THE GRADED GROTHENDIECK GROUP
AND THE CLASSIFICATION OF LEAVITT PATH ALGEBRAS
R. HAZRAT
Abstract. This paper is an attempt to show that, parallel to Elliott’s classification of AF C ∗ -algebras by
means of K-theory, the graded K0 -group classifies Leavitt path algebras completely. In this direction, we prove
this claim at two extremes, namely, for the class of acyclic graphs (graphs with no cycles) and comet and
polycephaly graphs (graphs which each head is connected to a cycle or a collection of loops).
1. Introduction
In [20] Leavitt considered the free associative K-algebra A generated by symbols {xi , yi | 1 ≤ i ≤ n}
subject relations
n
X
xi yj = δij , for all 1 ≤ i, j ≤ n, and
yi xi = 1,
(1)
i=1
where K is a field, n ≥ 2 and δij is the Kronecker delta. The relations guarantee the right A-module
homomorphism
φ : A −→ An
(2)
a 7→ (x1 a, x2 a, . . . , xn a)
has an inverse
ψ : An −→ A
(3)
(a1 , . . . , an ) 7→ y1 a1 + · · · + yn an ,
so, A ∼
as a right A-module. He showed that A is universal with respect to this property, of type (1, n − 1)
=
(see §2.4) and it is a simple ring. Modeled on this, Leavitt path algebras are introduced [1, 7], which attach to
a directed graph a certain algebra. In the case that the graph has one vertex and n loops, it recovers Leavitt’s
algebra (1) (where yi are the loops and xi are the ghost loops).
In [10] Cuntz considered the universal unital C ∗ -algebra On generated by isometries {si | 1 ≤ i ≤ n}
subject relation
n
X
si s∗i = 1,
An
i=1
where n ≥ 2. He showed that On is purely infinite, simple C ∗ -algebras. Modeled on this, graph C ∗ -algebras
are introduced [16, 23], which attach to a directed graph a certain C ∗ -algebras. In the case that the graph
has one vertex and n loops, it recovers Cuntz’s algebra.
The study of Leavitt path algebras and their analytic counterpart develop parallel and strikingly similar.
Cuntz computed K0 of On [11] and in [22] Rauburn and Szymański carried out the calculation for graph C ∗ algebras (see also [23, Ch. 7]). Ara, Brustenga and Cortinãs [8] calculated K-theory of Leavitt path algebras.
It was shown that when K = C, K0 of graph C ∗ -algebras and Leavitt path algebras coincide [7, Theorem 7.1].
The Grothendieck group K0 has long been recognized as an essential tool in classifying certian types of C ∗ algebras. Elliott proved that K0 as a “pointed” pre-ordered group classifies the AF C ∗ -algebras completely.
Since the “underlying” algebras of Leavitt path algebras are ultramatricial algebras (as in AF C ∗ -algebras)
Elliott’s work (see also [24, §8]) has prompted to consider K0 -group as a main tool of classifying certain type
of Leavitt path algebras [2].
This note is an attempt to justify that parallel to Elliott’s work on AF C ∗ -algebras, the “pointed” pre-order
graded Grothedieck group, K0gr , classifies the Leavitt path algebras completely. This note proves this claim at
2000 Mathematics Subject Classification. 16D70.
Key words and phrases. Leavitt path algebras, weighted Leavitt path algebras, graded Grothendieck group, K-theory.
1
2
R. HAZRAT
two extremes, namely for acyclic graphs (graphs with no cycle) and for graphs which each head is connected
to a cycle or collection of loops (see 6).
Leavitt’s algebra constructed in (1) has a natural grading; assigning 1 to yi and −1 to xi , 1 ≤ i ≤ n, since
the relations are homogeneous (of degree zero), the algebra A is a Z-graded algebra. The isomorphism (2)
induces a graded isomorphism
φ : A −→ A(−1)n
(4)
a 7→ (x1 a, x2 a, . . . , xn a),
where A(−1) is the suspension of A by −1. In the non-graded setting, the relation A ∼
= An translates to
(n − 1)[A] = 0 in K0 (A) . In fact for n = 2, one can show that K0 (A) = 0. However, when considering
the grading, for n = 2, A ∼
=gr A(−1) ⊕ A(−1) and A(i) ∼
=gr A(i − 1) ⊕ A(i − 1), i ∈ Z, and therefore
i
gr
2
∼
A =gr A(−i) , i ∈ N. This implies not only K0 (A) 6= 0 but one can also show that K0gr (A) ∼
= Z[1/2]. This
is an indication that graded K-groups can capture more information than the non-graded K-groups (see §6
for more examples). In fact, we will observe that there is a close relation between graded K-groups of Leavitt
path algebras and their graph C ∗ -algebra counterparts. For example, if the graph is finite with no sink, we
have isomorphisms in the middle of the diagram below which induce isomorphisms on the right and left hand
side of the diagram. This immediately implies K0 (C ∗ (E)) ∼
= K0 (LC (E)). (See Remark 5.3 for the general
case of row finite graphs.)
0
−1
/ K0 C ∗ (E ×1 Z) 1−β∗ / K0 C ∗ (E ×1 Z)
/ K1 (C ∗ (E))
0
/ ker(1 − N t )
∼
=
/ K gr (LC (E))
0
1−N t
∼
=
/ K gr (LC (E))
0
/ K0 (C ∗ (E))
/0.
/ K0 (LC (E))
/0
In the graded K-theory we study in this note, suspensions play a pivotal role. For example, Abrams [5,
Proposition 1.3] has given a number theoretic criteria when matrices over Leavitt algebras with no suspensions
are graded isomorphism. Abrams’ criteria shows
∼gr M4 (L2 ),
M3 (L2 ) 6=
where L2 is the Leavitt algebra constructed in (1) for n = 2. However using graded K-theory (Theorem 5.6
and Theorem 5.7), we shall see that the Leavitt path algebras of the following two graphs
•>
•>
>>
>
q
@•e
E:
>>
>
@•
F :
•
/ • qe
•
are graded isomorphic, i.e., L(E) ∼
=gr L(F ), which would then imply
M3 (L2 )(0, 1, 1) ∼
=gr M4 (L2 )(0, 1, 2, 2).
(5)
This also shows, by considering suspensions, we get much wider classes of graded isomorphisms of matrices
over Leavitt algebras (see Theorem 5.11).
Elliott showed that K0 -groups classify ultramatricial algebras completely ([14, §15]). Namely, for two
ultramatricial algebras R and S, if φ : K0 (R) → K0 (S) is an isomorphism such that φ([R]) = [S] and the
set of isomorphism classes of f.g projective modules of R are sent to the set of isomorphism classes of f.g
projective modules of S, i.e., φ is order preserving, then R and S are isomorphism.
Replacing K0 by K0gr , we will prove that a similar statement holds for the class of Leavitt path algebras
arising from acyclic, comet and polycephaly graphs (graphs of the following types) in Theorem 5.7.
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THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
3
This paves the way to pose the following conjecture.
Conjecture 1 (Weak classification conjecture). Let E and F be row-finite graphs. Then L(E) ∼
=gr L(F ) if
−1
and only if there is an Z[x, x ]-module isomorphism
gr
K0gr (L(E)), [L(E)] ∼
= K0 (L(F )), [L(F )] .
Whereas, in Elliott’s case, K0 is a functor to the category of pre-ordered abelian groups, K0gr is a functor
to Γ-pre-ordered abelian groups taking into account the Γ-grading of the rings (see §2.9).
Affirmation to this conjecture for any class of graphs (say graphs E and F ), combined with the diagram
above relating K0 -theory of a graph C ∗ -algebra with K0gr of its Leavitt path algebra, show that if C ∗ (E) ∼
=
C ∗ (F ) then LC (E) ∼
=gr LC (F ) (see Remark 5.9).
In fact the proof of Theorem 5.7 shows that an isomorphism between the graded Grothendieck groups
induces a K-algebra isomorphism between the Leavitt path algebras. Therefore starting from a ring isomorphism, passing through K-theory, we obtain K-algebra isomorphisms between Leavitt path algebras.
Therefore we obtain the following conjecture for the class of acyclic, comet and polycephally graphs (see
Corollary 5.8).
Conjecture 2. Let E and F be row-finite graphs. Then LK (E) ∼
=gr LK (F ) as rings if and only if LK (E) ∼
=gr
LK (F ) as K-algebras.
In fact we prove K0gr functor is a fully faithful functor from the category of acyclic Leavitt path algebras
to the category of pre-ordered abelian groups (see §2.9 and Theorem 2.12). From the results of the paper, the
following conjecture is plausible.
Conjecture 3 (Strong classification conjecture). The graded Grothendieck group K0gr is a fully faithful functor
from the category of Leavitt path algebras with graded homomorphisms modulo inner-automorphisms to the
category of pre-ordered abelian groups with order-units.
2. The graded Grothendieck group
In this note all modules
are considered right modules unless stated otherwise. For a set Γ by ZΓ we mean
L
Γ-copies of Z, i.e., γ∈Γ Zγ where Zγ = Z for each γ ∈ Γ.
L
2.1. Graded rings. A ring A = γ∈Γ Aγ is called a Γ-graded ring, or simply a graded ring, if Γ is an (abelian)
group, each Aγ is an additive subgroup of A and Aγ · Aδ ⊆ Aγ+δ for all γ, δ ∈ Γ. The elements of Aγ are called
S
homogeneous of degree γ and we write deg(a) = γ if a ∈ Aγ . We let Ah = γ∈Γ Aγ be the set of homogeneous
L
elements of A. We call the set ΓA = {γ ∈ Γ | Aγ 6= 0}, the support of A. A Γ-graded ring A = γ∈Γ Aγ is
called a strongly graded ring if Aγ Aδ = Aγ+δ for all γ, δ ∈ Γ.
Let A be a Γ-graded ring.
L A graded right A-module M is defined to be a right A-module with a direct
sum decomposition M = γ∈Γ Mγ , where each Mγ is an additive subgroup of M such that Mλ · Aγ ⊆ Mγ+λ
L
for all γ, λ ∈ Γ. For some δ ∈ Γ, we define the δ-suspended A-module M (δ) as M (δ) = γ∈Γ M (δ)γ where
M (δ)γ = Mγ+δ . For two graded A-modules M and N , a graded A-module homomorphism of degree δ, is an
A-module homomorphism f : M → N , such that f (Mγ ) ⊆ Nγ+δ for any γ ∈ Γ. By a graded homomorphism
we mean a graded homomorphism of degree 0. By gr-A, we denote the category of graded right A-modules
with graded homomorphisms. For graded modules M and N , if M is finitely generated, then HomA (M, N )
has a natural Z-grading
M
HomA (M, N ) =
Hom(M, N )γ ,
(7)
γ∈Γ
where Hom(M, N )γ is the set of all graded A-module homomorphisms of degree γ (see [21, §2.4]).
A graded A-module P is called a graded projective module if P is a projective module. One can check
that P is graded projective if and only of the functor Homgr-A (P, −) is an exact functor in gr-A if and only
if P is a graded isomorphic to a direct summands of a graded free A-module. In particular, if P is graded
f.g projective A-module, then there is a graded f.g projective A-module Q such that P ⊕ Q ∼
=gr An (α), where
α = (α1 , · · · , αn ), αi ∈ Γ. By Pgr(A) we denote the category of graded f.g projective right A-modules with
graded homomorphisms.
4
R. HAZRAT
L
L
Let A =
Aγ and B =
γ∈Γ
γ∈Γ Bγ be graded rings. Then A × B has a natural grading given by
L
A × B = γ∈Γ (A × B)γ where (A × B)γ = Aγ × Bγ . Similarly, if A and B are K-modules for a field K (where
L
here K has a trivial grading), then A ⊗K B has a natural grading given by A ⊗K B = γ∈Γ (A ⊗K B)γ where,
nX
o
(A ⊗K B)γ =
ai ⊗ bi | ai ∈ Ah , bi ∈ B h , deg(ai ) + deg(bi ) = γ .
(8)
i
Some of the rings we are dealing with in this note are of the form K[x, x−1 ] where K is a field. This is an
example of a graded field. A nonzero commutative Γ-graded ring A is called a graded field if every nonzero
homogeneous element has an inverse. It follows that A0 is a field and ΓA is an abelian group. Similar to the
non-graded setting, one can show that any Γ-graded module M over a graded field A is graded free, i.e., it is
generated by a homogeneous basis and the graded bases have the same cardinality (see [21, Proposition 4.6.1]).
Moreover, if N is a graded submodule of M , then
dimA (N ) + dimA (M/N ) = dimA (M ).
(9)
In this note, all graded fields have torsion free abelian group gradings (in fact, in all our statements A =
K[xn , x−n ], with ΓA = nZ, for some n ∈ N and Γ = Z). However, this assumption is not necessary for the
statements below.
2.2. Grading on matrices. Let M = M1 ⊕ · · · ⊕ Mn , where Mi are graded f.g right A-modules. So M
is also a graded A-module. Let πj : M → Mj and κj : Mj → M be the (graded) projection and injection
homomorphisms. Then there is a graded isomorphism
EndA (M ) → Hom[Mj , Mi ] 1≤i,j≤n
(10)
defined by φ 7→ [πi φκj ], 1 ≤ i, j ≤ n.
For a graded ring A, observe that HomA A(δi ), A(δj ) ∼
=gr A(δj − δi ) (see 7). If
V = A(−δ1 ) ⊕ A(−δ2 ) ⊕ · · · ⊕ A(−δn ),
then by (10),
EndA (V ) ∼
=gr A(δj − δi ) 1≤i,j≤n .
=gr Hom A(−δj ), A(−δi ) ∼
Denoting this graded martix ring by Mn (A)(δ1 , . . . , δn ), we have

A(δ1 − δ1 ) A(δ2 − δ1 ) · · ·
 A(δ1 − δ2 ) A(δ2 − δ2 ) · · ·

Mn (A)(δ1 , . . . , δn ) = 
..
..
..

