Skew-Zigzag Algebras
We investigate the skew-zigzag algebras introduced by Huerfano and Khovanov. In particular, we relate moduli spaces of such algebras with the cohomology of the corresponding graph.
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irk-123456789-1477592019-02-16T01:26:15Z Skew-Zigzag Algebras Couture, C. We investigate the skew-zigzag algebras introduced by Huerfano and Khovanov. In particular, we relate moduli spaces of such algebras with the cohomology of the corresponding graph. 2016 Article Skew-Zigzag Algebras / C. Couture // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16G20 DOI:10.3842/SIGMA.2016.062 http://dspace.nbuv.gov.ua/handle/123456789/147759 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We investigate the skew-zigzag algebras introduced by Huerfano and Khovanov. In particular, we relate moduli spaces of such algebras with the cohomology of the corresponding graph. |
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Couture, C. Skew-Zigzag Algebras Symmetry, Integrability and Geometry: Methods and Applications |
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Skew-Zigzag Algebras |
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Skew-Zigzag Algebras |
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Skew-Zigzag Algebras |
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Skew-Zigzag Algebras |
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Skew-Zigzag Algebras |
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skew-zigzag algebras |
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Skew-Zigzag Algebras / C. Couture // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 9 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 062, 19 pages
Skew-Zigzag Algebras
Chad COUTURE
Department of Mathematics and Statistics, University of Ottawa,
585 King Edward Ave, Ottawa, ON K1N 6N5, Canada
E-mail: ccout045@uottawa.ca
Received October 02, 2015, in final form June 17, 2016; Published online June 26, 2016
http://dx.doi.org/10.3842/SIGMA.2016.062
Abstract. We investigate the skew-zigzag algebras introduced by Huerfano and Khovanov.
In particular, we relate moduli spaces of such algebras with the cohomology of the corre-
sponding graph.
Key words: zigzag algebra; path algebra; Dynkin diagram; moduli space; graph cohomology
2010 Mathematics Subject Classification: 16G20
1 Introduction
Zigzag algebras were introduced by Huerfano and Khovanov in [5] in their categorification of the
adjoint representation of simply-laced quantum groups. The zigzag algebra A(Γ) is a quotient of
the path algebra of the double quiver associated to the Dynkin diagram Γ of the quantum group
in question. The Grothendieck group of the category of A(Γ)-modules is then naturally identified
with the weight lattice of g. Zigzag algebras have also recently appeared in categorifications of
the Heisenberg algebra. See [2] and [7, Remark 6.2(c)].
In addition to their importance in categorification, zigzag algebras have a variety of nice
features, as pointed out in [5]. Examples of such features include the following: they have
nondegenerate symmetric trace forms and are quadratic algebras (provided that Γ has at least 3
vertices); if Γ is a finite Dynkin diagram, then A(Γ) is of finite type and its indecomposable
representations are enumerated by roots of g; and if Γ is bipartite, then the quadratic dual of
the A(Γ) is the preprojective algebra of Γ for a sink-source orientation.
Skew -zigzag algebras, also introduced in [5], are a generalisation of zigzag algebras. They
involve coefficients vab,c for vertices a, b, c such that a is connected to both b and c. When all
coefficients are equal to one, they recover the zigzag algebras. The goal of the current paper
is to investigate some important properties of skew-zigzag algebras. We provide proofs of some
results mentioned in [5] without proof, in addition to proving some results that appear to be new.
We begin with a review of the concepts in graph theory necessary for the current paper in
Section 2. In Section 3, we recall the definition of the skew-zigzag algebras, find an explicit
basis for them (see Proposition 3.7), and prove that they are graded symmetric algebras (see
Proposition 3.9). In Section 4, we describe two moduli spaces of skew-zigzag algebras. The
first is the moduli space of skew-zigzag algebras up to isomorphism preserving vertices, which
we show to be isomorphic to the graph cohomology of Γ (see Theorem 4.8), as stated in [5]
without proof (see Remark 4.9). The second is the moduli space of skew-zigzag algebras up to
arbitrary isomorphism, which we show to be isomorphic to the quotient of the graph cohomology
of Γ by a natural action of the automorphism group of Γ (see Theorem 4.12). We should note
here that we consider the moduli space only as a group, and do not consider any geometric
structure. Finally, in Section 5, we discuss an alternate definition of skew-zigzag algebras that
has appeared in the literature. We show that this alternate definition is more restrictive (see
Proposition 5.3).
mailto:ccout045@uottawa.ca
http://dx.doi.org/10.3842/SIGMA.2016.062
2 C. Couture
2 Graph theory background
Recall that a graph Γ is a pair (V,E) where V is a finite set and E is a set consisting of two
element subsets of V . The elements of V are called vertices and the elements of E are called
edges. Note that this implies that we consider graphs with no loops or multiple edges. We will
often depict graphs as diagrams, with a node for each vertex and curves between nodes a and b
if {a, b} ∈ E. If there exist subsets A,B ⊆ V such that A t B = V and each element of E
contains one element of A and one element B, then Γ is said to be bipartite.
A quiver, Q, is a four-tuple (Q0,Q1, s, t) where Q0 and Q1 are both finite sets and s, t
are set maps from Q1 to Q0. The elements of Q0 are again called vertices and the elements
of Q1 are called directed edges. For each directed edge f , we call s(f) and t(f) the source
and target of f (respectively). We will often denote a directed edge with source a and target b
by (a | b). Throughout this paper we will consider quivers with no parallel edges, i.e., no directed
edges f1, f2 such that s(f1) = s(f2) and t(f1) = t(f2).
