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arXiv:1108.4994v1 [math.RA] 25 Aug 2011
SHIFT EQUIVALENCE AND A CATEGORY
EQUIVALENCE INVOLVING GRADED MODULES OVER
PATH ALGEBRAS OF QUIVERS
S. PAUL SMITH
Abstract. In this paper we associate an abelian category to a finite
directed graph and prove the categories arising from two graphs are
equivalent if the incidence matrices of the graphs are shift equivalent.
The abelian category is the quotient of the category of graded vector
space representations of the quiver obtained by making the graded rep-
resentations that are the sum of their finite dimensional submodules
isomorphic to zero.
Actually, the main result in this paper is that the abelian categories
are equivalent if t he inciden ce matrices are strong shift eq uivalent. That
result is combined with an earlier result of the author to prove that if
the incidence matrices are shift equivalent, then the associated abelian
categories are equivalent.
Given William’s Theorem that subshifts of finite type associated to
two directed graphs are conjugate if and only if the graphs are strong
shift equivalent, our main result can be reformulated as follows: if the
subshifts associated to two directed graphs are conjugate, then the cat-
egories associated to those graphs are equivalent.
1. Introduction
1.1. This paper proves a result of the following general type: if two graphs
are equivalent in an appropriate sense, then certain algebraic objects asso-
ciated to them are equivalent in a corresponding s ense. We associate an
abelian category to a directed graph and prove the categories arising f rom
two graphs are equivalent if the graphs are equivalent in an appropriate
sense.
A nice paper by Bates and Pask [3] contains r esults of this general type.
They prove various isomorphisms and Morita equivalences between graph
C
∗
-algebras that unify earlier results for graph C
∗
-algebras and Cuntz-
Kreiger algebras (r eferences for some of those earlier results can be found in
the opening paragraph of [3]). Raeburn’s monograph [7] is a comprehensive
treatment of graph C
∗
-algebras. Analogous, but algebraic as opposed to
1991 Mathematics Subject Classification. 05C20, 16B50, 16G20, 16W50, 37B10.
Key words and phrases. graded modules; directed graphs; representations of quivers;
quotient category; strong shift equivalence; shift equivalence.
S. P. Smith was partially su pported by NSF grant DMS 0602347.
1
2 S. PAUL SMITH
C
∗
-algebraic, results for Leavitt path algebras have been proved by Abrams
et al. See, for example, [1] and [2] and the references therein.
1.2. The graph equivalences alluded to in the op ening paragraph of this
introduction arise in th e theory of sub shifts of finite type.
A shift space (X, σ) over a finite alphabet A is a compact subset X of A
Z
that is stable under the shift map σ defined by σ(f )(n) = f(n + 1).
To a directed graph Q with finitely many ed ges and no sources or sinks one
may associate a shift sp ace X
Q
, called the edge shift of Q , whose alphabet
is the set of arr ows in Q (see [6, Defn. 2.2.5]). Ed ge shifts are subshifts of
finite type.
One may also associate a subshift of finite type X
A
to every square 0-1
matrix A having no zero row s or columns (see [6, Defn. 2.3.7]). The alphabet
is now the set of vertices for the directed graph whose incidence matrix is
A.
The equivalence between shift spaces that concerns us is conjugacy. Shift
spaces (X, σ
X
) and (Y, σ
Y
) are conjugate, or topologically conjugate, denoted
X
∼
=
Y , if there is a homeomorphism φ : X → Y such that σ
Y
◦ φ = φ ◦ σ
X
.
By [6, Prop. 2.3.9], every sub shift of fin ite ty pe is conjugate to an edge
shift X
Q
for some directed graph Q.
One may associate to Q a new graph Q
[2]
(see [6, Defn. 2.3.4]) whose
vertices are the edges of Q and Q
[2]
has an arrow from e to e
′
if e terminates
where e
′
begins. The incidence matrix for Q
[2]
, which we denote by B
Q
,
is a 0-1 matrix such that X
Q
∼
=
X
B
Q
. Thus every subshift of finite type
is conjugate to a shift described by a 0-1 m atrix. Conversely, every shift
described by a 0-1 matrix A is conjugate to an edge shift: if Q
A
is the
directed graph with incidence matrix A, then X
Q
A
is conjugate to X
Q
[6,
Exer. 1.5.6 an d Prop. 2.3.9]. Hence su bshifts of finite type are edge shifts
or shifts associated to 0-1 matrices.