.
.
.
A(δ1 − δn ) A(δ2 − δn ) · · ·

A(δn − δ1 )
A(δn − δ2 ) 

.
..

.
(11)
A(δn − δn )
Therefore for λ ∈ Γ, Mn (A)(δ1 , . . . , δn )λ , the λ-homogeneous elements, are the n × n-matrices over A with
the degree shifted (suspended) as follows:


Aλ+δ1 −δ1 Aλ+δ2 −δ1 · · · Aλ+δn −δ1
 Aλ+δ −δ Aλ+δ −δ · · · Aλ+δ −δ 
n
1
2
2
2
2

Mn (A)(δ1 , . . . , δn )λ = 
(12)
.
..
..
..
..


.
.
.
.
Aλ+δ1 −δn
Aλ+δ2 −δn
···
Aλ+δn −δn
This also shows that
deg(eij (x)) = deg(x) + δi − δj .
(13)
Setting δ = (δ1 , . . . , δn ) ∈ Γn , one denotes the graded matrix ring (11) as Mn (A)(δ). Consider the graded
A-bi-module
An (δ) = A(δ1 ) ⊕ · · · ⊕ A(δn ).
Then one can check that An (δ) is a graded right Mn (A)(δ)-module and An (−δ) is a graded left Mn (A)(δ)module, where −δ = (−δ1 , . . . , −δn ).
Note that if A has a trivial grading, this construction induces a good grading on Mn (A). These group
gradings on matrix rings have been studied by Dăscălescu et al. [12]. Therefore for x ∈ A,
deg(eij (x)) = δi − δj .
(14)
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
5
We will use these gradings on matrices to describe the graded structure of Leavitt path algebras of acyclic
and comet graphs.
For a Γ-graded ring A, α = (α1 , . . . αm ) ∈ Γm and δ = (δ1 , . . . , δn ) ∈ Γn , let


Aα1 −δ1 Aα1 −δ2 · · · Aα1 −δn

 Aα −δ
 2 1 Aα2 −δ2 · · · Aα2 −δn 
Mm×n (A)[α][δ] =  ..
.
..
.
.
..
..

 .
.
Aαm −δ1
Aαm −δ2
···
Aαm −δn
So Mm×n (A)[α][δ] consists of matrices with the ij-entry in Aαi −δj .
Proposition 2.1. Let A be a Γ-graded ring and let α = (α1 , . . . , αm ) ∈ Γm , δ = (δ1 , . . . , δn ) ∈ Γn . Then the
following are equivalent:
(1) Am (α) ∼
=gr An (δ) as graded right A-modules.
(2) Am (−α) ∼
=gr An (−δ) as graded left A-modules.
(3) There exist a = (aij ) ∈ Mn×m (A)[δ][α] and b = (bij ) ∈ Mm×n (A)[α][δ] such that ab = In and ba = Im .
Proof. (1) ⇒ (3) Let φ : Am (α) → An (δ) and ψ : An (δ) → Am (α) be graded right A-modules isomorphisms
such that φψ = 1 and ψφ = 1. Let ej denote the standard basis element of Am (α) with 1 in the j-th entry
and zeros elsewhere. Then let φ(ej ) = (a1j , a2j , . . . , anj ), 1 ≤ j ≤ m. Since φ is a graded map, comparing
the grading of both sides, one can observe that deg(aij ) = δi − αj . Note that the map φ is represented by
the left multiplication with the matrix a = (aij )n×m ∈ Mn×m (A)[δ][α]. In the same way one can construct
b ∈ Mm×n (A)[α][δ] which induces ψ. Now φψ = 1 and ψφ = 1 translate to ab = In and ba = Im .
(3) ⇒ (1) If a ∈ Mn×m (A)[δ][α], then multiplication from the left, induces a graded right A-module
homomorphism φa : Am (α) −→ An (δ). Similarly b induces ψb : An (δ) −→ Am (α). Now ab = In and ba = Im
translate to φa ψb = 1 and ψb φa = 1
(2) ⇐⇒ (3) This part is similar to the previous cases by considering the matrix multiplication from the
right. Namely, the graded left A-module homomorphism φ : Am (−α) → An (−δ) represented by a matrix
multiplication from right of the form Mm×n (A)[α][δ] and similarly ψ gives a matrix in Mn×m (A)[δ][α]. The
rest follows easily.
When m = n = 1 in the Proposition 2.1 we obtain,
Corollary 2.2. Let A be a Γ-graded ring and α ∈ Γ. Then the following are equivalent:
(1) A(α) ∼
=gr A as graded right A-modules.
(2) A(−α) ∼
=gr A as graded right A-modules.
∼
(3) A(α) =gr A as graded left A-modules.
(4) A(−α) ∼
=gr A as graded left A-modules.
(5) There is an invertible homogeneous element of degree α.
Proof. This follows from Proposition 2.1.
2.3. Leavitt algebras as graded rings. Let K be a field, n and k positive integers and A the free associative
K-algebra generated by symbols {xij , yji | 1 ≤ i ≤ n + k, 1 ≤ j ≤ n} subject relations (coming from)
Y · X = In,n
where

y11
 y21

Y =  ..
 .
y12
y22
..
.
...
...
..
.
yn,1 yn,2 . . .
and
X · Y = In+k,n+k ,


y1,n+k
x11
x12
...
 x21
x
.
..
y2,n+k 
22


..  , X =  ..
..
.
..
 .
.
. 
yn,n+k
xn+k,1 xn+k,2 . . .
x1,n
x2,n
..
.



.

(15)
xn+k,k
This algebra was studied by Leavitt in relation with its type in [18, 19, 20]. In [18, p.190], he studied this
algebra for n = 2 and k = 1, where he showed that the algebra has no zero divisors, in [19, p.322] for arbitrary
n and k = 1 and in [20, p.130] for arbitrary n and k and established that these algebras are of type (n, k) (see
6
R. HAZRAT
§2.4) and when n ≥ 2 they are domain. We denote this algebra by L(n, k + 1). (See Definition 3.6; Cohn’s
notation in [9] for this algebra is Vn,n+k .)
Assigning deg(yji ) = (0, . . . , 0, 1, 0 . . . , 0) and deg(xij ) = (0, . . . , 0, −1, 0 . . . , 0), 1 ≤ i ≤ n + k, 1 ≤ j ≤ n,
L
in n Z, where 1 and −1 are in j-th entries respectively, makes the free algebra generated by xij and yji a
graded ring. Furthermore, one can easily observe that the relations coming
from (15) are all homogeneous
L
with respect to this grading, so that the Leavitt algebra L(n, k + 1) is a n Z-graded ring. Therefore, L(1, n)
is a Z-graded ring. In fact this is a strongly graded ring by Theorem 5.1.
2.4. Graded IBN and graded type. A ring A has an invariant basis number (IBN), if any two bases of a
free (right) A-module have the same cardinality, i.e., if An ∼
= Am as A-modules, then n = m. When A does
not have IBN, the type of A is defined as a pair of positive integers (n, k) such that An ∼
= An+k as A-modules
0
and these are the smallest number with this property. It was shown that if A has type (n, k), then Am ∼
= Am
if and only if m = m0 or m, m0 > n and m ≡ m0 (mod k) (see [9, p. 225], [20, Theorem 1]).
A graded ring A has a graded invariant basis number (gr-IBN), if any two homogeneous bases of a graded
free (right) A-module have the same cardinality, i.e., if Am (α) ∼
=gr An (δ), where α = (α1 , . . . , αm ) and
δ = (δ1 , . . . , δn ), then m = n. Note that contrary to non-graded case, this does not imply that two graded
free modules with bases of the same cardinality are graded isomorphic (see Proposition 2.1). A graded ring
∼ n
A has IBN in gr-A, if Am ∼
=gr An then m = n. If A has IBN in gr-A, then A0 has IBN. Indeed, if Am
0 = A0
m
m
n
n
as A0 -modules, then A ∼
=gr A0 ⊗A0 A ∼
= A0 ⊗A0 A ∼
=gr A , so n = m (see [21, p. 215]).
When the graded ring A does not have gr-IBN, the graded type of A is defined as a pair of positive integers
(n, k) such that An (δ) ∼
=gr An+k (α) as A-modules, for some δ = (δ1 , . . . , δn ) and α = (α1 , . . . , αn+k ) and these
are the smallest number with this property.
In Proposition 2.3 we show that the Leavitt algebra L(n, k + 1) has graded type (n, k). Using graded
K-theory we will also show that (see Corollary 5.13)
0
L(1, n)k (λ1 , . . . , λk ) ∼
=gr L(1, n)k (γ1 , . . . , γk0 )
if and only if
Pk
i=1 n
λi
=
Pk0
i=1 n
γi .
Let A be a Γ-graded ring such that Am (α) ∼
=gr An (δ), where α = (α1 , . . . , αm ) and δ = (δ1 , . . . , δn ).
Then there is a universal Γ-graded ring R such that Rm (α) ∼
=gr Rn (δ), and a graded ring homomorphism
R → A which induces the graded isomorphism Am (α) ∼
=gr Rm (α) ⊗R A ∼
=gr Rn (δ) ⊗R A ∼
=gr An (δ). Indeed,
by Proposition 2.1, there are matrices a = (aij ) ∈ Mn×m (A)[δ][α] and b = (bij ) ∈ Mm×n (A)[α][δ] such that
ab = In and ba = Im . The free ring generated by symbols in place of aij and bij subject to relations imposed
by ab = In and ba = Im is the desired universal graded ring. In details, let F be a free ring generated by
xij , 1 ≤ i ≤ n, 1 ≤ j ≤ m and yij , 1 ≤ i ≤ m, 1 ≤ j ≤ n. Assign the degrees deg(xP
ij ) = δi − αj and
deg(yij ) = αi − δjP
. This makes F a Γ-graded ring. Let R be a ring F modulo the relations m
s=1 xis ysk = δik ,
1 ≤ i, k ≤ n and nt=1 yit xtk = δik , 1 ≤ i, k ≤ m, where δik is the Kronecker delta. Since all the relations are
homogeneous, R is a Γ-graded ring. Clearly the map sending xij to aij and yij to bij induces a graded ring
homomorphism R → A. Again Proposition 2.1 shows that Rm (α) ∼
=gr Rn (δ).
Proposition 2.3. Let R = L(n, k + 1) be the Leavitt algebra of type (n, k). Then
L
(1) R is a universal n Z-graded ring which does not have gr-IBN.
(2) R has graded type (n, k)
(3) For n = 1, R has IBN in R-gr.
L
Proof. (1) Consider the Leavitt algebra L(n, k + 1) constructed in (15), which is a
n Z-graded ring and
n
∼
is universal. Furthermore (15) combined by Proposition 2.1(3) shows that R =gr Rn+k (α). Here α =
(α1 , . . . , αn+k ), where αi = (0, . . . , 0, 1, 0 . . . , 0) and 1 is in i-th entry. This shows that R = L(n, k + 1) does
not have gr-IBN.
(2) By [9, Theorem 6.1], R is of type (n, k). This immediately implies the graded type of R is also (n, k).
∼gr Rm . Then Rn =
∼ Rm as R0 -modules. But R0 is an ultramatricial algebra, i.e., direct
(3) Suppose Rn =
0
0
limit of matrices over a field. Since IBN respects direct limits ([9, Theorem 2.3]), R0 so has IBN. Therefore,
n = m.
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
7
Remark 2.4. Assigning deg(yij ) = 1 and deg(xij ) = −1, for all i, j, makes R = L(n, k + 1) a Z-graded
algebra of graded type (n, k) with Rn ∼
=gr Rn+k (1).
Proposition 2.5 (Morita Equivalence). Let A be a Γ-graded ring and let δ = (δ1 , . . . , δn ), where δi ∈ Γ,
1 ≤ i ≤ n. Then the functors
ψ : Pgr(Mn (A)(δ)) −→ Pgr(A)
P 7−→ P ⊗Mn (A)(δ) An (−δ)
and
ϕ : Pgr(A) −→ Pgr(Mn (A)(δ))
Q 7−→ Q ⊗A An (δ)
respect the action of Z[Γ] and form equivalences of categories.
Proof. One can check that there is a graded A-module isomorphism
θ : An (δ) ⊗Mn (A)(δ) An (−δ) −→ A
(a1 , . . . , an ) ⊗ (b1 , . . . , bn ) 7−→ a1 b1 + · · · + an bn .
Furthermore, there is a graded Mn (A)(δ)-isomorphism
θ0 : An (−δ) ⊗A An (δ) −→ Mn (A)(δ)