Example 2.1 (a graph and a quiver). The leftmost diagram below represents the graph (V,E)
with V = {a, b, c, d} and E = {{a, b}, {b, c}, {a, c}, {a, d}}. The rightmost diagram depicts the
quiver Q = (Q0,Q1, s, t) with Q0 = {a, b, c, d}, Q1 = {(a | b), (a | d), (b | c), (c | d), (d | b)} and
s, t : Q1 → Q0 given by
s((a | b)) = s((a | d)) = a, s((b | c)) = b, s((c | d)) = c, s((d | b)) = d,
t((a | b)) = t((d | b)) = b, t((a | d)) = t((c | d)) = d, t((b | c)) = c.
d
a b
c d
a b
c
We define a path in a graph Γ to be a sequence of vertices (a1, a2, . . . , an) such that {ai, ai+1} ∈
E for i = 1, . . . , n − 1. Analogously, we define a path, P , in Q to be a sequence of directed
edges (f1, . . . , fn) such that the source of fi+1 is equal to the target of fi for 1 ≤ i ≤ n− 1. The
source and the target of P are the source of f1 and the target of fn (respectively). Furthermore,
for any path P = (a1, a2, . . . , an) (respectively P = (f1, . . . , fm)), the length, `(P ), of P is equal
to n− 1 (respectively m). We also consider paths of length 0 which start and end at the same
vertex a, called empty paths or trivial paths and denoted (a). We shall use Qi to denote the
paths of length i in Q. In addition, we shall use (a1 | a2 | . . . | an) (n ≥ 1) to denote a path
(f1, . . . , fn−1) such that s(fi) = ai and t(fi) = ai+1 for 1 ≤ i ≤ n − 1 in Q. We say that Γ is
connected if for any vertices a and b, there exits a path P between a and b.
Example 2.2 (paths). In the graph of Example 2.1, (a, b, c, a, d) is a path from a to d of length 4.
In the quiver of Example 2.1, (a | b | c | d) is a path of length 3. However, (a | b | d) is not a path
in the quiver of Example 2.1.
We define the double graph of Γ, denoted DΓ, to be the quiver consisting of the vertices
of Γ and for each edge {a, b} of Γ, DΓ has two edges f1 and f2 with s(f1) = t(f2) = a and
s(f2) = t(f1) = b.
Example 2.3 (double graph). Let Γ be the graph in Example 2.1. Its double graph, DΓ, is the
following quiver.
Skew-Zigzag Algebras 3
d
a b
c
A path C = (a1, a2, . . . , an) in Γ is said to be a cycle if a1 = an. Then we define VC and EC
to be the sets {ai | 1 ≤ i ≤ n− 1} and {{ai, ai+1} | 1 ≤ i ≤ n− 1} respectively. Similarly, a path
C = (a1 | a2 | . . . | an) in Q is said to be a cycle if a1 = an. Then we define VC and EC to
be the sets {ai | 1 ≤ i ≤ n − 1} and {{ai, ai+1} | 1 ≤ i ≤ n − 1} respectively. For each vertex
a ∈ V , we define the degree of a, denoted deg(a), to be the cardinality of the set {e ∈ E | a ∈ e}.
Furthermore, for any cycle C (in a quiver or a graph), we define the degree in C of a vertex
a ∈ VC , denoted degC(a), to be the cardinality of the set {e ∈ EC | a ∈ e}. Finally, we say
that C is a simple cycle if for all a ∈ VC we have degC(a) = 2.
We say that a graph Γ′ = (V ′, E′) is a subgraph of Γ = (V,E) if V ′ ⊆ V and E′ ⊆ E. If T
is a connected graph that does not contain any simple cycles, then T is said to be a tree. Let
T = (VT , ET ) be a subgraph of Γ. If VT = V and if T is a tree, then T is called a spanning tree
of Γ. Every connected graph has at least one spanning tree (see, for instance, [4, Section 1.5]).
If Γ = (V,E) is a connected graph and T = (VT , ET ) is a spanning tree of Γ, then |ET | = |V |−1.
3 Skew-zigzag algebras
In this section we introduce our main objects of study, the skew-zigzag algebras, and prove some
basic facts about them that are stated without proof in the literature. Throughout this section,
we fix a field k and a connected graph Γ = (V,E).
3.1 Definitions
We first recall the definition of the path algebra of a quiver. We refer the reader to Chapter 2
of [1] for further details. The path algebra of a quiver Q, denoted kQ, is the vector space
with basis consisting of all paths. We define the concatenation of two paths (a1 | a2 | . . . | an)
and (a′1 | a′2 | . . . | a′m) to be the path (a1 | a2 | . . . | an = a′1 | a′2 | . . . | a′m) when an = a′1 and 0
otherwise. Then, we define the multiplication of two paths P1 ∗ P2 to be the concatenation of
paths. The path algebra is an N-graded algebra, i.e.,
kQ =
∞⊕
i=0
kQi,
where each kQi is the k-vector space spanned by all paths of length i, and kQikQj ⊆ kQi+j for
all i, j. We call an element of kQ homogeneous of degree i if it lies in kQi. We will be mostly
interested in the case where Q = DΓ.