1.3. The results. Throughout k is a field and Q a directed graph, or quiver,
with a finite number of vertices and arrows—loops and multiple arrows
between vertices are allowed.
We write kQ for the path algebra of Q. The finite paths, including the
trivial path at each ver tex, in Q form a basis for kQ and multiplication is
given by concatenation of paths.
We adop t the convention that the incidence matrix of Q is C = (c
ij
)
where
c
ij
= the number of arrows from j to i.
Given a square N-valued matrix we write Q
C
for the directed graph with
incidence matrix C.
We make kQ an N-graded algebra by declaring that a path is homogeneous
of degree equal to its length. Th e category of Z-graded left kQ-modules
with degree-preserving homomorphisms is denoted by GrkQ and we write
FdimkQ for its full subcategory of consisting of modules that are the sum of
SHIFT EQUIVALENCE AND MODULES OVER PATH ALGEBRAS 3
their finite-dimensional submod ules. Since FdimkQ is a Serr e subcategory
of GrkQ (it is, in fact, a localizing subcategory) we may form the quotient
category
QGr kQ :=
GrkQ
FdimkQ
.
The main results in this paper is the following theorem an d its conse-
quences.
Theorem 1.1. Let L and R be N-valued matrices such that LR and RL
make sense. Let Q
LR
be the quiv er with incidence matrix LR and Q
RL
the
quiver with incidence matrix RL. There is an equivalence of categories
QGr kQ
LR
≡ QGr kQ
RL
.
Strong shift equivalence (see section 2.1 for its d efinition) is an equiv-
alence relation on square matrices with entries in N th at is important in
symbolic dynamics (see section 2.1 below). By interpreting a square matrix
with entries in N as an incidence matrix, an equivalence relation on square
matrices with entries in N is the same thing as an equivalence relation on
finite directed graphs.
Theorem 1.2. If the incidence matrices for Q and Q
′
are strong shift equiv-
alent, then QGr(kQ) ≡ QGr(kQ
′
).
Given William’s Theorem (see Theorem 2.1 below), T heorem 1.2 can be
restated as follows: Let (X, σ) and (X, σ
′
) be subshifts of finite typ e, and Q
and Q
′
directed graphs such that (X, σ) = X
Q
and (X, σ
′
) = X
Q
′
. If (X, σ)
and (X, σ
′
) are conjugate, then the categories QGr(kQ) and QGr(kQ
′
) are
equivalent.
It is difficult to decide if two given matrices are strong shift equivalent.
It is not know n whether the strong s hift equivalence problem is decidable.
However, there is a weaker notion, shift equivalence (see section 2.4 for its
definition), and Kim and Rous h [4] have shown that the shift equivalence
problem is decidable. Strong shift equivalence implies shift equivalence but
the question of whether the two notions were the same was open for over
twenty years before K im and Roush [5 ] gave an example in 1999 showing
shift equivalence does not imply strong shif t equivalence.
If two incidence matrices A and B are shift equivalent, then A
ℓ
is strong
shift equivalent to B
ℓ
for some integer ℓ. Theorem 1.8 in [9] says that if Q
(ℓ)
is the directed graph whose incidence matrix is the ℓ
th
power of the incidence
matrix for Q, then QGr kQ is equivalent to QGr(kQ
(ℓ)
).
1
Combining [9, Thm.
1.8] with Theorem 1.2 gives th e following.
Corollary 1.3. If the incidence matrices for Q and Q
′
are shift equivalent,
then QGr(kQ) ≡ QGr(kQ
′
).
1
In symbolic dynamics Q
(ℓ)
is called the ℓ
th
higher power graph of Q [6, Defn. 2.3.10].
4 S. PAUL SMITH
Given Q, define Q
[n]
to be the following qu iver: its vertices are the paths
of length n in Q; if p and q are paths of length n in Q there is an arrow in
Q
[n]
from p to q if there is a path of length n + 1 in Q that begins with p
and ends with q.