   
a1 b1 · · ·
b1
a1
 ..
 ..   .. 
 .  ⊗  .  7−→  .
an
bn
an b1 · · ·

a1 bn
.. 
. 
an bn
Now it easily follows that φψ and ψφ are equivalent to identity functors.
2.5. Graded K0 -group. Let A be a Γ-graded ring with identity and let Vgr (A) denote the monoid of isomorphism classes of graded f.g projective modules over A. Then the graded Grothendieck group, K0gr (A), is
defined as the group completion of Vgr (A). Note that for α ∈ Γ, the α-suspension functor Tα : gr-A → gr-A,
M 7→ M (α) is an isomorphism with the property Tα Tβ = Tα+β , α, β ∈ Γ. Furthermore, Tα restricts to the
category of graded f.g projective modules. This induces a structure of Z[Γ]-module on K0gr (A), defined by
α[P ] = [P (α)] on the generators and extended naturally. In particular if A is a Z-graded then K0gr (A) is a
Z[x, x−1 ]-module for i ≥ 0.
Recall that there is a description of K0 in terms of idempotent matrices. In the following, we can always
enlarge matrices of different sizes by adding zeros in the lower right hand corner, so that they can be considered
in a ring Mk (A) for a suitable k ∈ N. Any idempotent matrix p ∈ Mn (A) gives rise to the f.g projective right
A-module pAn . On the other hand any f.g projective module gives rise to an idempotent matrix. We say two
idempotent matrices p and q are equivalent if (after suitably enlarging them) there are matrices x and y such
that xy = p and yx = q. One can show that p and q are equivalent if and only if they are conjugate if and
only if the corresponding f.g projective modules are isomorphic. Therefore K0 (A) can be defined as
the group
p 0
completion of the monoid of equivalence classes of idempotent matrices with addition, [p] + [q] =
. In
0 q
fact, this is the definition one adopts for K0 when the ring A does not have identity.
A similar construction can be given in the setting of graded rings. Since this does not seem to be documented in literature, we give the details here for the convenient of the reader.
Let A be a Γ-graded ring and α = (α1 , . . . , αn ), where αi ∈ Γ. In the following if we need to enlarge a
homogeneous matrix p ∈ Mn (A)(α), by adding zeroes in the lower right hand corner, then we add zeros in the
right hand side of α = (α1 , . . . , αn ) as well accordingly (and call it α again) so that p is a homogeneous matrix
in Mk (A)(α), where k ≥ n. Recall the definition of Mk (A)[α][δ] from §2.2 and note that if x ∈ Mk (A)[α][δ]
and y ∈ Mk (A)[δ][α] then xy ∈ Mk (−α)0 and yx ∈ Mk (−δ)0 .
Definition 2.6. Let A be a Γ-graded ring, α = (α1 , . . . , αn ) and δ = (δ1 , . . . , δm ), where αi , δj ∈ Γ. Let p ∈
Mn (A)(α)0 and q ∈ Mm (A)(δ)0 be idempotent matrices (which are homogeneous of degree zero). Then p and
q are grade equivalent if (after suitably enlarging them) there are x ∈ Mk (A)[−α][−δ] and y ∈ Mk (A)[−δ][−α]
such that xy = p and yx = q.
8
R. HAZRAT
Lemma 2.7.
(1) Any graded f.g projective module gives rise to a homogeneous idempotent matrix of degree zero.
(2) Any homogeneous idempotent matrix of degree zero gives rise to a graded f.g projective module.
(3) Two homogeneous idempotent matrices of degree zero are graded equivalent if and only if the corresponding
graded f.g projective modules are graded isomorphic.
Proof. (1) Let P be a graded f.g projective (right) A-module. Then there is a graded module Q such that
P ⊕Q ∼
=gr An (−α) for some n ∈ N and α = (α1 , . . . , αn ), where αi ∈ Γ. Define the homomorphism
f ∈ EndA (An (−α)) which sends Q to zero and acts as identity on P . Clearly, f is an idempotent and graded
homomorphism of degree 0. Thus (see 7 and §2.2)
f ∈ EndA (An (−α))0 ∼
= Mn (A)(α)0 .
(2) Let p ∈ Mn (A)(α)0 , α = (α1 , . . . , αn ), where αi ∈ Γ. Then 1 − p ∈ Mn (A)(α)0 and
An (−α) = pAn (−α) ⊕ (1 − p)An (−α).
This shows that pAn (−α) is a graded f.g projective A-module.
(3) Let p ∈ Mn (A)(α)0 and q ∈ Mm (A)(δ)0 be graded equivalent idempotent matrices. By Definition 2.6,
there are x0 ∈ Mk (A)[−α][−δ] and y 0 ∈ Mk (A)[−δ][−α] such that x0 y 0 = p and y 0 x0 = q. Let x = px0 q and
y = qy 0 p. Then xy = px0 qy 0 p = p(x0 y 0 )2 p = p and similarly yx = q. Furthermore x = px = xq and y = yp =
qy. Now the left multiplication by x and y induce right graded A-homomorphisms qAk (−δ) → pAk (−α) and
pAk (−α) → qAk (−δ), respectively, which are inverse of each other. Therefore pAk (−α) ∼
=gr qAk (−δ).
On the other hand if f : pAk (−α) ∼
=gr qAk (−δ), then extend f to Ak (−α) by sending (1 − p)Ak (−α) to
zero and thus define a map
θ : Ak (−α) = pAk (−α) ⊕ (1 − p)Ak (−α) −→ qAk (−δ) ⊕ (1 − q)Ak (−δ) = Ak (−δ).
Similarly, extending f −1 to Ak (−δ), we get a map
φ : Ak (−δ) = qAk (−δ) ⊕ (1 − q)Ak (−δ) −→ pAk (−α) ⊕ (1 − p)Ak (−α) = Ak (−α)
such that θφ = p and φθ = q. It follows (see the proof of Proposition 2.1) θ ∈ Mk (A)[−α][−δ] whereas
φ ∈ Mk (A)[−δ][−α]. This gives that p and q are equivalent.
moniod
Lemma 2.7 shows that K0gr (A) can be defined as the group completion of the of equivalent classes
p 0
of homogeneous idempotent matrices of degree zero with addition, [p] + [q] =
. In fact, this is the
0 q
gr
definition we adopt for K0 when the graded ring A does not have identity. Note that the action of Γ on
the idempotent matrices (so that K0gr (A) becomes a Z[Γ]-module with this definition) is as follows: For a
γ ∈ Γ and p ∈ Mn (A)(α)0 , γp is represented by the same matrix as p but considered in Mn (A)(α + γ)0 where
α + γ = (α1 + γ + · · · , αn + γ). A quick inspection of the proof of Lemma 2.7 shows that the action of Γ is
compatible in both definitions of K0gr .
In the case of graded fields, one can compute the graded Grothendieck group completely. This will be used
to compute K0gr of acyclic and comet graphs in Theorems 5.4 and 5.5.
Proposition 2.8. Let A be a Γ-graded field with the support the subgroup ΓA . Then the monoid of isomorphism classes of Γ-graded f.g projective A-modules is P
N[Γ/ΓA ] and K0gr (A) ∼
= Z[Γ/ΓA ] as a Z[Γ]-modules.
gr
n
Furthermore, [A (δ1 , . . . , δn )] ∈ K0 (A) corresponds to ni=1 δ i in Z[Γ/ΓA ], where δ i = ΓA + δi . In particular
if A is a (trivially graded) field, Γ a group and A considered asPa graded field concentrated in degree zero, then
gr
n
K0gr (A) ∼
= Z[Γ] and [An (δ1 , . . . , δn )] ∈ K0 (A) corresponds to i=1 δi in Z[Γ].
Proof. By Proposition 2.1, A(δ1 ) ∼
=gr A(δ2 ) if and only if δ1 − δ2 ∈ ΓA . Thus any graded free module of rank 1
is graded isomorphic to some A(δi ), where {δi }i∈I is a complete set of coset representative of the subgroup ΓA
in Γ, i.e., {ΓA + δi , i ∈ I} represents Γ/ΓA . Since any graded f.g module M over A is graded free (see §2.1),
(16)
M∼
=gr A(δi )r1 ⊕ · · · ⊕ A(δi )rk ,
1
k
where δi1 , . . . δik are distinct elements of the coset representative. Now suppose
M∼
=gr A(δi0 )s1 ⊕ · · · ⊕ A(δi0 0 )sk0 .
1
k
(17)
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
9
Since A is commutative, k = k 0 . Now considering the A0 -module Mδ1 , from (16) we have Mδ1 = Ar01 . This
implies one of δi0 j , 1 ≤ j ≤ k, say, δi0 1 , has to be δi1 and so r1 = s1 as A0 is a field. Repeating the same
argument for each δij , 1 ≤ j ≤ k, we see δi0 j = δij and rj = sj , for all 1 ≤ j ≤ k. Thus any graded f.g
projective A-module can be written uniquely as M ∼
=gr A(δi1 )r1 ⊕ · · · ⊕ A(δik )rk , where δi1 , . . . δik are distinct
elements of the coset representative. The (well-defined) map
Vgr (A) → N[Γ/ΓA ]
[A(δi1 )r1 ⊕ · · · ⊕ A(δik )rk ] 7→ r1 (ΓA + δi1 ) + · · · + rk (ΓA + δik ),
gives a Z[Γ]-monoid isomorphism between the monoid of isomorphism classes of Γ-graded f.g projective Amodules Vgr (A) and N[Γ/ΓA ]. The rest of the proof follows easily.
2.6. Dade’s theorem. Let A be a strongly graded ring. By Dade’s Theorem [21, Thm. 3.1.1], the functor
(−)0 : gr-A → mod-A0 , M 7→ M0 , is an additive functor with an inverse − ⊗A0 A : mod-A0 → gr-A so
that it induces an equivalence of categories. This implies that Kigr (A) ∼
= Ki (A0 ), for i ≥ 0. Furthermore,
∼
since Aα ⊗A0 Aβ = Aαβ as A0 -bimodule, the functor Tα on gr-A induces a functor on the level of mod-A0 ,
Tα : mod-A0 → mod-A0 , M 7→ M ⊗A0 Aα such that Tα Tβ = Tαβ , α, β ∈ Γ, so that the following diagram is
commutative.
gr-A
(−)0
mod-A0
Tα
/ gr-A
(−)0
/ mod-A0
Therefore Ki (A0 ) is also a Z[Γ]-module and
Kigr (A0 ) ∼
= Ki (A),
(18)
as Z[Γ]-modules.
2.7. Let A be a graded ring and f : A → A an inner-automorphism defined by f (a) = rar−1 , where r is a
homogeneous and invertible element of A of degree τ . Then considering A as a graded left A-module via f , it
is easy to observe that there is a right graded A-module isomorphism P (−τ ) → P ⊗f A, p 7→ p ⊗ r (with the
inverse p ⊗ a 7→ pr−1 a). This induces an isomorphism between the functors − ⊗f A : Pgr(A) → Pgr(A) and
τ -suspension functor Tτ : Pgr(A) → Pgr(A). Recall that a Quillen’s Ki -group, i ≥ 0, is a functor from the
category of exact categories with exact functors to the category of abelian groups. Furthermore, isomorphic
functors induce the same map on the K-groups [Q, p.19]. Thus Kigr (f ) = Kigr (Tτ ). Therefore if r is a
homogeneous element of degree zero, i.e., τ = 0, then Kigr (f ) : Kigr (A) → Kigr (A) is the identity map. This
will be used in Theorem 5.7.
2.8. Homogeneous idempotent. The following facts about idempotents are well known in a non-graded
setting and one can check that they translate in the graded setting with the similar proofs (cf. [17, §21]). Let
Pi , 1 ≤ i ≤ l, be homogeneous right ideals of A such that A = P1 ⊕ · · · ⊕ Pl . Then there are homogeneous
orthogonal idempotents ei (hence of degree zero) such that 1 = e1 + · · · + el and ei A = Pi .
Let e and f be homogeneous idempotent elements in the graded ring A. (Note that, in general, there are
non-homogeneous idempotents in a graded ring.) Let θ : eA → f A be a graded A-module homomorphism.
Then θ(e) = θ(e2 ) = θ(e)e = f ae for some a ∈ A and for b ∈ eA, θ(b) = θ(eb) = θ(e)b. This shows there is a
map
HomA (eA, f A) → f Ae
(19)
θ 7→ θ(e)
and one can prove this is a graded group isomorphism. Now if θ : eA → f A is a graded A-isomorphism, then
x = θ(e) ∈ f Ae and y = θ−1 (f ) ∈ eAf , where x and y are homogeneous of degree zero, such that yx = e and
xy = f . Finally, for f = e, the map (19) gives a graded ring isomorphism EndA (eA) ∼
=gr eAe. These facts
will be used in Theorem 2.12.
10
R. HAZRAT
2.9. Γ-pre-ordered groups. Here we define the category of Γ-pre-ordered abelian groups. Classically, the
objects are abelian groups with a pre-ordering. However, since we will consider graded Grothendieck groups,
our groups are Z[x, x−1 ]-modules, so we need to adopt the definitions to this setting.
Let Γ be a group and G be a (left) Γ-module. Let ≤ be a reflexive and transitive relation on G which
respects the group and the module structures, i.e., for γ ∈ Γ and x, y, z ∈ G, if x ≤ y, then γx ≤ γy and
x + z ≤ y + z. We call G a Γ-pre-ordered group. We call G a pre-ordered group when Γ is clear from the
context. The cone of G is defined as {x ∈ G | x ≥ 0} and denoted by G+ . The set G+ is a Γ-submonoid of G,
i.e., a submonoid which is closed under the action of Γ. In fact, G is a Γ-pre-ordered group if and only if there
exists a Γ-submonoid of G. (Since G is a Γ-module, it can be considered as a Z[Γ]-module.) AnP
order-unit in
G is an element u ∈ G+ such that for any x ∈ G, there are α1 , . . . , αn ∈ Γ, n ∈ N, such that x ≤ ni=1 αi u. As
usual, we only consider homomorphisms which preserve the pre-ordering, i.e., a Γ-homomorphism f : G → H,
