ Q_0',
    α⊗ u     if α∈ Q_1',
    α⊗ u^-1    if α∈ (Q_1')^*.


These morphisms are ℤ-graded for the standard grading on
L_k(Q), L_k(Q'), L_k(Q”), and the gradings on
L_k(Q')⊗ k[u,u^-1] and L_k(Q”)⊗ k[u,u^-1] 
given by the standard grading of the rightmost tensorand. Furthermore, π_1
and π_2 are surjective, and f and δ are injective.





§ A GRADED PULLBACK STRUCTURE


To prove the theorem of the paper, we need a general lemma along the lines of <cit.>.
We omit  its routine proof.

Let A_1, A_2, B and P be abelian groups. A commutative diagram of group homomorphisms

    P[ld]_p_1[rd]^p_2   
    
    A_1[dr]_q_1       A_2
    [dl]^q_2
       B

is a pullback diagram if and only if the following conditions hold:

    (p_1)∩(p_2)={0},
    
    q_1^-1(q_2(A_2))=p_1(P),
    
    p_2((p_1))=(q_2).



Recall that to prove that an algebra P is a pullback algebra, one can proceed as follows. The first step is to establish
the existence of a commutative diagram of algebra homomorphisms as above. 
This implies that p_1 and p_2 define an algebra homomorphism p into the pullback algebra of A_1 and A_2 over B. 
Then one only needs to prove that the three conditions of Lemma <ref> are satisfied to conclude that p is an
isomorphism. Note that (<ref>) is equivalent to the injectivity of p, whereas the conjunction of (<ref>) and (<ref>)
is equivalent to the surjectivity of p.

Much as Lemma <ref>, the theorem  of the paper is an algebraic incarnation
of a graph-C*-algebraic theorem proved in <cit.>. 
This time, the only difference between their proofs  is that instead of
using  <cit.> we use <cit.> (see (<ref>), cf. <cit.>).

Let π_1, π_2, f and δ be as in Lemma <ref>.
Then the commutative diagram

    L_k(Q)[ld]_π_1[rd]^f   
    
    L_k(Q')[dr]_δ         
    L_k(Q”)⊗ k[u,u^-1]
    [dl]^  π_2⊗𝕀
       L_k(Q')⊗ k[u,u^-1]

of graded algebra homomorphisms is a pullback diagram.

Representing pictorially the Leavitt path algebras by their respective graphs, the above diagram becomes:





    < g r a p h i c s >
    [ld]_π_1  [rd]^  f   
    
        < g r a p h i c s >
    [dr]_δ        
        < g r a p h i c s >
    ⊗ X
        < g r a p h i c s >
    [dl]^  π_2⊗𝕀
       
        < g r a p h i c s >
    ⊗ X
        < g r a p h i c s >
    .



Note that the only non-standard map in this diagram is f. It can be described verbally by  the assignment

    vertex   ⟼vertex⊗ 1,
    v_0-emitted edge   ⟼ v_0⊗ u,
    other edge   ⟼other edge⊗ u.




§ ACKNOWLEDGEMENT
The authors are very grateful to Francesco D'Andrea for sharing his mathematical insight with us, and  for designing the pictures of graphs.
 It is also a pleasure to thank Søren Eilers, Adam Skalski and Wojciech Szymański for discussions. This
work was partially supported by NCN grant UMO-2015/19/B/ST1/03098.

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