Corollary 1.4. For all integers n ≥ 2, QGr(kQ) ≡ QGr(kQ
[n]
).
Acknowledgements. I wish to thank Doug Lind for introducing me
to the notion of shift equivalence and for useful discussions about symbolic
dynamics and related matters.
2. (Strong) shift equivalence
2.1. Strong shift equivalence and Williams’s Theorem. Let A and B
be square matrices with entries in N. An elementary strong shift equivalence
between A and B is a pair of matrices L and R with non-negative integer
entries such that
A = LR and B = RL.
We say A and B are strong shift equivalent if there is a chain of elementary
strong shift equivalences from A to B.
The following fundamental r esult explains the importance of strong shift
equivalence.
Theorem 2.1 (Williams). [10, Thm. A] Let A and B be square N-valued
matrices and X
A
and X
B
the associated subshifts of finite type. Then X
A
∼
=
X
B
if and only if A and B are strong shift equivalent.
A proof of T heorem 2.1 can also be found at [6, Thm. 7.2.7].
2.2. The matrices
A =
1 1
1 1
=
1
1
1 1
and B = (2) =
1 1
1
1
are strong shift equivalent. The corresponding quivers are
Q
A
=
•
**
99
•
jj
ee
Q
B
=
•
99
ee
By Theorem 1.1, QGr kQ
A
≡ QGr kQ
B
.
By [9], there are ultramatricial k-algebras S(Q
A
) and S(Q
B
) such that
QGr kQ
A
≡ ModS(Q
A
) and QGr kQ
B
≡ ModS(Q
B
). The Bratteli diagram
for S(Q
A
) is
1
//
>
>
>
>
>
>
>
2
//
>
>
>
>
>
>
>
4
//
>
>
>
>
>
>
>
8
//
@
@
@
@
@
@
@
@
16
//
!!
C
C
C
C
C
C
C
C
· · ·
1
//
@@
2
//
@@
4
//
@@
8
//
??
~
~
~
~
~
~
~
~
16
//
==
{
{
{
{
{
{
{
{
· · ·
and that f or S(Q
B
) is
1
//
//
2
//
//
4
//
//
8
//
//
16
//
// · · ·
SHIFT EQUIVALENCE AND MODULES OVER PATH ALGEBRAS 5
2.3. No general procedure is known to decide if two matrices are strong
shift equivalent. The shortest known sequence of elementary s trong shift
equivalences prov ing that
1 3
2 1
and
1 6
1 1
are strong shift equivalent was found by a computer search [6, Ex. 7.3.12]
and en route from the first to the second m atrix one passes through the
incidence matrix for the graph
Q =
•
xxp
p
p
p
p
p
p
p
p
p
p
p
p
//
&&
N
N
N
N
N
N
N
N
N
N
N
N
N
•
zz
oo
•
%%
DD
•
ZZ
88
p
p
p
p
p
p
p
p
p
p
p
p
p
dd
jj
Thus Theorem 1.2 shows that
QGr(kQ) ≡ QGr(kQ
′
) ≡ QGr(kQ
′′
)
where
Q
′
=
•
%%
//
99
BB
•
yy
ee
and Q
′′
=
•
%%
99
//
%%
•
\\
ee
2.4. Shift equivalence. Two square matrices A an d B with non-negative
integer entries are shift equivalent if there is a positive integer ℓ and matrices
L and R with non-negative integer entries such that
AL = LB, RA = RB, A
ℓ
= LR, and B
ℓ
= RL.
3. Proof of Theorem 1.1
3.1. Notation for quivers and path algebras. Let k be a field and Q a
finite quiver, i.e., a finite directed graph. We write Q
0
for its set of vertices
and Q
1
for its set of arrows . If the arrow a ends where the arrow b starts
we write ba for th e path “first traverse a then traverse b”. We write Q
n
for
the set of paths of length n in Q.
If p is a path we write s(p) for the vertex at wh ich it starts and t(p) for
the vertex at which it terminates.