such that f (G+ ) ⊆ H + . We denote by PΓ the category of pointed Γ-pre-ordered abelian groups with pointed
ordered preserving homomorphisms, i.e., the objects are the pairs (G, u), where G is a Γ-pre-ordered abelian
group and u is an order-unit, and f : (G, u) → (H, v) is an ordered-preserving Γ-homomorphism such that
f (u) = v. Note that when Γ is a trivial group, we are in the classical setting of pre-ordered abelian groups.
In this paper, Γ = Z and so Z[Γ] = Z[x, x−1 ]. We simply write P for PZ throughout.
Example 2.9. Let A be a Γ-graded ring. Then K0gr (A) is a Γ-pre-ordered abelian group with the set of
isomorphic classes of graded f.g projective right A-modules as the cone of ordering and [A] as an order-unit.
Indeed, if x ∈ K0gr (A), then there are graded f.g projective modules P and P 0 such that x = [P ] − [P 0 ]. But
there is a graded module Q such that P ⊕ Q ∼
= An (α), where α = (α1 , . . . , αn ), αi ∈ Γ (see §2.1). Now
[An (α)] − x = [P ] + [Q] − [P ] + [P 0 ] = [Q] + [P 0 ] = [Q ⊕ P 0 ] ∈ K0gr (A)+ .
P
This shows that ni=1 αi [A] = [An (α)] ≥ x.
2.10. Graded matricial algebras. We will see that Leavitt path algebras of acyclic and Cn -comet graphs
turn out to be a finite direct product of matrix rings over graded fields of the form K and K[xn , x−n ],
respectively. In this section we study the K-theory of these algebras.
Definition 2.10. Let A be a Γ-graded field. A Γ-graded matricial A-algebra is a graded A-algebra of the
form
Mn1 (A)(δ 1 ) × · · · × Mnl (A)(δ l ),
(i)
(i)
(i)
where δ i = (δ1 , . . . , δni ), δj ∈ Γ, 1 ≤ j ≤ ni and 1 ≤ i ≤ l.
Lemma 2.11. Let A be a Γ-graded field and R be a Γ-graded matricial A-algebra. Let P and Q be graded f.g
projective R-modules. Then [P ] = [Q] in K0gr (R), if and only if P ∼
=gr Q.
Proof. Since K0 respects the direct sum, it suffices to prove the statement for a matricial algebra of the
form R = Mn (A)(δ). Let P and Q be graded f.g projective R-modules such that [P ] = [Q] in K0gr (R). By
Proposition 2.5, R = Mn (A)(δ) is graded Morita equivalent to A. So there are equivalent functors ψ and
gr
gr
gr
φ such that ψφ ∼
= 1 and φψ ∼
= 1, which also induce an isomorphism K0 (ψ) : K0 (R) → K0 (A) such that
gr
[P ] 7→ [ψ(P )]. Now since [P ] = [Q], it follows [ψ(P )] = [ψ(Q)] in K0 (A). But since A is a graded field, by
Proposition 2.8, ψ(P ) ∼
=gr ψ(Q). Now applying the functor φ to this we obtain P ∼
=gr Q.
Let A be a Γ-graded field (with the support ΓA ) and C be a category consisting of Γ-graded matricial A-algebras as objects and A-graded algebra homomorphisms as morphisms. We consider the quotient
category Cout obtained from C by identifying homomorphisms which differ up to a degree zero graded innerautomorphim. That is, the graded homomorphisms φ, ψ ∈ HomC (R, S) represent the same morphism in Cout
if there is an inner-automorpshim θ : S → S, defined by θ(s) = xsx−1 , where deg(x) = 0, such that φ = θψ.
The following theorem shows that K0gr “classifies” the category of Cout . This will be used in Theorem 5.7
where Leavitt path algebras of acyclic and comet graphs are classified by means of their K-groups.
Theorem 2.12. Let A be a Γ-graded field and Cout be the category consisting of Γ-graded matricial A-algebras
as objects and A-graded algebra homomorphisms modulo graded inner-automorphisms as morphisms. Then
K0gr : Cout → P is a fully faithful functor. Namely,
(1) (well-defined and faithful) For any graded matricial A-algebras R and S and φ, ψ ∈ HomC (R, S), we have
φ(s) = xψ(s)x−1 , s ∈ S, for some invertible homogeneous element x of S, if and only if K0gr (φ) = K0gr (ψ).
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
11
(2) (full) For any graded matricial A-algebras R and S and the morphism f : (K0gr (R), [R]) → (K0gr (S), [S])
in P, there is φ ∈ HomC (R, S) such that K0gr (φ) = f .
Proof. (1) (well-defined.) Let φ, ψ ∈ HomC (R, S) such that φ = θψ, where θ(s) = xsx−1 for some invertible
homogeneous element x of S of degree 0. Then K0gr (φ) = K0gr (θψ) = K0gr (θ)K0gr (ψ) = K0gr (ψ) since K0gr (θ) is
the identity map (see §2.7).
(faithful.) The rest of the proof is similar to the non-graded version with an extra attension given to
the grading (cf. [14, p.218]). We give it here for the completeness. Let K0gr (φ) = K0gr (ψ). Let R =
(i)
(i)
(i)
(i)
Mn1 (A)(δ 1 ) × · · · × Mnl (A)(δ l ) and set gjk = φ(ejk ) and hjk = ψ(ejk ) for 1 ≤ i ≤ l and 1 ≤ j, k ≤ ni , where
(i)
(i)
(i)
ejk are the standard basis for Mni (A). Since φ and ψ are graded homomorphism, deg(ejk ) = deg(gjk ) =
(i)
(i)
(i)
deg(hjk ) = δj − δk .
(i)
Since ejj are pairwise graded orthogonal idempotents (of degree zero) in R and
same is also the case for
(i)
gjj
and
(i)
hjj .
Pl
i=1
(i)
j=1 ejj
Pni
= 1, the
Then
(i)
(i)
(i)
(i)
[g11 S] = K0gr (φ)([e11 R]) = K0gr (ψ)([e11 R]) = [h11 S].
(i)
(i)
By Lemma 2.11, g11 S ∼
=gr h11 S. Thus there are homogeneous elements of degree zero xi and yi such that
P P i (i) (i)
P P i (i)
(i)
(i)
(i)
hj1 yi g1j . Note that
xi yi = g11 and yi xi = h11 (see §2.8). Let x = li=1 nj=1
gj1 xi h1j and y = li=1 nj=1
x and y are homogeneous elements of degree zero. One can easily check that xy = yx = 1. Now for 1 ≤ i ≤ l
and 1 ≤ j, k ≤ ni we have
(i)
xhjk =
=
=
ns
l X
X
(s)
s=1 t=1
(i)
(i) (i)
gj1 xi h1j hjk
ns
l X
X
(s) (i)
gt1 xs h1t hjk
(i) (i)
(i)
= gjk gk1 xi h1k
(i) (s)
(i)
(i)
gjk gt1 xs h1t = gjk x
s=1 t=1
(i)
(i)
(i)
(i)
Let θ : S → S be the graded inner-automorphism θ(s) = xsy. Then θψ(ejk )) = xhjk y = gjk = φ(ejk ). Since
(i)
ejk , 1 ≤ i ≤ l and 1 ≤ j, k ≤ ni , form a homogeneous A-basis for R, θψ = φ.
(2) Let R = Mn1 (A)(δ 1 ) × · · · × Mnl (A)(δ l ). Consider Ri = Mni (A)(δ i ), 1 ≤ i ≤ l. Each Ri is a graded
f.g projective R-module, so f ([Ri ]) is in the cone of K0gr (S), i.e., there is a graded f.g projective S-module Pi
such that f ([Ri ]) = [Pi ]. Then
[S] = f ([R]) = f ([R1 ] + · · · + [Rl ]) = [P1 ] + · · · + [Pl ] = [P1 ⊕ · · · ⊕ Pl ].
Since S is a graded matricial algebra, by Lemma 2.11, P1 ⊕ · · · ⊕ Pl ∼
=gr S as a right S-module. So there are
homogeneous orthogonal idempotents g1 , . . . , gl ∈ S such that g1 + · · · + gl = 1 and gi S ∼
=gr Pi (see §2.8).
(i)
(i)
Note that each of Ri = Mni (A)(δ i ) are graded simple algebras. Set δ i = (δ1 , . . . , δn ) (here n = ni ). Let
(i)
(i)
ejk ,1 ≤ j, k ≤ n, be matrix units of Ri and consider the graded f.g projective (right) Ri -module, V = e11 Ri =
(i)
(i)
(i)
(i)
(i)
(i)
A(δ1 − δ1 ) ⊕ A(δ2 − δ1 ) ⊕ · · · ⊕ A(δn − δ1 ). Then (11) shows
(i)
(i)
(i)
(i)
(i)
Ri ∼
=gr V (δ1 − δ1 ) ⊕ V (δ1 − δ2 ) ⊕ · · · ⊕ V (δ1 − δn(i) ),
as graded R-module. Thus
(i)
(i)
(i)
(i)
(i)
[Pi ] = [gi S] = f ([Ri ]) = f ([V (δ1 − δ1 )]) + f ([V (δ1 − δ2 )]) + · · · + f ([V (δ1 − δn(i) )]).
(i)
(20)
(i)
There is a graded f.g projective S-module Q such that f ([V ]) = f ([V (δ1 − δ1 )]) = [Q]. Since f is a
Z[Γ]-module homomorphism, for 1 ≤ k ≤ n,
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
f ([V (δ1 − δk )]) = f ((δ1 − δk )[V ]) = (δ1 − δk )f ([V ]) = (δ1 − δk )[Q] = [Q(δ1 − δk )].
From (20) and Lemma 2.11 now follows
(i)
(i)
(i)
(i)
(i)
gi S ∼
=gr Q(δ1 − δ1 ) ⊕ Q(δ1 − δ2 ) ⊕ · · · ⊕ Q(δ1 − δn(i) ).
(21)
12
R. HAZRAT
(i)
Let gjk ∈ End(gi S) ∼
=gr gi Sgi maps the j-th summand of the right hand side of (21) to its k-th summand
(i)
(i)
(i)
(i)
and everything else to zero. Observe that deg(gjk ) = δj − δk and gjk , 1 ≤ j, k ≤ n, form the matrix units.
(i)
(i)
(i)
(i)
(i)
(i)
(i)
Furthermore, g11 + · · · + gnn = gi and g11 S = Q(δ1 − δ1 ) = Q. Thus [g11 S] = [Q] = f ([V ]) = f ([e11 Ri ]).
(i)
(i)
Now for 1 ≤ i ≤ l, define the A-algebra homomorphism Ri → gi Sgi , ejk 7→ gjk . This is a graded
(i)
(i)
homomorphism, and induces a graded homorphism φ : R → S such that φ(ejk ) = gjk . Clearly
(i)
(i)
(i)
(i)
K0gr (φ)([e11 Ri ]) = [φ(e11 )S] = [g11 S] = f ([e11 Ri ]),
(i)
for 1 ≤ i ≤ l. Now K0gr (R) is generated by [e11 Ri ], 1 ≤ i ≤ l, as Z[Γ]-module. This implies that K0gr (φ) =
f.
3. (Weighted) Leavitt path algebras
The weighted Leavitt path algebras are introduced [15] as a generalization of Leavitt path algebras. When
the graph has one vertex and n + k loops of weight n, it recovers the Leavitt algebra constructed in §2.3.
When the weights are one, they reduce to the case of Leavitt path algebras. In this section we recall the
definition of weighted Leavitt path algebras. In the next section we calculate the (non graded) K0 -group of
these algebras before specializing to the case of weight one, i.e., Leavitt path algebras.
Definition 3.1. A weighted graph E = (E 0 , E st , E 1 , r, s, w) consists of three countable sets, E 0 called vertices,
E st structured
edges and E 1 edges and maps s, r : E st → E 0 , and a weight map w : E st → N such that
`
E 1 = α∈E st {αi | 1 ≤ i ≤ w(α)}, i.e., for any α ∈ E st , with w(α) = k, there are k distinct elements
{α1 , . . . , αk }, and E 1 is the disjoint union of all such sets for all α ∈ E st .
We sometimes write (E, w) to emphasis the graph is weighted. If s−1 (v) is a finite set for every v ∈ E 0 ,
then the graph is called row-finite. In this note we will consider only the row-finite graphs. In this setting, if
the number of vertices, i.e., |E 0 |, is finite, then the number of edges, i.e., |E 1 |, is finite as well and we call E
a finite graph.
Definition 3.2. Weighted Leavitt path algebras.
For a weighted graph E and a ring R with identity, we define the weighted Leavitt path algebra of E, denoted
∗
by LR (E, w), to be the algebra generated by the sets {v | v ∈ E 0 }, {α1 , . . . αw(α) | α ∈ E st } and {α1∗ , . . . αw(α)
|
st
α ∈ E } with the coefficients in R, subject to the relations
(1)
(2)
(3)
(4)
vi vj = δij vi for every vi , vj ∈ E 0 .
∗
∗
∗
st
s(α)α
1 ≤ i ≤ w(α).
P i = αi r(α) = α∗ i and r(α)αi = αi s(α) = αi for all α ∈ E and st
α
α
=
δ
s(α)
for
fixed
1
≤
i,
j
≤
max{w(α)
|
α
∈
E
, s(α) = v}.
st
P{α∈E |s(α)=v} i j ∗ ij0
0
st
1≤i≤max{w(α),w(α0 )} αi αi = δαα0 r(α), for all α, α ∈ E .
Here the ring R commutes with the generators {v, α, α∗ | v ∈ E 0 , α ∈ E 1 }. Also in relations (3) and (4),
we set αi and αi∗ zero whenever i > w(α). When the coefficient ring R is clear from the context, we simply
write L(E, w) instead of LR (E, w). In this note the coefficient ring is always a field K.
Example 3.3. We compare the relations of the weighted Leavitt path algebra L(E, w) and the usual Leavitt
path algebras L(E 0 ) and L(E 00 ) in the following:
α1 ,α2
(E, w) =
(
6v
u
β1 ,β2
E0
=
u
α2
α1
β1
β2
(
6? v
α1
E 00
=
u
β1
(
6v
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
α1 α1∗ + β1 β1∗ = u
α1 α1∗ + α2 α2∗ + β1 β1∗ + β2 β2∗ = u
α1 α1∗ + β1 β1∗ = u
α2 α2∗ + β2 β2∗ = u
α1∗ α1 = α2∗ α2 = β1∗ β1 = β2∗ β2 = v
α1∗ α1 = β1∗ β1 = v
α1 α2∗ + β1 β2∗ = 0
αi∗ αj = βi∗ βj = 0 if i 6= j
α1∗ β1 = β1∗ α1 = 0
α2 α1∗ + β2 β1∗ = 0
αi∗ βj = βj∗ αi = 0 for all i, j
13
α1∗ α1 + α2∗ α2 = v
β1∗ β1 + β2∗ β2 = v
α1∗ β1 + α2∗ β2 = 0
β1∗ α1 + β2∗ α2 = 0
Note that in L(E, w), relations (3) and (4) in Definition 3.2 amounts to
∗
∗
α1 β1
α1 α2∗
u 0
α1 α2∗
α1 β1
v 0
=
and
=
.
α2 β2
β1∗ β2∗
0 u
β1∗ β2∗
α2 β2
0 v
Example 3.4. Leavitt Path Algebras.
Let the weight map w : E st → N be the constant function w(α) = 1 for any α ∈ E st . Then LR (E, w) is the
usual Leavitt path algebra (with the coefficients in the ring R) as defined in [1] and [7].
Example 3.5. The Leavitt algebra of type (n, k).
Let R be a division ring. For positive integers n and k, let the structured edges E st of a graph E consist of