Recall that an ideal I of a graded algebra A =
⊕∞
i=0Ai is called graded or homogeneous if it
is generated by homogeneous elements of A. Equivalently, I is homogeneous if I =
⊕∞
i=0(I∩Ai).
Definition 3.1 (skew-zigzag coefficients). A set v =
(
vab,c ∈ k | {a, b}, {a, c} ∈ E
)
is a collection
of skew-zigzag coefficients for the graph Γ = (V,E) if it satisfies the following three conditions:
• vab,b=1 for all {a, b} ∈ E,
• vab,cvac,b = 1 for all {a, b}, {a, c} ∈ E, and
• vab,cvac,dvad,b = 1 for all {a, b}, {a, c}, {a, d} ∈ E.
4 C. Couture
From now on, whenever we say that two vertices are connected, we shall mean that there is
an edge between these two vertices. We are now ready to recall the definition of the algebras
defined originally in [5, p. 527].
Definition 3.2 (skew-zigzag algebra). Let v be a collection of skew-zigzag coefficients.
• If Γ only contains one vertex, then we define Av(Γ) to be the algebra generated by 1 and X
with X2=0.
• If Γ contains two vertices, then we define Av(Γ) to be the quotient algebra of the path
algebra of DΓ by the two-sided ideal generated by all paths of length greater than two.
• If Γ has at least three vertices, we define Iv to be the ideal generated by
a) paths of the form (a1 | a2 | a3) for all a1, a2, a3 in Γ such that a1 is connected to a2, a2
is connected to a3 and a1 6= a3,
b) elements of the form (a1 | a2 | a1)−va1a2,a3(a1 | a3 | a1) for all a1, a2, a3 ∈ V such that a1
is connected to a2, a3.
We then define Av(Γ) to be the quotient algebra of the path algebra of DΓ by the ideal Iv.
We call Av(Γ) the skew-zigzag algebra of Γ. When va1a2,a3 = 1 for all vertices a1, a2, a3 such
that a1 is connected to both a2 and a3, then we call Av(Γ) the zigzag algebra of Γ and denote
it A(Γ).
Remark 3.3. Notice that Iv is generated by homogeneous elements of kΓ and hence is a graded
ideal. Therefore, the skew-zigzag algebra inherits a grading of its own. More precisely, we have
Av(Γ) =
∞⊕
i=0
kQi/(Iv ∩ kQi) =
∞⊕
i=0
(kQi + Iv)/Iv.
Moreover, if Γ contains at least two vertices, Iv is generated by elements of kQ2. Thus Iv is
contained in
⊕
i≥2 kQi. Consequently, any path of length less than two cannot sit inside Iv.
For a path P in DΓ, we let [P ] denote the equivalence class of P in Av(Γ). Similarly, we
shall use the notation [a1 | a2 | . . . | an] to denote the equivalence class of (a1 | a2 | . . . | an), n ≥ 1,
in Av(Γ).
3.2 Bases
Our next goal is to describe an explicit basis of the skew-zigzag algebra. We begin with a few
technical results. Throughout this subsection, we fix a collection v of skew-zigzag coefficients
for the connected graph Γ = (V,E).
Lemma 3.4. Let a, a1, . . . , an ∈ V with n ≥ 2, and suppose that a is connected to a1, . . . , an.
Then
vaa1,a2v
a
a2,a3 · · · v
a
an−1,an = vaa1,an . (3.1)
Proof. We shall proceed by induction on n. For n = 2, this trivially holds. Now suppose
that (3.1) holds for some integer n ≥ 2. Let us prove that it holds for n+ 1. We have
n∏
i=1
vaai,ai+1
=
(
n∏
i=1
vaai,ai+1
)
vaan+1,an−1
vaan−1,an+1
=
(
n−2∏
i=1
vaai,ai+1
)
vaan−1,an+1
= vaa1,an+1
,
where the last equality follows from the induction hypothesis. �
Skew-Zigzag Algebras 5
Corollary 3.5. For all n ≥ 2, we have
vaa1,a2v
a
a2,a3 · · · v
a
an−1,anv
a
an,a1 = 1.
Proof. Suppose n ≥ 2. By Lemma 3.4, we have
vaa1,a2v
a
a2,a3 · · · v
a
an−1,anv
a
an,a1 = vaa1,anv
a
an,a1 = 1. �
Lemma 3.6. Let P1, . . . , Pn be paths in a quiver, no two of which have the same source and
target. If [Pi] 6= 0 for all i = 0, . . . , n then [P1], . . . , [Pn] are linearly independent.
Proof. Suppose there exist α1, . . . , αn ∈ k such that
α1[P1] + · · ·+ αn[Pn] = 0.
Then, for all i = 1, . . . , n, we obtain
0 = α1[s(Pi)][P1][t(Pi)] + · · ·+ αi[s(Pi)][Pi][t(Pi)] + · · ·+ αn[s(Pi)][Pn][t(Pi)] = αi[Pi].
Hence, we must have αi = 0 for all i = 1, . . . , n. Thus, [P1], . . . , [Pn] are linearly independent. �
We are now in a position to determine a basis of the skew-zigzag algebra. In particular, this
gives us the dimension of the zigzag algebra, which was stated in [5, Section 3] without proof.
Proposition 3.7 (basis of the skew-zigzag algebra). Recall that Γ = (V,E) is a connected graph
and v is a collection of skew-zigzag coefficients.
• If Γ only has one vertex, then {1, X} is a basis for Av(Γ).