The path algebra kQ has a basis given by the set of all finite paths, includ-
ing the empty path and the trivial paths at each vertex. The multiplication
6 S. PAUL SMITH
in kQ is the linear extension of that given by concatenation of paths, i.e.,
q × p =
(
qp if s(q) = t(p)
0 if s(q) 6= t(p).
The algebra kQ is N-graded with degree n component equal to kQ
n
, the
linear span of the paths of length n. The subalgebra kQ
0
of kQ is isomorphic
to a product of |Q
0
| copies of k and is therefore a semisimple ring. Each
kQ
n
is a kQ
0
-bimodule. The multiplication in kQ gives an isomorphism
(kQ
1
)
⊗n
∼
=
kQ
n
of kQ
0
-bimodules where the tensor product on the left-hand side is taken
over kQ
0
. It follows that kQ is isomorphic to the tensor algebra over kQ
0
of kQ
1
,
kQ
∼
=
T
kI
(kQ
1
).
3.2. Let i and j be positive integers.
Let kI denote the ring of k-valued functions on [i] = {1, . . . , i} with
pointwise add ition and multiplication. Similarly, kJ denotes the ring of k-
valued fu nctions on [j] = {1, . . . , j}. We identify kI ⊗
k
kJ with the ring
of k-valued functions on the Cartesian product [i] × [j]. The category of
kI-kJ-bimodules is equivalent to the category of kI ⊗
k
kJ-modules and we
write E
pq
for the simple kI-kJ-bimodule correspond ing to (p, q) ∈ [i] × [j];
i.e., E
pq
is a copy of k supported at (p, q).
3.3. Let L = (ℓ
pq
) be an i × j matrix over N and R = (r
st
) a j × i matrix
over N. Let Q
LR
be the directed graph with inciden ce matrix LR and Q
RL
the directed graph with incidence matrix RL.
Since it is unlikely to cause confusion we will also use the letter L to
denote the kI-kJ-bimodule
L :=
M
p∈[i]
q∈[j]
(E
pq
)
⊕ℓ
pq
.
In a s im ilar way we define the kJ-kI-bimodule
R :=
M
t∈[i]
s∈[j]
(E
st
)
⊕r
st
.
The linear span in kQ
LR
of the arrows in Q
LR
is isomorphic to L ⊗
kJ
R
as a kI-kI-bimodule. We identify the path algebras kQ and kQ
′
with the
followin g tensor algebras:
kQ
LR
= T
kI
(L ⊗
kJ
R) =
∞
M
n=0
(L ⊗
kJ
R)
⊗n
SHIFT EQUIVALENCE AND MODULES OVER PATH ALGEBRAS 7
and
kQ
RL
= T
kJ
(R ⊗
kI
L) =
∞
M
n=0
(R ⊗
kI
L)
⊗n
.
We give kQ
LR
its standard grading by declaring that kI is its degree-zero
component and L ⊗
kJ
R its degree-one component.
3.4. Since kQ
LR
is the tensor algebra of the kI-bimodule L⊗
kJ
R, a graded
left kQ
LR
-module is a pair (M, λ) consisting of a graded left kI-module M
and a homomorphism
λ : L ⊗
kJ
R ⊗
kI
M → M
of left kI-modules such that λ(L ⊗
kJ
R ⊗
kI
M
n
) ⊂ M
n+1
for all n. A homo-
morphism (M, λ) → (M
′
, λ
′
) of graded kQ
LR
-modules is a homomorphism
θ : M → M
′
of graded kI-modules such that
θ ◦ λ = λ
′
◦ (id
L
⊗ id
R
⊗θ).
3.5. We now define functors
Gr
kQ
LR
F
,,
Gr
kQ
RL
.
F
′
ll
If M is a graded left kQ
LR
-module we define
F (M, λ) := (R ⊗
kI
M, id
R
⊗λ)
with the grad ing
(R ⊗
kI
M)
n
:= R ⊗
kI
M
n
.
The action of the degree-one component of kQ
RL
, which is R ⊗
kI
L, on the
dgree n component F (M, λ)
n
is
(id
R
⊗λ)
(R ⊗
kI
L) ⊗
kJ
(R ⊗
kI
M)
n
= R ⊗
kI
λ(L ⊗
kJ
R ⊗
kI
M
n
)
⊂ R ⊗
kI
M
n+1
= (R ⊗
kI
M)
n+1
so R ⊗
kI
M really is a graded kQ
′
-module.