n + k loops, i.e., s(y) = r(y) for y ∈ E st and let the weight function be the constant map w(y) = n for all
y ∈ E st . We visualize this data as follows:
y1,n+k ,...,yn,n+k
D •Q e
y11 ,...,yn1
y12 ,...,yn2
y13 ,...,yn3
Then the weighted Leavitt path algebra associated to E, LR (E, w), is the algebra L(n, k + 1) constructed by
Leavitt (see §2.3).
To recover Leavitt’s algebra from Definition 3.2 (and to arrive to his notations), let E st = {y1 , . . . , yn+s }
∗ =x
be the structured edges and denote (ys )r = yrs ∈ E 1 , for 1 ≤ r ≤ n and 1 ≤ s ≤ n + k. Denote yrs
sr and
arrange the y’s and x’s in the matrices X and Y as in (15). Then Condition (3) of Definition 3.2 precisely
says that Y · X = In,n and Condition (4) is equivalent to X · Y = In+k,n+k which is how Leavitt defines his
algebra.
Definition 3.6. A weighted rose with k-petals, is a weighted graph which consists of one vertex, k structured
loops and the weight map w. We denote this weighted graph by L(k,w) . If w is a constant map 1, then the
weighted rose reduces to the usual Lk , that is a graph with one vertex and k loops (see Example 3.4) and if
there are n + k − 1 petals and w is the constant map assigning n to each petal, then the wLPA of L(n+k−1,w)
is the Leavitt algebra of type (n, k − 1), denoted by L(n, k) (see Example 3.5). The weighted Leavitt path
algebra of L(n,w) with the coefficients in R is denoted by LR (n, w)
Definition 3.7. A weighted polycephaly graph E consists of an unweighted finite acyclic graph E1 with sinks
{v1 , . . . , vt } together with weighted ns -petal graphs L(ns ,ws ) , 1 ≤ s ≤ t, attached to vs , where ns ∈ N. If
ns = 0 for all 1 ≤ s ≤ t, then the graph is finite acyclic. Note that by definition the weighted map w of E is
w(α) = 1 if α ∈ E1st and w(α) = ws (α) if α ∈ Lst
(ns ,ws ) .
4. K0 of (weighted) Leavitt path algebras
Definition 4.1. Let E be an acyclic graph with a unique sink and F a weighted graph. Denote by E ⊗ F a
graph obtained by attaching E from the sink to each vertex of F .
Example 4.2. Consider the graphs
14
R. HAZRAT
/•
uu
u
u
zu
•
•D
E:
•
DD
D! •
y11 ,...,yn1
/•
D
F :•
and
/ • qe
D
/•
y12 ,...,yn2
y1,n+k ,...,yn,n+k
Then the graph E ⊗ F is
/•
~
~
~
/•
~
~
~
/•
~
~
~
/•
~
~
~
•
•
•
•
•@ •
•@ •
•@ •
•@ •
@@
@@
/•
D
•
@@
/•
y11 ,...,yn1
@@
/ • qe
D
y12 ,...,yn2
y1,n+k ,...,yn,n+k
Lemma 4.3. Let E be an acyclic graph with a unique sink and F a weighted graph. Then
L(E ⊗ F ) ∼
=gr L(E) ⊗K L(F ).
Proof. Let u be the sink of the acyclic graph E and let {pi | 1 ≤ i ≤ n} be the set of all paths in E which end
in u. Then in E ⊗ F for each v ∈ F 0 , by definition, we have one copy of the above set with the range v, which
we denote them by {pvi | 1 ≤ i ≤ n}. Recall from [15, Thm. 2.23], that the set of pi p∗j , 1 ≤ i 6= j ≤ n(v), is the
basis of L(E) and
L(E) ∼
=gr Mn(v) (K)(|p1 |, . . . , |pn(v) |).
s(x)
r(x) ∗
Define φ : L(E) ⊗K L(F ) → L(E ⊗ F ) by φ(pi p∗j ⊗ x) = pi xpj
, where x is a word in L(F ) and extend
it linearly. Since the set of pi p∗j is a basis of L(E) and pvj ∗ pw
=
δ
jt δvw v, it is an easy exercise to see that
t
this map is well-defined, and is a K-algebra isomorphism. Now considering the grading defined on the tensor
product of graded algebras (see 8), the map φ is clearly graded as well.
Example 4.4. If E and F are acyclic graphs with unique sinks, then since tensor products in the level of
algebras commute, we have
L(E ⊗ F ) ∼
=gr L(E) ⊗K L(F ) ∼
=gr L(F ) ⊗K L(E) ∼
=gr L(F ⊗ E).
For example, if E is •
are graded isomorphic.
/ • and F is •
/•
/ • then the Leavitt path algebras of the following two graphs
•
E⊗F :
•
•
•
/•
/•
•
F ⊗E :
•
•
•
/•
•
In fact by [15, Thm. 2.23], the Leavitt path algebras of both graphs are M6 (K)(0, 1, 1, 2, 2, 3).
Example 4.5. Consider the oriental line graphs E1 and E2 with n and k + 1 vertices respectively and the
graph E consisting of n(k + 1) loops, all of weight n. Then by Lemma 4.3
y11 ,...,yn1
E1 ⊗ E :
•
/•
/•
/ • qe
D
y12 ,...,yn2
y1,n(k+1) ,...,yn,n(k+1)
and L(E1 ⊗ E) ∼
= L(E1 ) ⊗ L(E) = Mn (A), where A = Vn,n(k+1) . Furthermore E2 ⊗ E1 ⊗ E is
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
•
•
•
•
•
•
•
•
•
•
•
•
•
/•
/•
15
y11 ,...,yn1
/ • qe
D
y12 ,...,yn2
y1,n+k ,...,yn,n+k
and L(E2 ⊗ E1 ⊗ E) ∼
= L(E2 ) ⊗ L(E1 ⊗ E) = Mk+1 (A) ⊗ Mn (A) ∼
= Mn(k+1) (A). But because A = Vn,n(k+1) ,
n
n(k+1)
∼
we have A = A
. Considering the endomorphism rings, we get
Mn (A) ∼
= Mn (Mk+1 (A)).
= Mn(k+1) (A) ∼
But A is not isomorphic to Mk+1 (A) as the first ring is a domain. This gives an example of two rings R and
S such that Mn (R) ∼
6 S (see [9, p.225]).
=
= Mn (S) for some n, but R ∼
For an abelian monoid M , we denote by M + the group completion of M . This gives a left adjoint functor
to the forgetful functor from the category of abelian groups to abelian monoids. Let
P F be a free abelian
monoid generated by a countable set X. The nonzero elements of F can be written as nt=1 xt where xt ∈ X.
Let ri , si , i ∈ I ⊆ N, be elements of F . We define an equivalence
relation on
P
P F denoted by hri = si | i ∈ Ii
as follows: Define a binary operation → on F \{0}, ri + nt=1 xt → si + nt=1 xt , i ∈ I and generate the
equivalence relation on F using this binary operation. Namely, a ∼ a for any a ∈ F and for a, b ∈ F \{0},
a ∼ b if there is a sequence a = a0 , a1 , . . . , an = b such that for each t = 0, . . . , n − 1 either at → at+1 or
at+1 → at . We denote the quotient monoid by F/hri = si | i ∈ Ii. Then one can see that there is a canonical
group isomorphism
+
F+
F
∼
.
(22)
=
hri = si | i ∈ Ii
hri − si | i ∈ Ii
0
0
Definition 4.6. Let (E, w) be a weighted graph and N 0 be the adjacency matrix (nij ) ∈ ZE ⊕E where nij
is the number of structured edges from vi to vj . Furthermore let Iw0 be the weighted identity matrix defined
as (aij ) where aij = 0 for i 6= j and aii = nvi , where for every v ∈ E 0 that emits edges nv = max{w(α) | α ∈
E st , s(α) = v} and zero otherwise. Let N and Iw be the matrices obtained from N 0 and Iw0 by removing the
rows corresponding to sinks respectively.
Clearly the adjacency matrix depends on the ordering we put on E 0 . We usually fix an ordering on E 0
such that the elements of E 0 \ Sink appear first in the list follow with elements of the Sink.
0
0
Multiplying the matrix N t − Iw from left defines a homomorphism ZE \ Sink −→ ZE0 where ZE \ Sink and
Z are the direct sum of copies of Z indexed by E 0 \ Sink and E 0 , respectively. The next theorem shows
that the cokernel of this map gives the Grothendieck group of weighted Leavitt path algebras. Clearly, for
the graph with weights one (unweighted), this recovers K0 group of Leavitt path algebras.
E0
Theorem 4.7. Let (E, w) be a weighted graph and L(E, w) a weighted Leavitt path algebra. Then
0
0
K0 (L(E, w)) ∼
= coker N t − Iw : ZE \ Sink −→ ZE .
(23)
Proof. Let ME be the abelian monoid generated by {v | v ∈ E 0 } subject to the relations
X
nv v =
r(α),
(24)
{α∈E st |s(α)=v}
for every v ∈ E 0 \ Sink, where nv = max{w(α) | α ∈ E st , s(α) = v}. Arrange E 0 in a fixed order such that the
elements of E 0 \ Sink appear first in the list follow with elements of Sink. The relations (24) can be written
as N t v i = Iw v i where vi ∈ E 0 \ Sink and v i is the (0, . . . , 1, 0, . . . ) with 1 in i-th component. Therefore,
ME ∼
=
hN t v
i
F
,
= Iw v i , vi ∈ E 0 \ Sinki
16
R. HAZRAT
where F is the free abelian monoid generated by the vertices of E. By Theorem 4.19 in [15] there is a natural
monoid isomorphism V(LK (E, w)) ∼
= ME . So using (22) we have,
K0 (L(E, w)) ∼
=
= V(LK (E, w))+ ∼
= ME+ ∼
F+
.
h(N t − Iw )v i , vi ∈ E 0 \ Sinki
0
Now F + is ZE and it is easy to see that the denominator in (25) is the image of N t −Iw : ZE
(25)
0 \ Sink
0
−→ ZE . Example 4.8. Consider the following weighted graph where all the edges have weight one except the top
right edge which has weight two.
2
E:
-1 •
W
•Q qe
2 1
1 0
0 0
Then we have N =
and Iw =
. Therefore the matrix N − Iw is equivalent to
.
1 3
0 2
0 1
Thus by Theorem 4.7, K0 (L(E, w)) ∼
= Z.
Example 4.9. Consider the following weighted graph consisted of an acyclic graph (of weight one) and the
Leavitt algebra Vn,n+k attached to the only sink of the acyclic graph.
•>
E:
•
@ • >>
>>
>>
>
q
•
@ >>
@D • e
>>
•
y11 ,...,yn1
y12 ,...,yn2
y1,n+k ,...,yn,n+k
By Lemma 4.3, L(E, w) ∼
= M9 (Vn,n+k ). Therefore by Morita theory and Theorem 4.7,
K0 (L(E, w)) ∼
= K0 (Vn,n+k ) ∼
= Z/kZ.
Example 4.10. Here is an example of a ring A such that K0 (A) 6= 0 but [A] = 0. Let R = Vn,n+k where
k ≥ 2. By Theorem 4.7, K0 (R) = Z/kZ and k[R] = 0. The ring A = Mk (R) is Morita equivalent to R
(using the assignment Mk (R) P 7→ Rk ⊗Mk (R) P ). Thus K0 (A) ∼
= K0 (R) ∼
= Z/kZ. Under this assignment,
[A] = [Mk (R)] is sent to k[R] which is zero, thus [A] = 0.
5. Graded K0 of Leavitt path algebras
For a strongly graded ring R, Dade’s Theorem implies that K0gr (R) ∼
= K0 (R0 ) (§2.6). In [15], we determined
the strongly graded Leavitt path algebras. In particular we proved
Theorem 5.1 ([15], Thm. 3.12). Let E be a finite graph. The Leavitt path algebra L(E) is strongly graded if
and only if E does not have a sink.
This theorem shows that many interesting classes of Leavitt path algebras fall into the category of strongly
graded LPAs, such as purely infinite simple LPAs.
The following theorem shows that there is a close relation between graded K0 and the non-graded K0 of
strongly graded LPAs.
Theorem 5.2. Let E be a row-finite graph. Then there is an exact sequence
N t −I
K0 (L(E)0 ) −→ K0 (L(E)0 ) −→ K0 (L(E)) −→ 0.
(26)
Furthermore if E is a finite graph with no sinks, the exact sequence takes the form
N t −I
K0gr (L(E)) −→ K0gr (L(E)) −→ K0 (L(E)) −→ 0.
(27)
In this case,
K0gr (L(E)) ∼
ZE
= lim
−→
0
Nt
0
Nt
0
Nt
of the inductive system ZE −→ ZE −→ ZE −→ · · · .
0
(28)
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
17
0
Proof. First let E be a finite graph. Write E 0 = V ∪ W where W is the set of sinks and
S∞ V = E \W . The
ring L(E)0 can be described as follows (see the proof of Theorem 5.3 in [7]): L(E)0 = n=0 L0,n , where L0,n
is the linear span of all elements of the form pq ∗ with r(p) = r(q) and |p| = |q| ≤ n. The transition inclusion
L0,n → L0,n+1 is the identity on all elements pq ∗ where r(p) ∈ W and is to identify pq ∗ with r(p) ∈ V by
X
pα(qα)∗ .
{α|s(α)=v}
Note that since V is a set of vertices which are not sinks, for any v ∈ V the set {α|s(α) = v} is not empty.
For a fixed v ∈ E 0 , let Lv0,n be the linear span of all elements of the form pq ∗ with |p| = |q| = n and
r(p) = r(q) = v. Arrange the paths of length n with the range v in a fixed order pv1 , pv2 , . . . , pvknv , and observe
that the correspondence of pvi pvj ∗ to the matrix unit eij gives rise to a ring isomorphism Lv0,n ∼
= Mknv (K).
v
0
w
Furthermore, L0,n , v ∈ E and L0,m , m < n and w ∈ W form a direct sum. This implies that
L0,n ∼
=
M
Mknv (K) ×
v∈V
M
n−1
M
Mknv (K) ×
M
v∈W
m=0
v∈W
v (K)
M km
,
v , v ∈ E 0 and 0 ≤ m ≤ n, is the number of paths of length m with the range v. The inclusion map
where km
L0,n → L0,n is then identity on factors where v ∈ W and is
Nt :
M
Mknv (K) →
v∈V
M
v
Mkn+1
(K) ×
v∈V
M
v
Mkn+1
(K),
(29)
v∈W
L
where N is the adjacency matrix of the graph E (see Definition 4.6). This means (A1 , . . . , Al ) ∈ v∈V Mknv (K)
Pl
Pl
L
v
is sent to ( j=1 nj1 Aj , . . . , j=1 njl Aj ) ∈ v∈E 0 Mkn+1
(K), where nji is the number of edges connecting vj
to vi and