• If Γ has two vertices, a and b, then {[a], [b], [a | b], [b | a], [a | b | a], [b | a | b]} is a basis
for Av(Γ).
• If Γ has three or more vertices, for all x ∈ V we define Vx to be set of all vertices that are
connected to x and we fix a vertex yx ∈ Vx. Then
J := {[a], [b | c], [x | yx |x] | a, x ∈ V, {b, c} ∈ E} (3.2)
is a basis for Av(Γ). In particular, we have dimAv(Γ) = 2|V |+ 2|E|.
Proof. The author would like to thank a referee for bringing into light a much simpler proof
of this proposition.
The first two claims are obvious. Therefore, we assume that Γ has at least three vertices.
Recall that the ideal Iv is generated by the set
Xv :=
{
(a | b | c), (x | y |x)− vxy,z(x | z |x) | {a, b}, {b, c}, {x, y}, {x, z} ∈ E, a 6= c
}
,
and hence
[a | b | c] = 0, [x | y |x] = vxy,z[x | z |x], (3.3)
for all {a, b}, {b, c}, {x, y}, {x, z} ∈ E with a 6= c. Note that the first equality implies that any
path with three consecutive pairwise distinct vertices is equivalent to zero. Now, consider a path
of the form (a | b | a | b) where {a, b} ∈ E. Since Γ is connected and has at least three vertices,
either a or b is connected to a third vertex d 6= a, b. Suppose b is connected to d. (The case
that a is connected to d is analogous.) Then, (3.3) yields
[a | b | a | b] = [a | b][b | a | b] = vba,d[a | b][b | d | b] = [a | b | d][d | b] = 0.
6 C. Couture
Thus, any path of length 3 or greater has an equivalence class equal to 0. So Av(Γ) only contains
elements of degree 0, 1 or 2. It is clear that {[a] | a ∈ V } and {[a | b] | {a, b} ∈ E} are bases for
(Av(Γ))0 and (Av(Γ))1 respectively since Iv is concentrated in degrees 2 and higher. Now, let
x ∈ V . Notice that for any a ∈ Vx, we have [x | a |x] = vxa,yx [x | yx |x]. So any element in
([x]Av(Γ)[x])2 can be written as some nonzero scalar times [x | yx |x]. Thus, {[x | yx |x] |x ∈ V }
is a spanning set for (Av(Γ))2 and so, by Lemma 3.6, it is a basis of (Av(Γ))2. Consequently,
J is a basis for Av(Γ).
Finally, notice that we have
J = {[a] | a ∈ V } t {[a | b] | {a, b} ∈ E} t {[x | yx |x] |x ∈ V }.
Thus, |J | = |{[a] | a ∈ V }|+ |{[a | b] | {a, b} ∈ E}|+ |{[x | yx |x] |x ∈ V }| = 2|V |+ 2|E|. �
3.3 Skew-zigzag algebras as Frobenius algebras
We begin by recalling the concept of a Frobenius algebra, referring the reader to [6] for further
details. Let f be a bilinear form
f : V × V → k,
where V is a vector space of finite dimension over the field k. A trace map gives rise to a bilinear
form (x, y) 7→ tr(xy).
Let A be a k-algebra. Let tr be a k-linear map
tr : A→ k.
We call tr a trace map. A trace map gives rise to a bilinear form (x, y) 7→ tr(xy).
Definition 3.8 (Frobenius and symmetric algebra). Let A be a finite-dimensional unital asso-
ciative k-algebra. If there exists a nondegenerate trace map tr : A → k, then A is said to be
a Frobenius algebra. Moreover, if there exists a nondegenerate symmetric trace map, then A is
said to be a symmetric Frobenius algebra or simply a symmetric algebra.
Recall that Γ = (V,E) is a connected graph. Let v be a collection of skew-zigzag coefficients,
and let P be a path in DΓ. Throughout this article, we shall define the source and the target
of the equivalence class [P ], s([P ]) and t([P ]), to be [s(P )] and [t(P )] respectively. If P1 and P2
are both trivial paths or paths of length 1 then we have [P1] = [P2] if and only if P1 = P2. If P1
and P2 are paths of length 2 then [P1] is a scalar multiple of [P2] if and only if s(P1) = s(P2)
and t(P1) = t(P2). Thus [s(P )] and [t(P )] are well-defined.
If [P ] 6= 0, we define the length of [P ], denoted `([P ]), to be `(P ). If [P ] = 0, then we
simply define `([P ]) to be 0. Since Iv is generated by homogeneous elements of the same degree,
`([P ]) is well-defined.
Let J be as in (3.2) and define the k-linear map tr : Av(Γ) → k on the elements of J as
follows:
tr([P ]) =
{
1 if `([P ]) = 2,
0 otherwise.
For any path P = (a | b | a), we let vP = vab,ya where ya is defined as in Proposition 3.7.
Proposition 3.9. Recall that Γ = (V,E) is a connected graph, and let v be a collection of skew-
zigzag coefficients. Then Av(Γ) is a graded Frobenius algebra. In addition, A(Γ) is a graded
symmetric algebra.