If θ : (M, λ) → (M, λ
′
) is a homomorphism of graded kQ
LR
-modules we
define
F (θ) := id
R
⊗θ.
It is easy to check that F θ : F (M, λ) → F (M
′
, λ
′
) is a homomorphism of
graded kQ
RL
-modules. Hence F is a functor.
The functor F
′
is defin ed in a s im ilar way.
Since kI and kJ are semisimple rings F and F
′
are exact functors.
Theorem 3.1. Let L be an i × j matrix over N and R a j × i matrix over
N. Then the functors F and F
′
induce mutually quasi-inverse equivalences
of categories
QGr
kQ
LR
≡ QGr
kQ
RL
.
8 S. PAUL SMITH
Proof. Let (M, λ) ∈ Gr
kQ
LR
. Then
F
′
F (M, λ) = F
′
(R ⊗
kI
M, id
R
⊗λ)
= (L ⊗
kJ
R ⊗
kI
M, id
L
⊗ id
R
⊗λ).
We define τ
M
: F
′
F M → M by τ
M
(x⊗y ⊗m) := λ(x⊗y ⊗m ), Since τ
M
= λ
it is a tautology th at
τ
M
◦ (id
L
⊗ id
R
⊗λ) = λ ◦ (id
L
⊗ id
R
⊗τ
M
)
whence τ
M
is a homomorphism of kQ
LR
-modules. Since
(F
′
F M )
n
= L ⊗
kJ
(F M)
n−1
= L ⊗
kJ
R ⊗
kI
M
n−1
we have
τ
M
(F
′
F M )
n
= λ(L ⊗
kJ
R ⊗
kI
M
n−1
) ⊂ M
n
.
Hence τ
M
is a homomorphism of graded kQ
LR
-modules.
The above shows that
τ : F
′
F → id
Gr(kQ
LR
)
is a natural transformation.
Since F and F
′
are exact functors that send finite dimensional m odules
to finite dim ensional modules they induce functors between the quotient
categories QGr(kQ
LR
) and QGr(kQ
RL
), say f and f
′
. It follows that τ
induces a natur al transformation from f
′
f to id
QGr(kQ
LR
)
. We will now
show this induced natural transformation is an isomorphism of functors. A
similar argument will show f f
′
is isomorphic to id
QGr(kQ
RL
)
. The proof of
the theorem will then be complete.
Write V = L ⊗
kJ
R.
Claim
: If M ∈ Gr(kQ
LR
), then F
′
F M
∼
=
(kQ
LR
)
≥1
⊗
kQ
LR
M as graded
left kQ
LR
-modules. Proof
: Let (M, λ) be a graded left kQ
LR
-module. Then
F
′
F (M, λ) = (V ⊗
kI
M, id
V
⊗λ).
We make the identification (kQ
LR
)
≥1
= V ⊗
kI
kQ
LR
so the formula
θ(v ⊗ a ⊗ m) := v ⊗ am
for v ∈ V , a ∈ kQ
LR
, and m ∈ M, defines an isomorphism of left kI-mod ules
θ : (kQ
LR
)
≥1
⊗
kQ
LR
M = V ⊗
kI
kQ
LR
⊗
kQ
LR
M −→ V ⊗
kI
M.