A1
l


X
A1


kj Aj = 

..


.
j=1
Al
in which each matrix is repeated kj times down the leading diagonal and if kj = 0, then Aj is omitted. This
can be seen as follows: for v ∈ E 0 , arrange the the set of paths of length n + 1 with the range v as
{pv11 α1v1 v , . . . , pvk1v1 α1v1 v , pv11 α2v1 v , . . . , pvk1v1 α2v1 v , . . . , pv11 αnv11vv , . . . , pvk1v1 αnv11vv ,
pv12 α1v2 v , . . . , p
n
v2
v
kn2
α1v2 v , pv12 α2v2 v , . . . , p
n
v2
v
kn2
α2v2 v , . . . , pv12 αnv22vv , . . . , p
n
v2
v
kn2
(30)
αnv22vv ,
...}
where {pv1i , . . . , pvkivi } are paths of length n with the range vi and {α1vi v , . . . , αnviivv } are all the edges with
n
i
the source vi ∈ V and range v. Now this shows that if pvsi (pvt i )∗ ∈ Lv0,n
(which corresponds to est ), then
Pniv vi vi v v1 vi v ∗
v
j=1 ps αj (pt αj ) ∈ L0,n+1 corresponds to the unit matrix est which is repeated down the diagonal niv
times.
Writing L(E)0 = limn L0,n , since K0 respects the direct limit, we have K0 (L(E)0 ) ∼
K (L ).
= lim
−→
−→n 0 0,n
Since K0 of (Artinian) simple algebras are Z, the ring homomorphism L0,n → L0,n+1 induces the group
homomorphism
ZV × ZW ×
n−1
M
m=0
N t ×I
ZW −→ ZV × ZW ×
n
M
ZW ,
m=0
t B
where N t : ZV → ZV × ZW is multiplication of the matrix N t =
from left which is induced by the
Ct
L
Ln
W →
W is the identity map. Consider the commutative
homomorphism (29) and In : ZW × n−1
m=0 Z
m=0 Z
18
R. HAZRAT
diagram
/ K0 (L0,n+1 )
K0 (L0,n )
∼
=
/ ···
∼
=
L
t
L
n
W
W N ⊕In/ ZV × ZW ×
ZV × ZW ×  n−1
Z
 _ m=0 Z
_ m=0
N t ⊕I∞
ZV × Z W × Z W × · · ·
/ ···
/ ZV × Z W × Z W × · · ·
/ ···
where

N t ⊕ I∞



=


Bt
Ct
0
0
0
0
1
0
0
0
0
1
0
0
0
0
·
·
·
·
·
·
·
·
..
.




.


It is easy to see that the direct limit of the second row in the diagram is isomorphic to the direct limit of the
third row. Thus K(L(E)0 ) ∼
(ZV × Z W × Z W × · · · ).
= lim
−→
Consider the commutative diagram on the left below which induces a natural map, denoted by N t again,
on the direct limits.
ZV × ZW × · · ·
Nt
/ ZV × ZW × · · ·
Nt
ZV × ZW × · · ·
Nt
Nt
/ ···
K (L(E)0 )
lim ∼
−→ = 0
Nt
/ ZV × ZW × · · ·
Nt
/ ···
Nt
K (L(E)0 )
lim ∼
−→ = 0
Now Lemma 7.15 in [23] implies that
N t −1
N t −1
coker K0 (L(E)0 ) −→ K0 (L(E)0 ∼
= coker(ZV × ZW × · · · −→ ZV × ZW × · · · ).
And Lemma 3.4 in [22] implies that the
t
t
N −1
N −1
coker(ZV × ZW × · · · −→ ZV × ZW × · · · ) ∼
= coker(ZV −→ ZV × ZW ).
(31)
N t −1
Finally by Theorem 4.7, replacing coker(ZV −→ ZV ×ZW ) by K0 (L(E)) in (31) we obtain the exact sequence
N t −I
K0 (L(E)0 ) −→ K0 (L(E)0 ) −→ K0 (L(E)) −→ 0
(32)
when the graph E is finite. Now a row-finite graph can be written as a direct limit of finite complete
subgraphs [7, Lemmas 3.1, 3.2], i.e., E = lim Ei , L(E) ∼
L(Ei ) and L(E)0 ∼
L(Ei )0 . Since the direct
= lim
= lim
−→
−→
−→
limit is an exact functor, applying lim to the exact sequence (32) for Ei , we get the first part of the theorem.
−→
When the graph E is finite with no sinks, then L(E) is strongly graded (Theorem 5.1). Therefore
K0gr (L(E)) ∼
= K0 (L(E)0 ).
This gives the exact sequence (27).
Remark 5.3. Let E be a row finite graph. The K0 of graph C ∗ -algebra of E was first computed in [22], where
Raeburn and Szymański obtained the top exact sequence of the following commutative diagram. Here E ×1 Z
is a graph with (E ×1 Z)0 = E 0 × Z, (E ×1 Z)1 = E 1 × Z and s(e, n) = (s(e), n − 1)) and r(e, n) = (r(e), n)).
Looking at the proof of [22, Theorem 3.2], and Theorem 5.2, one can see that, K0 C ∗ (E ×1 Z) ∼
= K0 (LC (E)0 )
and the following diagram commutes.
0
/ K1 (C ∗ (E))
0
/ ker(1 − N t )
−1
/ K0 C ∗ (E ×1 Z) 1−β∗ / K0 C ∗ (E ×1 Z)
∼
=
/ K0 (LC (E)0 )
1−N t
∼
=
/ K0 (LC (E)0 )
/ K0 (C ∗ (E))
/0
/ K0 (LC (E))
/ 0.
(33)
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
19
This immediately shows that for a row-finite graph E, K0 (C ∗ (E)) ∼
= K0 (LC (E)). This was also proved in
[7, Theorem 7.1]. In particular if E is a finite graph with no sinks, then
gr
K0 C ∗ (E ×1 Z) ∼
= K0 (LC (E)).
The graded structure of Leavitt path algebras of acyclic, Cn -comet and polycephaly graphs were classified
in [15]. This, coupled by the graded Morita theory and (18) enable us to calculate graded K0 of these graphs.
We record them here.
Theorem 5.4 (K0gr of acyclic graphs). Let E be a finite acyclic graph with sinks {v1 , . . . , vt }. For any sink vs ,
let {pvi s | 1 ≤ i ≤ n(vs )} be the set of all paths which end in vs . Then there is a Z[x, x−1 ]-module isomorphism
t
M
K0gr (L(E)), [L(E)] ∼
Z[x, x−1 ], (d1 , . . . , dt ) ,
=
(34)
s=1
where ds =
Pn(vs )
i=1
vs
x−|pi | .
Proof. By [15, Theorem 3.23],
L(E) ∼
=gr
t
M
s
Mn(vs ) (K) |pv1s |, . . . , |pvn(v
| .
s)
(35)
s=1
Since K0gr is an exact functor, K0gr (L(E)) is isomorphic to the direct sum of K0gr of matrix algebras of (35).
The graded Morita equivalence (Proposition 2.5), induces a Z[x, x−1 ]-isomorphism
s
K0gr Mn(vs ) (K) |pv1s |, . . . , |pvn(v
| −→ K0gr (K),
s)
with
s
s
Mn(vs ) (K) |pv1s |, . . . , |pvn(v
|
7→ [K(−|pv1s |) ⊕ . . . , ⊕K(−|pvn(v
|)].
s)
s)
Now the theorem follows from Proposition 2.8, by considering K as a graded field concentrated in degree zero,
i.e., ΓK = 1, and Γ = Z.
Leavitt path algebras of Cn -comets were classified in [15, Theorem 3.15, Corollary 3.18] as follows: Let E
be a Cn -comet with the cycle C of length n ≥ 1. Let u be a vertex on the cycle C. Eliminate the edge in the
cycle whose source is u and consider the set {pi | 1 ≤ i ≤ m} of all paths which end in u. Then
L(E) ∼
(36)
=gr Mm K[xn , x−n ] |p1 |, . . . , |pm | .
Let dl , 0 ≤ l ≤ n − 1, be the number of i such that |pi | represents l in Z/nZ. Then
L(E) ∼
=gr Mm K[xn , x−n ] 0, . . . , 0, 1, . . . , 1, . . . , n − 1, . . . , n − 1 ,
(37)
where 0 ≤ l ≤ n − 1 occurring dl times. It is now easy to see
L(E)0 ∼
= Md0 (K) × · · · × Mdn−1 (K).
(38)
Furthermore, let F be another Cn0 -comet with the cycle C 0 of length n0 ≥ 1 and u0 be a vertex on the
cycle C 0 . Eliminate the edge in the cycle whose source is u0 and consider the set {qi | 1 ≤ i ≤ m0 } of all paths
which end in u0 . Then L(E) ∼
=gr L(F ) if and only if n = n0 , m = m0 and for a suitable permutation π ∈ Sm ,
and r ∈ N, r + |pi | + nZ = |qπ(i) | + nZ, where 1 ≤ i ≤ m.
Theorem 5.5 (K0gr of Cn -comet graphs). Let E be a Cn -comet with the cycle C of length n ≥ 1. Then there
is a Z[x, x−1 ]-module isomorphism
n−1
M
K0gr (L(E)), [L(E)] ∼
Z, (d0 , . . . , dn−1 ) ,
=
i=0
where dl , 0 ≤ l ≤ n − 1, are as in (37).
(39)
20
R. HAZRAT
gr
Proof. By (36), K0gr (L(E)) ∼
=gr K0 Mm K[xn , x−n ] |p1 |, . . . , |pm | . The graded Morita equivalence (Proposition 2.5), induces a Z[x, x−1 ]-isomorphism
K0gr Mm K[xn , x−n ] |p1 |, . . . , |pm | −→ K0gr (K[x, x−n ]),
with
Mm K[xn , x−n ] |p1 |, . . . , |pm | 7→ K[xn , x−n ](−|p1 |) ⊕ . . . , ⊕K[xn , x−n ](−|pm |) .
Now the theorem follows from Proposition 2.8, by considering K as a graded field with ΓK = nZ and Γ = Z.
Theorem 5.6 (K0gr of polycephaly graphs). Let E be a polycephaly graph consisting of an acyclic graph E1
with sinks {v1 , . . . , vt } which are attached to Ln1 , . . . , Lnt , respectively. For any vs , let {pvi s | 1 ≤ i ≤ n(vs )}
be the set of all paths in E1 which end in vs . Then there is a Z[x, x−1 ]-module isomorphism
t
M
K0gr (L(E)), [L(E)] ∼
Z[1/ns ], (d1 , . . . , dt ) ,
=
(40)
s=1
where ds =
Pn(vs )
i=1
−|pvi s |
ns
.
Proof. By [15, Theorem 3.21]
L(E) ∼
=gr
t
M
s
Mn(vs ) L(1, ns ) |pv1s |, . . . , |pvn(v
| .
s)
(41)
s=1
Since K0gr is an exact functor, K0gr (L(E)) is isomorphic to the direct sum of K0gr of matrix algebras of (41).
The graded Morita equivalence (Proposition 2.5), induces a Z[x, x−1 ]-isomorphism
s
K0gr Mn(vs ) (L(1, ns )) |pv1s |, . . . , |pvn(v
| −→ K0gr (L(1, ns )),
s)
s
s
with Mn(vs ) (L(1, ns )) |pv1s |, . . . , |pvn(v
| 7→ [L(1, ns )n(vs ) (−|pv1s |, . . . , −|pvn(v
|)]. Now by Theorem 5.2(28),
s)
s)
gr
gr
∼
K0 (L(1, ns )) = Z[1/ns ]. Observe that [L(1, ns )] represents 1 in K0 (L(1, ns )). Since for any i ∈ Z,
M
L(1, ns )(i) ∼
L(1, ns )(i − 1),
=
ns
Pn(vs ) −|pvi s |
s
it immediately follows that L(1, ns )n(vs ) (−|pv1s |, . . . , −|pvn(v
|)
maps
to
in Z[1/ns ]. This also
i=1 ns
)
s
−1
shows that the isomorphism is a Z[x, x ]-module isomorphism.
Theorem 5.7. Let E and F be acyclic, comet or polycephaly graphs. Then L(E) ∼
=gr L(F ) if and only if
−1
there is a Z[x, x ]-module isomorphism
gr
K gr (L(E)), [L(E)] ∼
= K (L(F )), [L(F )] .
0
0
Proof. One direction is straightforward. For the other direction, suppose there is an order-preserving Z[x, x−1 ]isomorphism φ : K0gr (L(E)) → K0gr (L(F )) such that φ([L(E)]) = [L(F )]. Comparing K0gr -groups, Theorems 5.4, 5.5 and 5.6 immediately imply that E and F have to be of the same type, i.e., both either acyclic,
comet or polycephaly. We consider each case separately.
(i) E and F are acyclic: In this case L(E) and L(F ) are K-algebras (see 35). The isomorphism between
Leavitt path algebras now follows from Theorem 2.12, by setting A = K and ΓA = 1.
(ii) E and F are comets: If E and F are Cn and Cn0 -comets, respectively, then by Theorem 5.5,
Ln−1
gr
gr
∼ Ln0 −1 Z. Tensoring with Q, the isomorphism φ extends to an
K0 (L(E)) ∼
=
i=0 Z and K0 (L(F )) =
i=0
isomorphism of Q-vector spaces, which then immediately implies n = n0 . This shows that L(E) and L(F ) are
both A-algebras where A = K[xn , x−n ] (see 36). The isomorphism between Leavitt path algebras now follows
from Theorem 2.12, by setting ΓA = nZ and Γ = Z.
Lt
Lt0
gr
(iii) E and F are polycephaly graphs: By (40) K0gr (L(E)) ∼
= s=1 Z[1/ns ] and K0 (L(F )) ∼
= s=1 Z[1/n0s ].
Tensoring with Q, the isomorphism φ extends to an isomorphism of Q-vector spaces, which then immediately
implies t = t0 . Since the group of n-adic fractions Z[1/n] is a flat Z-module, φ can be represented by a
t × t matrix with entries from Q. Write the sequences (n1 , . . . , nt ) and (n01 , . . . , n0t ), in an ascending order,
respectively (changing the labels if necessary). We first show that ns = n0s , 1 ≤ s ≤ t and φ is a block
diagonal matrix. Write the ascending sequence (n1 , . . . , nt ) = (n1 , . . . , n1 , n2 , . . . , n2 , . . . , nk , . . . , nk ), where
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
21
each nl occurring rl times, 1 ≤ l ≤ k. Let φ be represented by the matrix a = (aij )t×t . Choose 1 ≤ s1 ≤ t such
that starting from s1 -th row all entires below the s1 -th row and between the 1st and r1 -th column (including
the column r1 ) are zero. Clearly s1 ≥ 1, as there is at least one nonzero entry on the r1 -th column, as φ
is an isomorphism so a is an invertible matrix. Formally, s1 is chosen such that aij = 0 for all s1 < i ≤ t
and 1 ≤ j ≤ r1 and there is 1 < j ≤ r1 such that as1 j 6= 0. We will show that s1 = r1 . Pick as1 j 6= 0, for
P
L
L
1 < j ≤ r1 . Then φ(ej ) = ts=1 asj e0s , where es ∈ ts=1 Z[1/ns ] and e0s ∈ ts=1 Z[1/n0s ], with 1 in s-th entry
and zero elsewhere, respectively. This shows as1 j e0s1 ∈ Z[1/n0s1 ]. But φ is a Z[x, x−1 ]-module homomorphism.
Thus
t
t
X
X
0
0
ns asj es = xφ(ej ) = φ(xej ) = φ(n1 ej ) =
n1 asj e0s .
(42)
Since as1 j 6= 0, we obtain
below).
s=1
n0s1 =
s=1
n1 . Next we show aij = 0 for all 1 ≤ i ≤ s1 and r1 < j ≤ t (see the matrix
a11 · · ·
..
 ..
.
 .