Skew-Zigzag Algebras 7
Proof. As noted in [5, Proposition 1], zigzag algebras are symmetric algebras. Furthermore,
as stated in [5, Section 4.5], skew-zigzag algebras are Frobenius algebras. This follows from the
fact that a basic finite-dimensional algebra over an algebraically closed field is Frobenius if the
socle of any projective indecomposable module is simple, and the map P/ rad(P ) 7→ soc(P ) is
a bijection onto the set of isomorphism classes of simple modules. The fact that the Frobenius
form is as defined above then follows from Propositions 1.10.18 and 3.6.14 of [9]. It is also
possible to prove directly that the trace map defined above has the desired properties. �
4 Moduli spaces of skew-zigzag algebras
In this section we describe the moduli spaces of skew-zigzag algebras up to various types of
isomorphism. We will see that such moduli spaces are related to the cohomology of the corre-
sponding graph. As noted in the introduction, we will only consider the group structure, and
not any geometric structure, on the moduli spaces to be introduced below. Throughout this
section we fix a field k that contains square roots.
Let Γ = (V,E) be a connected graph and v a collection of skew-zigzag coefficients. Let
P = (a1 | . . . | an) be a path in Γ. Define
vpath
P =
n−1∏
i=2
vaiai−1,ai+1
.
We call vpath
P the product of v along P . If P is a cycle, then we also define
vcycle
P = va1an−1,a2
n−1∏
i=2
vaiai−1,ai+1
= va1an−1,a2v
path
P ,
and call vcycle
P the cycle product of v along P . Furthermore, we define P ∗ = (an | . . . | a1). It is
easily seen that
(
vpath
P
)−1
= vpath
P ∗ .
Let P1 = (a1 | . . . | an) and P2 = (an | . . . | am) be two paths. Notice that we have
vpath
P1P2
=
(
m−1∏
i=2
vaiai−1,ai+1
)
=
(
n−1∏
i=2
vaiai−1,ai+1
)
vanan−1,an+1
(
m−1∏
i=n+1
vaiai−1,ai+1
)
= vpath
P1
vanan−1,an+1
vpath
P2
. (4.1)
If, in addition, P1 and P2 are cycles, then P1P2 is also a cycle and thus,
vcycle
P1P2
= va1am−1,a2v
path
P1P2
= va1am−1,a2v
an
an−1,an+1
vpath
P1
vpath
P2
= va1am−1,an+1
va1an−1,a2v
an
an−1,an+1
vpath
P1
vpath
P2
= va1an−1,a2v
an
am−1,an+1
vpath
P1
vpath
P2
= vcycle
P1
vcycle
P2
. (4.2)
In particular, the second equality of (4.2) implies that
vcycle
P1P ∗2
= va1an+1,a2v
an
an−1,am−1
vpath
P1
vpath
P ∗2
. (4.3)
Example 4.1 (product of v along a path and a cycle product). Consider the graph Γ = (V,E)
where V = {a, b, c, d} and E = {{a, b}, {a, d}, {d, c}, {b, c}, {b, d}} and let k = C. Its associated
double graph is the following quiver.
8 C. Couture
d
a b
c
It is easy to show that the following are skew-zigzag coefficients for Γ using straightforward
calculations:
vab,d = vcb,d = 2, vad,b = vcd,b = 1/2, vda,c = vba,c = 5, vdc,a = vbc,a = 1/5, vdb,c = vbd,c = 7,
vdc,b = vbc,d = 1/7, vda,b = vba,d = 5/7, vdb,a = vbd,a = 7/5, vxy,y = 1 for all {x, y} ∈ E.
Now, consider the cycles P1 = (d | b | c | d) and P2 = (d | a | b | d). Then, we have
vpath
P1
= 14, vpath
P2
= 5/14, vpath
P1P2
= 1, vcycle
P1
= 2, vcycle
P2
= 1/2, and vcycle
P1P2
= 1.
Remark 4.2. We note that Av(Γ) is both a k-algebra and a kQ0-algebra. Moreover, note
that a homomorphism of kQ0-modules is precisely a homomorphism of k-modules that fixes the
vertices.
Lemma 4.3. Let Γ be a connected graph with at least 3 vertices and v and u be two collections
of skew-zigzag coefficients. Suppose that
φ : Av(Γ)→ Au(Γ),
is an isomorphism of graded algebras such that φ([a]) = [a] for all a ∈ V . Then
ucycle
P = vcycle
P ,
for any cycle P .
Proof. For all {a, b} ∈ E, we have φ([a | b]) = αa,b[a | b] for some αa,b ∈ k∗. In addition, for any
{a, b} ∈ E we have
φ([a | b | a]) = φ([a | b][b | a]) = φ([a | b])φ([b | a]) = αa,b[a | b]αb,a[b | a] = αa,bαb,a[a | b | a].
Let a, b, c ∈ V be such that a is connected to both b and c. Then
αa,bαb,au
a
b,c[a | c | a] = αa,bαb,a[a | b | a] = αa,bαb,a[a | b][b | a]
= φ([a | b][b | a]) = φ([a | b | a]) = φ(vab,c[a | c | a]) = vab,cαa,cαc,a[a | c | a].
Thus, we have
uab,c =
αa,cαc,a
αa,bαb,a
vab,c.
Therefore, if P = (a1 | . . . | an) is a cycle, we have
ucycle
P =
αa1,an−1αan−1,a1
αa1,a2αa2,a1
(
n−2∏
i=2
αai,ai−1αai−1,ai
αai,ai+1αai+1,ai
)
αan−1,an−2αan−2,an−1
αan−1,a1αa1,an−1
vcycle
P = vcycle
P ,
as required. �
The following proposition shows that the converse to Lemma 4.3 holds.