If v
′
∈ V and v ⊗ a ⊗ m ∈ V ⊗
kI
kQ
LR
⊗
kQ
LR
M, then
θ
v
′
.(v ⊗ a ⊗ m)
= θ(v
′
⊗ va ⊗ m)
= v
′
⊗ vam
= (id
V
⊗λ)(v
′
⊗ v ⊗ am)
= v
′
.(v ⊗ am)
= v
′
.θ(v ⊗ a ⊗ m)
SHIFT EQUIVALENCE AND MODULES OVER PATH ALGEBRAS 9
so θ is a homomorph ism, and therefore an isomorphism, of left kQ
LR
-
modules. In fact,
θ : (kQ
LR
)
≥1
⊗
kQ
LR
M → F
′
F (M, λ)
is an isomorp hism of g raded kQ-modules because if v ⊗ a ⊗ m is a homoge-
neous element of V ⊗
kI
kQ ⊗
kQ
M = (kQ)
≥1
⊗
kQ
M, then
deg(v ⊗ a ⊗ m) = 1 + deg(am);
however, F
′
F (M, λ)
n
= V ⊗
kI
M
n−1
so, as an element of F
′
F (M, λ), deg(v⊗
am) = 1 + deg(am) so
deg θ(v ⊗ a ⊗ m) = deg(v ⊗ am) = deg(v ⊗ a ⊗ m);
i.e., θ is a degree-preserving map so an isomorphism in GrkQ. This completes
the proof of the claim. ♦
The claim shows that the homomorp hisms
η
M
: F
′
F (M, λ) = (V ⊗
kI
M, id
V
⊗λ) → (kQ)
≥1
⊗
kQ
M
η
M
(v ⊗ m) = v ⊗ m
produce an isomorphism of functors
η : F
′
F → (kQ)
≥1
⊗
kQ
−.
Consider the diagram
F
′
F M
τ
M
//
η
M
M
0
//
Tor
kQ
1
(kI, M)
//
(kQ)
≥1
⊗
kQ
M
µ
//
M
//
kI ⊗
kQ
M
//
0
where the bottom row is the exact sequence obtained by applying − ⊗
kQ
M
to the exact sequence of kQ-bimodules 0 → (kQ)
≥1
→ kQ → kI → 0 and µ
is the multiplication in kQ. Since τ
M
(v ⊗ m) = λ(v ⊗ m) = vm, the square
commutes. Hence
ker τ
M
∼
=
Tor
kQ
1
(kI, M) and coker τ
M
∼
=
kI ⊗
kQ
M.
Both these modules are annih ilated by (kQ )
≥1
so belong to Fdim(kQ).
Therefore, after passing to QGr kQ, the diagram yields a commutative square
in which µ and η
M
become isomorphisms. It follows that τ
M
becomes an iso-
morphism in QGr kQ, and hence that τ : F
′
F → id
QGr kQ
is an isomorphism
of functors as claimed.
Given the symmetry of the situation we can reverse the roles of L and R
and repeat the previous argument to produce an isomorphism of functors
F F
′
→ id
QGr kQ
′
. This completes the proof th at QGr kQ is equivalent to
QGr kQ.
10 S. PAUL SMITH
4. Proof of Theorem 1.2
Suppose the N-valued matrices A and B are strong shift equ ivalent. By
definition, there is a sequence of matrices
A = A
1
, A
2
, . . . , A
n
= B
and elementary strong shift equivalences between A
i
and A
i+1
for 1 ≤ i ≤
n − 1. If Q
A
i
is the quiver with incidence matrix A
i
, then repeated applica-
tions of Theorem 1.1 show that
QGr kQ
A
1
≡ QGr kQ
A
2
≡ · · · ≡ QGr kQ
A
n
thereby proving Theorem 1.2.
4.1. In-splitting and out-splitting. I am grateful to Min Wu for telling
me that Theorem 1.1 applies to in-splittings and out-splittings.
Lind and Marcus define and discuss in-splittings and ou t-splittings of
a directed graph in section 2.4 of [6]. As the name suggests, in-splitting
involves replacing one vertex v by several, say n, vertices v
1
, . . . , v
n
and
replacing each arrow a ending at v by n arrow s a
1
, . . . , a
n
where a
i
starts
where a does and ends at v
i
. Actually, in-splitting is a more general process
than this, but the basic idea is along the lines just described. Out-splitting
is an analogous process, now based on the arrows leaving a vertex.
The important point for us is that if Q
′
is obtained from Q by an in-
splitting or an out-splitting there is an elementary str on g shift equivalence
between their incidence matrices (see [6, Thm. 2.4.12] and [6, Exer. 2.4.9]).
Theorem 1.1 therefore yields the following result.
Corollary 4.1. If Q
′
is obtained from Q by an in-splitting or out-splitting,
then QGr kQ ≡ QGr kQ
′
.