···
 a
a =  s1 ,1
0
 0
 .
..
 ..
.
0
0
a1r1
..
.

0
···
0

0
0
0
0








as1 ,r1
0
..
.
as1 +1,r1 +1
..
.
···
···
···
..
.
0
at,r1 +1
···
as1 +1,t
0
at,t
.
(43)
t×t
Suppose aij 6= 0 for some 1 ≤ i ≤ s1 and r1 < j ≤ t. Since j > r1 , nj > n1 . Applying φ on ej , a similar
calculation as (42) shows that, since aij 6= 0, n0i = nj . But the sequence (n01 , . . . , n0t ) is arrange in an ascending
order, so if i ≤ s1 , then nj = n0i ≤ n0s1 = n1 , which is a contradiction. Therefore the matrix a, consists of
two blocks as in (43). Next we show for 1 ≤ i ≤ s1 n0i = n0s1 = n1 . If 1 ≤ i ≤ s1 , there is aij 6= 0, for some
1 ≤ j ≤ r1 (as all aij = 0, j > r1 , and a is invertible). Then consider φ(ej ) and again a similar calculation as
in (42) shows that n0i = n1 .
Now restricting φ to the first r1 summands of K0gr (L(E)), we have a homomorphism
φr1 :
r1
M
s=1
Z[1/n1 ] −→
s1
M
Z[1/n1 ].
(44)
s=1
Being the restriction of the isomorphism φ, φr1 is also an isomorphism. Tensoring (44) with Q, we deduce
s1 = r1 as claim. Repeating the same argument (k − 1 times) for the second block of the matrix a in (43), we
obtain that a is a diagonal block matrix consisting of k blocks of ri ×ri , 1 ≤ i ≤ k matrices. Since φ is an order
preserving homomorphism, all the entires of the matrix a are positive numbers. Next we show each of φri is a
permutation matrix. Consider the matrix ar1 = (aij ), 1 ≤ i, j ≤ r1 , representing the isomorphism φr1 . Since
φr1 is an isomorphism, ar1 is an invertible matrix, with the inverse br1 = (bij )P
with bij ≥ 0, 1 ≤ i, j ≤ r1 , as
1
br1 is also order preserving. Thus there is 1 ≤ j ≤ r1 such that a1j 6= 0. Since ri=1
a1i bil = 0, for 1 < l ≤ r1 ,
a1j 6= 0 and all the numbers involved are positive, it follows bij = 0 for all 1 < i ≤ r1 . That is the entires of
j-th row of the matrix br1 are all zero except b1j , which has to be nonzero. If there is another j 0 6= j such that
a1j 0 6= 0, a similar argument shows that all bij 0 = 0 for all 1 < i ≤ r1 . But this then implies that the matrix
br1 is singular which is not the case. That is there is only one non zero entry in the first row of the matrix
ar1 . Furthermore, since a1j bj1 = 1, a1j = n1 c1 , where c1 ∈ Z. Repeating the same argument for other rows of
the matrix ar1 , we obtain that ar1 consists of exactly one number of the form n1 c , c ∈ Z, in each row and in
column, i.e., it is a permutation matrix with entries invertible in Z[1/n1 ]. As in Theorem 5.6, we can write
L(E) ∼
=gr
t
M
s
| .
Mn(vs ) L(1, ns ) |pv1s |, . . . , |pvn(v
s)
s=1
Furthermore, one can re-group this into k summands, with each summand matrices over the Leavitt algebras
Lk Lr u
L(1, nu ), 1 ≤ u ≤ k, i.e.,
u=1
s=1 M∗ (L(1, nu )). Do the same re-grouping for the L(F ). On the level
of K0 , since the isomorphism φ decomposes into a diagonal block matrix, it is enough to consider each φri
separately. For i = 1,
r1
r1
M
M
0vs0
0vs0
gr
s
0
φr1 :
K0gr Mn(vs ) R |pv1s |, . . . , |pvn(v
|
−→
K
M
R
|p
|,
.
.
.
,
|p
|
(45)
n(vs )
1
0
n(v 0 )
s)
s
s=1
s=1
22
R. HAZRAT
with
φr1
r1 h
M
r1 h
i M
i
0vs0
0vs0
s
0
Mn(vs ) R |pv1s |, . . . , |pvn(v
|
=
M
|
,
R
|p
|,
.
.
.
,
|p
0
n(vs )
1
n(v )
s)
s
s=1
s=1
where R = L(1, n1 ). Using the graded Morita theory, (as in Theorem 5.6), we can write
r1
r1
M
M
0v 0
0vs0
s
φr1
R(−|pv1s |) ⊕ · · · ⊕ R(−|pvn(v
|)
=
R(−|p1 s |) ⊕ · · · ⊕ R(−|pn(v
0 ) |) .
s)
s
s=1
s=1
But φr1 is a permutation matrix (with power of n1 as entires). Thus we have (by suitable relabeling if
necessary), for any 1 ≤ s ≤ r1 ,
0vs0
0vs0
s
nc1s [R(−|pv1s |) ⊕ · · · ⊕ R(−|pvn(v
(46)
|)
=
R(−|p
|)
⊕
·
·
·
⊕
R(−|p
0 ) |) ,
1
n(v
)
s
s
where cs ∈ Z. Since for any i ∈ Z,
L(1, n1 )(i) ∼
=
M
L(1, n1 )(i − 1),
n1
if cs > 0, we have
0v 0
0vs0
s
R(−|pv1s | + cs ) ⊕ · · · ⊕ R(−|pvn(v
| + cs ) = R(−|p1 s |) ⊕ · · · ⊕ R(−|pn(v
0 ) |) ,
s)
(47)
s
otherwise
0v 0
0vs0
s
R(−|pv1s |) ⊕ · · · ⊕ R(−|pvn(v
|) = R(−|p1 s | + cs ) ⊕ · · · ⊕ R(−|pn(v
0 ) | + cs ) .
s)
(48)
s
Suppose, say, (47) holds. Recall that by Dade’s theorem (see §2.6), the functor (−)0 : gr-A → mod-A0 ,
M 7→ M0 , is an equivalent so that it induces an an isomorphism K0gr (R) → K0 (R0 ). Applying this to (47),
we obtain
R−|pv1s |+cs ⊕ · · · ⊕ R−|pvs |+cs = R 0vs0 ⊕ · · · ⊕ R 0vs0
(49)
−|p1 |
n(vs )
−|p
0)
n(vs
|
in K0 (R0 ). Since R0 is an ultramatricial algebra, it is unit-regular and by Proposition 15.2 in [14],
∼
R vs
⊕ · · · ⊕ R vs
= R 0v0 ⊕ · · · ⊕ R 0v0 .
−|p1 |+cs
−|pn(v ) |+cs
−|p1 s |
s
s
0 )|
n(vs
−|p
(50)
as R0 -modules. Tensoring this with the graded ring R over R0 , since R is strongly graded (Theorem 5.1), and
so Rα ⊗R0 R ∼
=gr R(α) as graded R-modules ([21, 3.1.1]), we have a (right) graded R-module isomorphism
0
0
0v
0vs
s
R(−|pv1s | + cs ) ⊕ · · · ⊕ R(−|pvn(v
| + cs ) ∼
=gr R(−|p1 s |) ⊕ · · · ⊕ R(−|pn(v
0 ) |).
s)
(51)
s
Applying End-functor to this isomorphism, we get (see §2.2),
0v0
0vs0
s
Mn(vs ) L(1, ns ) |pv1s | − cs , . . . , |pvn(v
| − cs ∼
=gr Mn(vs ) L(1, ns ) |p1 s |, . . . , |pn(v
0 )| ,
s)
s
which is a K-algebra isomorphism. But clearly (see §2.2)
vs
vs
s
∼
Mn(vs ) L(1, ns ) |pv1s | − cs , . . . , |pvn(v
|
−
c
M
L(1,
n
)
|p
|,
.
.
.
,
|p
|
.
=
s
gr
s
n(v
)
1
s
)
n(v
)
s
s
So
0vs0
0vs0
s
∼
|,
.
.
.
,
|p
Mn(vs ) L(1, ns ) |pv1s |, . . . , |pvn(v
|
M
L(1,
n
)
|p
=
gr
s
0 )| .
n(v
)
1
s
n(v
)
s
s
Repeating this argument for all 1 ≤ s ≤ r1 and then for all ri , 1 ≤ i ≤ k, we get
L(E) ∼
=gr L(F )
as a K-algebra.
Corollary 5.8. Conjecture 2 holds for the category of graphs consisting of acyclic, comet and polycephaly
graphs.
Proof. Let E be and F be two graphs of the above type such that L(E) ∼
=gr L(F ). As in Part 1 of the proof
of Theorem 5.7, E and F have to be of the same type. This isomorphism then induces
gr
K gr (L(E)), [L(E)] ∼
= K (L(F )), [L(F )] .
0
0
Now if E and F are acyclic or comets, then the combination of part (1) and (2) of Theorem 2.12 shows
that L(E) ∼
=gr L(F ) as K-algebras. (In fact if E and F are Cn -comets, then there is an K[xn , x−n ]-graded
isomorphism.) If E and F are polycephaly graph, then the K-isomorphism of Leavitt path algebras was
proved in Theorem 5.7.
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
23
Remark 5.9. Let E and F be graphs with no sinks such that the weak isomorphism Conjecture 1 holds.
Then C ∗ (E) ∼
= C ∗ (F ) implies LC (E) ∼
=gr LC (F ). Indeed, if C ∗ (E) ∼
= C ∗ (F ), then C ∗ (E) oγ T ∼
= C ∗ (F ) oγ T.
By [22, Lemma 3.1], C ∗ (E ×1 Z) ∼
= C ∗ (E) oγ T. From sequence (33) now follows that
gr
K gr (LC (E)) ∼
= K0 (C ∗ (E ×1 Z)) ∼
= K0 (C ∗ (F ×1 Z)) ∼
= K (LC (F )).
0
0
So if Conjecture 1 holds, we then have LC (E) ∼
=gr LC (F ). Thus by Theorem 5.7, if E and F are comet or
∗
∗
∼
polycephaly graphs and C (E) = C (F ) then LC (E) ∼
=gr LC (F ).
Remark 5.10. In Theorem 5.7, the assumption that the isomorphism between K-groups is Z[x, x−1 ]-module
homomorphism is used in significant way. The following example shows one can not relax this assumption.
Consider R = L2 and S = L4 . By Theorem 5.6, (K0gr (R), [R]) = (Z[1/2], 1) and (K0gr (S), [S]) = (Z[1/4], 1).
gr
So the identity map, gives an isomorphism (K0gr (R), [R]) ∼