Skew-Zigzag Algebras 9
Proposition 4.4. Let Γ = (V,E) be a connected graph with at least 3 vertices and let v and u
be two collections of skew-zigzag coefficients. Then vcycle
P = ucycle
P for every cycle P if and only
if there exists an isomorphism of graded algebras φ : Av(Γ)
∼=−→ Au(Γ) such that φ([a]) = [a] for
all a ∈ V .
Proof. ⇒ : Fix an edge {a, b} ∈ E. For any {d, e} ∈ E, consider a path P = (a1 | . . . | an) with
a1 = a, an−1 = d and an = e. Note that since Γ is connected, there is at least one such path.
Now, choose αd,e = αe,d ∈ k∗ such that
α2
d,e = uab,a2v
a
a2,bu
path
P vpath
P ∗ . (4.4)
We will show that since we have vcycle
C = ucycle
C for any cycle C, (4.4) is independent of the choice
of the path. Let P1 = (a1 | . . . | an) and P2 = (b1 | . . . | bm) be two paths such that a = a1 = b1,
d = an−1 = bm−1 and e = an = bm. Since P1P
∗
2 is a cycle, we have
vcycle
P1P ∗2
= va1b2,a2v
an
an−1,bm−1
vpath
P1
vpath
P ∗2
= vab2,bv
a
b,a2v
path
P1
vpath
P ∗2
,
where the first equality follows from (4.3) and the second equality follows from Lemma 3.4 and
the fact that an−1 = bm−1. Similarly, we obtain
ucycle
P1P ∗2
= uab2,bu
a
b,a2u
path
P1
upath
P ∗2
.
Since vcycle
P1P ∗2
= ucycle
P1P ∗2
, we have
vab2,bv
a
b,a2v
path
P1
vpath
P ∗2
= uab2,bu
a
b,a2u
path
P1
upath
P ∗2
.
Consequently,
uab,b2v
a
b2,bu
path
P2
vpath
P ∗2
= uab,a2v
a
a2,bu
path
P1
vpath
P ∗1
,
as required. Thus the coefficients αd,e are path independent. Now, define a map φ as follows
φ : kDΓ→ kDΓ, (d) 7→ (d) for all d ∈ V, (d | e) 7→ αd,e(d | e) for all {d, e} ∈ E.
By definition this map is k-linear. We then extend the map to longer paths by requiring it to be
an algebra homomorphism. We will now show that φ(Iv) ⊆ Iu. Now suppose {x, y}, {x, z} ∈ E.
Let P1 = (a1 | . . . | an) and P2 = (b1 | . . . | bm) be two paths such that a = a1 = b1, x = an−1 =
bm−1, y = an and z = bm. We have
α2
x,y
α2
x,z
vxz,y =
uab,a2v
a
a2,b
upath
P1
vpath
P ∗1
uab,b2v
a
b2,b
upath
P2
vpath
P ∗2
vxz,y =
vaa2,b2v
path
P ∗1
vpath
P2
uaa2,b2u
path
P ∗1
upath
P2
vxz,y. (4.5)
The path P2(z |x | y)P ∗1 is a cycle. Thus,
vcycle
P2(z |x | y)P ∗1
= vaa2,b2v
path
P2(z |x | y)P ∗1
= vaa2,b2v
z
x,xv
y
x,xv
path
P2
vpath
(z |x | y)v
path
P ∗1
= vaa2,b2v
path
P2
vpath
P ∗1
vxz,y,
where the second equality uses (4.1). Similarly, we obtain
ucycle
P2(z |x | y)P ∗1
= uaa2,b2u
path
P2
upath
P ∗1
uxz,y.
Since we have vcycle
P2(z |x | y)P ∗1
= ucycle
P2(z |x | y)P ∗1
, we must also have
vaa2,b2v
path
P2
vpath
P ∗1
vxz,y = uaa2,b2u
path
P2
upath
P ∗1
uxz,y. (4.6)
10 C. Couture
Combining (4.5) and (4.6) gives us
α2
x,y
α2
x,z
vxz,y =
vaa2,b2v
path
P ∗1
vpath
P2
uaa2,b2u
path
P ∗1
upath
P2
vxz,y =
uaa2,b2u
x
z,yu
path
P ∗1
upath
P2
uaa2,b2u
path
P ∗1
upath
P2
= uxz,y.
Hence, we can now deduce that φ(Iv) ⊆ Iu, and thus φ induces an algebra homomorphism
φ̄ : Av(Γ)→ Au(Γ),
[d] 7→ [d], d ∈ V,
[d | e] 7→ αd,e[d | e], {d, e} ∈ E,
[d | e | d] 7→ α2
d,e[d | e | d], {d, e} ∈ E.
Since φ is surjective, φ̄ is also surjective. Since Av(Γ) and Au(Γ) have the same dimension, φ̄ is
also injective. Consequently, φ̄ is an isomorphism and hence Av(Γ) ∼= Au(Γ).
The converse of this proposition is Lemma 4.3. �
We will now introduce the concept of graph cohomology. Let Γ = (V,E) be a connected graph
and DΓ = (V,E′) its double graph. Let
ZV =
{∑
a∈V
αaa |αa ∈ Z for all a ∈ V
}
, ZE′ =
{∑
e∈E′
αee |αe ∈ Z for all e ∈ E′
}
.