5. Proof of Corollary 1.4
5.1. Let Q be a finite qu iver. Define Q
[2]
by setting
Q
[2]
0
:= Q
1
and Q
[2]
1
:= Q
2
with a length-two path ba in Q being considered as an arrow in Q
[2]
from a
to b.
If a and b are arrows in Q, there is at most one arrow in Q
[2]
from a to b
so the incidence matrix for Q
[2]
is a 0-1 matrix.
The following example from [7, Example 2.7] and [6, Example 1.4.2] il-
lustrates the construction:
Q =
1
v
**
w
99
2
u
jj
Q
[2]
=
w
wv
A
A
A
A
A
A
A
ww
u
wu
>>
}
}
}
}
}
}
}
}
vu
++
v
uv
kk
SHIFT EQUIVALENCE AND MODULES OVER PATH ALGEBRAS 11
Returning to the general case, define a |Q
0
| × |Q
1
| matrix L by
L
ia
=
(
1 if t(a) = i
0 if t(a) 6= i
for i ∈ Q
0
and a ∈ Q
1
, and a |Q
1
| × |Q
0
| matrix R by
R
ai
=
(
1 if s(a) = i
0 if s(a) 6= i.
Then LR is the |Q
0
| × |Q
0
| matrix with entries
(LR)
ij
=
{a ∈ Q
1
| t(a) = i and s(a) = j}
= the number of arrows in Q from j to i
and RL is the |Q
1
| × |Q
1
| matrix with entries
(RL)
ab
=
{i ∈ Q
0
| s(a) = i = t(b)}
=
(
1 if s(a) = t(b)
0 if s(a) 6= t(b).
Therefore LR is the incidence matrix for Q and RL is the incidence matrix
for Q
[2]
. Theorem 1.1 therefore gives an equivalence
(5-1) QGr kQ ≡ QGr kQ
[2]
.
5.2. Let Q be a finite qu iver and define Q
[n]
by setting
Q
[n]
0
:= Q
n
and Q
[n]
1
:= Q
n+1
with a path a
n
. . . a
0
of length n + 1 in Q being considered as an arrow in
Q
[n]
from a
n−1
. . . a
0
to a
n
. . . a
1
.
It is an easy exercise to show that
Q
[n−1]
[2]
= Q
[n]
.
Repeatedly applying the equivalence in (5-1) gives a chain of equivalences
QGr kQ ≡ QGr kQ
[2]
≡ · · · ≡ QGr kQ
[n]
thereby proving Corollary 1.4.
References
[1] G. Abrams, A. Louly, E. Pardo, and C. Smith, Flow invariants in the classification
of Leavitt path algebras, Journal of Algebra, 333 (2011) 202-231.
[2] G. Abrams and M. Tomforde, Isomorphism and Morita equivalence of graph algebras,
Trans. Amer. Math. Soc., 363 (2011) 3733-3767.
[3] T. Bates and D. Pask, Flow equivalence of graph algebras, Ergodi c Theory and Dy-
namical Systems, 24 (2004) 367-382.
[4] K. H. Kim and F. W. Roush, Decidability of shift equivalence, Proceedings of Mary-
land Special Year in Dynamics 1986-87, Springer-Verlag Lecture Notes in Math. 1342
(1988), 374-424.
[5] K. H. Kim and F. W. Roush, Williams’ conjecture is false for irreducible subshifts,
Annals of Math, 149 (199) 545-558.
12 S. PAUL SMITH
[6] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Camb.
Univ. Press, Cambridge, 1995.
[7] I. Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics, 103,
Published for the Conference Board of t he Mathematical Sciences, Washington, DC,
by the Amer. Math. Soc., Providence, RI, 2005. MR 2135030 (2005k:46141)
[8] S.P. Smith, The non-commutative scheme having a free algebra as a homogeneous
coordinate ring, arXiv:1104.3822
[9] S.P. Smith, Category equivalences involving graded modules over path algebras of
quivers, arXiv:1107.3511
[10] R.F. Williams, Classification of subshifts of finite type, Ann. of Math., 98 (1973)
120-153.
Department of Mathematics, Box 354350, Univ. Washington, Seattle, WA
98195
E-mail address: smith@math.washington.edu