6 L4 , since by
= (K0 (S), [S]). But we know L2 ∼
=
Theorem 4.7, K0 (L2 ) = 0 and K0 (L4 ) = Z/3Z. Theorem 5.7 can’t be applied here, as this isomorphism is
not Z[x, x−1 ]-module isomorphism. For, otherwise 2[S(1)] = 2[R(1)] = [R] = [S] = 4[S(1)]. This implies
[R] = 2[S(1)] = 0 which is clearly absurd.
In [5], Abrams gives the necessary and sufficient conditions for matrices over Leavitt path algebras to be
graded isomorphic. For a Γ-graded ring A, he considers grading induced on Mk (A) by setting Mk (A)γ =
Mk (Aγ ), for γ ∈ Γ, i.e., with (0, . . . , 0)-suspension. For positive integers k and n, factorization of k along
n, means k = td where d | ni for some positive integer i and gcd(t, n) = 1. Then Theorem 1.3 in [5] states
that for positive integers k, n, n0 , n ≥ 2, Mk (Ln ) ∼
=gr Mk0 (Ln ) if and only if t = t0 , where k = td, respectively
k 0 = t0 d0 , are the factorization of k, respectively k 0 , along n. With this classification,
M3 (L2 ) ∼
6 gr M4 (L2 ).
=
However combining Theorem 5.6 and Theorem 5.7, we shall see that the following two graphs
•>
E:
>>
>
q
@•e
•
give isomorphic Leavitt path algebras, i.e.,
•>
>>
>
@•
F :
/ • qe
•
M3 (L2 )(0, 1, 1) ∼
=gr M4 (L2 )(0, 1, 2, 2).
(52)
This shows, by considering suspensions, we get much wider classes of graded isomorphisms of matrices
over Leavitt algebras. Indeed, by Theorem 5.6, (K0gr (L(E), [L(E)]) = (Z[1/2], 2) and (K0gr (L(F ), [L(F )]) =
(Z[1/2], 2). Now the idenitity map gives an ordering preserving isomorphisms between these K-groups and so
by Theorem 5.7, we have L(E) ∼
=gr L(F ) which then applying (40) we obtain the graded isomorphism (52).
We can now give a criteria when two matrices over Leavitt algebras are graded isomorphic. It is an easy
exercise to see this covers Abrams’ theorem [5, Proposition 1.3], when there is no suspension.
Theorem 5.11. Let k, k 0 , n, n0 be positive integers, where n is prime. Then
Mk (Ln )(λ1 , . . . , λk ) ∼
=gr Mk0 (Ln0 )(γ1 , . . . , γk0 )
P
P 0
if and only if n = n0 and there is j ∈ Z such that nj ki=1 n−λi = ki=1 n−γi .
(53)
Proof. By Theorem 5.6 (and its proof), the isomorphism (53), induces an Z[x, x−1 ]-isomorphism φ : Z[1/n] →
P
P 0
Z[1/n0 ] such that φ( ki=1 n−λi ) = ki=1 n0 −γi . Now
nφ(1) = φ(n1) = φ(x1) = xφ(1) = n0 φ(1)
which implies n = n0 . (This could have also been obtained by applying non-graded K0 to isomorphism (53)
and applying Theorem 4.7.) Since φ is an isomorphism, it has to be multiplication by nj , for some j ∈ Z.
P
P
P 0
Thus ki=1 n−γi = φ( ki=1 n−λi ) = nj ki=1 n−λi .
P
P 0
Conversely suppose there is j ∈ Z, such that nj ki=1 n−λi = ki=1 n−γi . Set A = L(1, n). Since Leavitt
algebras are strongly graded (Theorem 5.1), it follows R = Mk (A)(λ1 , . . . , λk ) and S = Mk0 (A)(γ1 , . . . , γk0 ) are
strongly graded ([21, 2.10.8]). Again Theorem 5.6 shows that the multiplication by nj gives an isomorphism
gr
(K0gr (R), [R]) ∼
= (K0 (S), [S]). Therefore nj [R] = [S]. Using the graded Morita, we get
nj [A(−λ1 ) ⊕ · · · ⊕ A(−λk )] = [A(−γ1 ) ⊕ · · · ⊕ A(−γk0 )].
24
R. HAZRAT
Passing this equality to A0 using the Dade’s theorem, we get
nj [A−λ1 ⊕ · · · ⊕ A−λk ] = [A−γ1 ⊕ · · · ⊕ A−γk0 ].
If j ≥ 0 (the case j < 0 is argued similarly), since A0 is an ultramatricial algebra, it is unit-regular and by
Proposition 15.2 in [14],
j
j
An−λ1 ⊕ · · · ⊕ An−λk = A−γ1 ⊕ · · · ⊕ A−γk0 .
Tensoring this with the graded ring A over A0 , since A = L(1, n) is strongly graded (Theorem 5.1), and so
Aλ ⊗A0 A ∼
=gr A(λ) as graded A-modules ([21, 3.1.1]), we have a (right) graded R-module isomorphism
j
j
A(−λ1 )n ⊕ · · · ⊕ A(−λk )n = A(−γ1 ) ⊕ · · · ⊕ A(−γk0 ).
L
Since for any i ∈ Z, A(i) ∼
= n A(i − 1). we have
A(−λ1 + j) ⊕ · · · ⊕ A(−λk + j) = A(−γ1 ) ⊕ · · · ⊕ A(−γk0 ).
Now applying End-functor to this isomorphism, we get (see §2.2),
Mk (A)(λ1 − j, . . . , λk − j) ∼
=gr Mk0 (A)(γ1 , . . . , γk0 )
which gives the isomorphism (53).
Remarks 5.12.
(1) The if part of Theorem 5.11 does not need the assumption of n being prime. In fact, the same proof
shows that if Mk (Ln )(λ1 , . . . , λk ) ∼
=gr Mk0 (Ln0 )(γ1 , . . . , γk0 ) then n = n0 and there is j ∈ Z such that
P
P
0
k
k
pδ i=1 n−λi = i=1 n−γi , where p|nj and δ = ±1. By setting k = 1, λ1 = 0 and γi = 0, 1 ≤ i ≤ k 0 ,
we recover Theorem 6.3 of [3].
(2) The proof of Theorem 5.11 shows that the isomorphism (53) is in fact an induced graded isomorphism.
Recall that for graded (right) modules N1 and N2 over the graded ring A, the graded ring isomorphism
Φ : End(N1 ) → End(N2 ) is called induced, if there is a graded A-module isomorphism φ : N1 → N2
which lifts to Φ (see [5, p.4]).
Imitating a similar proof as in Theorem 5.11 we can obtain the following result.
Corollary 5.13. There is a graded L(1, n)-isomorphism
0
L(1, n)k (λ1 , . . . , λk ) ∼
=gr L(1, n)k (γ1 , . . . , γk0 )
if and only if
Pk
i=1 n
λi
=
Pk0
i=1 n
γi .
Theorem 5.14. Let E and F be finite graphs with no sinks. Then the following are equivalent:
gr
(1) K gr (L(E)) ∼
= K (L(F )) as partially ordered groups.
0
0
(2) L(E) is gr-Morita equivalent to L(F ).
(3) L(E)0 is Morita equivalent to L(F )0 .
Proof. The equivalence of the statements follow from Theorems 5.1 and 5.2 and Dade’s Theorem (§2.6) in
combination with Corollary 15.27 in [14] which states that two ultramatricial algebras are Morita equivalent
if their K0 are isomorphism as partially ordered groups.
6. Graded K0 of small graphs
In [2], using a ‘change the graph’ theorem, it was shown that the following graphs give rise to isomorphic
(purely infinite simple) Leavitt path algebras.
E1
9•b
"
•
E2 :
9•b
"
•e
E3 :
-1 • o
•
As it is noted [2], K0 of these graphs are all zero. However this change the graph theorem does not respect
the grading. We calculate K0gr of these graphs and show that the Leavitt path algebras of these graphs are
not graded isomorphic.
THE GRADED GROTHEDIECK GROUP AND LEAVITT PATH ALGEBRAS
Graph E1 : The adjacency matrix of this graph is N =
25
1 1
. By Theorem 5.2
1 0
K0gr (L(E1 )) ∼
Z⊕Z
= lim
−→
Nt
Nt
Nt
of the inductive system Z ⊕ Z −→ Z ⊕ Z −→ Z ⊕ Z −→ · · · . Since det(N ) 6= 0, one can easily see that
h 1 i
h 1 i
lim Z ⊕ Z ∼
Z
⊕Z
.
=
−→
det(N )
det(N )
Thus
K0gr (L(E1 )) ∼
= Z ⊕ Z.
In fact this algebra produces a so called the Fibonacci algebra (see [13, Example IV.3.6]).
1 1
Graph E2 : The adjacency matrix of this graph is N =
. By Theorem 5.2
1 1
1
K0gr (L(E2 )) ∼
Z⊕Z∼
= lim
= Z[ ]
−→
2
Graph E3 : By Theorem 5.6, K0gr (L(E3 )) ∼
= Z[ 21 ].
This shows that L(E1 ) is not isomorphic to L(E2 ) and L(E3 ). Finally we show that L(E2 ) 6∼
=gr L(E3 ).
First note that the graph E2 is the out-split of the graph
E4 :
-1 •
(see [2, Definition 2.6]). Therefore by [2, Theorem 2.8] L(E2 ) ∼
=gr L(E4 ). But by Theorem 5.6
1 K0gr (L(E4 ), [L(E4 )] = Z[ ], 1
2
whereas
1 3
K0gr (L(E3 ), [L(E3 )] = Z[ ], .
2 2
6 gr L(E3 ). Therefore L(E2 ) ∼
6 gr L(E3 ).
Now Theorem 5.7 shows that L(E4 ) ∼
=
=
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Department of Pure Mathematics, Queen’s University, Belfast BT7 1NN, United Kingdom
E-mail address: [email protected]