Define the map δ by
δ : ZE′/{(a | b) + (b | a) | {a, b} ∈ E} → ZV, e 7→ s(e)− t(e),
where we extend the map by linearity. For any path (a1 | . . . | an), we associate the element
n−1∑
i=1
(ai|ai+1) ∈ ZE′ to it. Notice that if a1 = an, then δ
(
n−1∑
i=1
(ai|ai+1)
)
= 0. Let C = ker δ.
We call C the space of cycles of Γ. The space of cycles is a Z-submodule of the free Z-module
ZE′/{(a | b) + (b | a) | {a, b} ∈ E} and thus it is a free Z-module. Consequently, it has a Z-basis.
Lemma 4.5 ([8, Section 4.4]). Let Γ = (V,E) be a connected graph. Then rank(C) = |E| −
|V |+ 1.
Let T = (VT , ET ) be a spanning tree of a graph Γ = (V,E) and let e ∈ V \VT . We shall denote
by T + e the subgraph of Γ with vertex set V and edge set ET ∪ {e}. Moreover, for any cycle
C = (a1 | . . . | an = a1), we say that an edge (a | b) is in C if (a | b) ∈ {(ai | ai+1)} | 1 ≤ i ≤ n−1}.
Lemma 4.6. Let Γ = (V,E) be a connected graph with |V | = n and |E| = m. There exists
a basis of C, B = {C1, . . . , Cm−n+1}, such that, for all 1 ≤ i ≤ m − n + 1, there exists (bi | ci)
in Ci such that (bi | ci) is not in Ck for k 6= i.
Proof. Let Γ = (V,E) be a connected graph with |V | = n and |E| = m. Pick a spanning tree
T = (VT , ET ) of Γ. Let E \ET = {e1, . . . , em−n+1}. Notice that for each i ∈ {1, . . . ,m−n+ 1},
T + ei contains one simple cycle. Thus, the quiver D(T + ei) contains two corresponding simple
cycles, Ci and −Ci. Let
B = {Ci | i = 1, . . . ,m− n+ 1}.
Clearly the elements of B are linearly independent. Moreover, |B| = m − n + 1. Thus, by
Lemma 4.5, B is a Q-basis of C ⊗Z Q. But since the coefficient of ei in Ci is one, it follows
that B is a Z-basis of C. Finally, it is also clear that for all i = 1, . . . ,m− n+ 1 we have (bi | ci)
in Ck if and only if k = i, where (bi | ci) is the corresponding directed edge of ei in Ci. �
Skew-Zigzag Algebras 11
Let Γ = (V,E) be a connected graph and DΓ = (V,E′) be its double graph. The graph
cohomology of Γ is defined to be the space of group homomorphisms from C to k∗:
H1(Γ, k∗) = Homgroup(C,k∗).
Note that the operation in this group is pointwise multiplication.
Let z ∈ C. Take a representative
∑
e∈E′
αee of z in NE′ =
{ ∑
e∈E′
αee |αe ∈ N for all e ∈ E′
}
.
Since z ∈ C = ker δ, for each a ∈ V we have
∑
e∈S
αe =
∑
e∈T
αe where S = {e ∈ E′ | s(e) = a} and
T = {e ∈ E′ | t(e) = a}. Thus, we can choose na ∈ N and vertices ba,1, . . . , ba,na , ca,1, . . . , ca,na
such that
∑
a∈V
na∑
i=1
((ba,i | a) + (a | ca,i)) = 2z.
Now, for a collection of skew-zigzag coefficients v, define
fv,a(z) =
na∏
i=1
vaba,i,ca,i .
Lemma 4.7. Let Γ = (V,E) be a connected graph, v be a collection of skew-zigzag coefficients,
a ∈ V and z ∈ C. Set ba,1, . . . , ba,na and ca,1, . . . , ca,na as in the previous paragraph. Then,
fv,a(z) is independent of the order chosen for ba,1, . . . , ba,na and ca,1, . . . , ca,na and fv,a(z) is
independent of the representative of z chosen in NE′.
Proof. For the first claim, it suffices to show that fv,a(z) remains unchanged when we inter-
change ca,j and ca,j+1 for some j ∈ {1, . . . , na − 1}. Indeed, we have
vaba,j ,ca,j+1
vaba,j+1,ca,j
na∏
i=1
i 6=j,j+1
vaba,i,ca,i
= vaba,j ,ca,jv
a
ca,j ,ca,j+1
vaba,j+1,ca,j+1
vaca,j+1,ca,j
na∏
i=1
i 6=j,j+1
vaba,i,ca,i =
na∏
i=1
vaba,i,ca,i ,
where the second equality follows from Lemma 3.4 and the third equality follows from the second
condition of Definition 3.1.
For the second part of the lemma, it suffices to show that fv,a(z) remains unchanged when
you remove ba,n and ca,n when ba,n = ca,n. But this is obvious as vaba,n,ca,n = vaba,n,ba,n = 1. �
Now, for any collection of skew-zigzag coefficients, v, define the map fv to be
fv : C → k∗, z 7→
∏
a∈V
fv,a(z).
It is clear by the definition of fv that we have fv(z1 + z2) = fv(z1)fv(z2) for any z1, z2 ∈ C.
Thus, fv ∈ H1(Γ,k∗). In addition, if C is a cycle in DΓ, then it is clear that fv(C) = vcycle
C .
Let SZC = {v | v is a collection of skew-zigzag coefficients}. For u, v ∈ SZC, define u · v to
be the set of skew-zigzag coefficients defined as follows:
(u · v)ab,c = uab,cv
a
b,c,
12 C. Couture
for all {a, b}, {a, c} ∈ E. It is straightforward to check that this introduces a group structure
on SZC. We then define an equivalence relation on SZC as follows:
v ≡ u ⇐⇒ there exists an isomorphism φ : Av(Γ)→ Au(Γ)
such that φ([a]) = [a] for all a ∈ V.
By Lemma 4.3 and Proposition 4.4, we have
v ≡ u ⇐⇒ vcycle
P = ucycle
P for every cycle P in Γ.
Now let
Σ = {Av(Γ) | v ∈ SZC}.
Let ∼ be the equivalence relation on Σ defined by
Av(Γ) ∼ Au(Γ) ⇐⇒ there exists an isomorphism φ : Av(Γ)→ Au(Γ)
such that φ([a]) = [a] ∀ a ∈ V.
Let v ∈ SZC. From now on, we shall use [v] to denote the equivalence class of v in SZC/≡.
Furthermore, we shall use [Av(Γ)]∼ and [Av(Γ)]∼= to denote the equivalence classes of Av(Γ)
in Σ/∼ and Σ/∼= respectively, where ∼= denotes isomorphism of graded algebras. Note that Σ/∼
and SZC/≡ are naturally isomorphic sets via the map
φ : Σ/∼ → SZC/≡, [Av(Γ)]∼ 7→ [v]. (4.7)
Theorem 4.8. Let Γ = (V,E) be a connected graph. Then we have
Σ/∼ ∼= SZC/≡ ∼= H1(Γ,k∗),
where the first isomorphism is an isomorphism of sets and the second isomorphism is an iso-
morphism of groups.
Proof. Define the map ψ by
ψ : SZC→ H1(Γ,k∗), v 7→ fv.
It is clear that this map is a group homomorphism. Let v, u ∈ SZC. Then, we have
ψ(v) = ψ(u) ⇐⇒ fv = fu ⇐⇒ fv(P ) = fu(P ) for all P ∈ C
⇐⇒ vcycle
P = ucycle
P for all P ∈ C ⇐⇒ v ≡ u.
Hence, SZC/ kerψ = SZC/≡.
Let us now prove that ψ is surjective. By Lemma 4.6, there exists a basis of cycles B =
{C1, . . . , C|E|−|V |+1} of C such that, for all 1 ≤ i ≤ |E| − |V |+ 1, there exists (bi | ci) ∈ Ci such
that (bi | ci) ∈ Ck if and only if k = i.
Let f be a group homomorphism from C to k∗. We define coefficients as follows:
vab,b = 1 for all {a, b} ∈ E,
vab,c = 1 for all {a, b}, {a, c} ∈ E, a 6= bi for all i = 1, . . . , |E| − |V |+ 1.
Then, for all i = 1, . . . , |E| − |V |+ 1 we define
1/vbici,a = vbia,ci = f(Ci) for all a 6= ck, 1 ≤ k ≤ |E| − |V |+ 1,
vbick,ci = f(Ci)/f(Ck) for all k ∈ {1, . . . , |E| − |V |+ 1} such that bi = bk,
vbia,c = 1 for all {bi, a}, {bi, c} ∈ E, a 6= ci 6= c.
Skew-Zigzag Algebras 13
It is clear that the set v = (vab,c | {a, b}, {a, c} ∈ E) is a collection of skew-zigzag coefficients.
Moreover, since (bi | ci) is in Ck if and only if k = i for all i = 1, . . . , |E| − |V | + 1, it is also
clear that fv(Ci) = f(Ci) for all i = 1, . . . , |E| − |V | + 1. Thus, we must have ψ(v) = fv = f .
Consequently, ψ is surjective.
Since ψ is surjective, the first isomorphism theorem implies SZC/≡ ∼= H1(Γ, k∗). The result
then follows using (4.7). �
Remark 4.9. In [5, Section 4], the authors state that “the moduli space of skew-zigzag algebras
is naturally isomorphic to H1(Γ,C∗)”. In light of Theorem 4.8, we assume that the moduli space
they had in mind was Σ/ ∼.
Let Γ = (V,E) be a graph. We say that a permutation σ of the elements of V is a graph
automorphism of Γ if, for all a, b ∈ V , we have
{a, b} ∈ E ⇐⇒ {σ(a), σ(b)} ∈ E.
The identity permutation is called the trivial graph automorphism. A graph is said to be
asymmetric if it admits only the trivial graph automorphism, otherwise it is said to be symmetric.
Suppose σ is a graph automorphism of Γ. For any path P = (a1 | . . . | an) we let σ(P ) =
(σ(a1) | . . . |σ(an)). Notice that if P is a cycle, then σ(P ) is also a cycle. Moreover, if P1
and P2 are two paths, then it is clear that σ(P1P2) = σ(P1)σ(P2). Thus, the map σ induces an
automorphism of the path algebra kDΓ
φσ : kDΓ→ kDΓ, P 7→ σ(P ).
Fix a collection of skew-zigzag coefficients v. We define (σv)ab,c for all {a, b}, {a, c} ∈ E as
follows:
(σv)ab,c = v
σ−1(a)
σ−1(b),σ−1(c)
.
Then we define σv = ((σv)ab,c | {a, b}, {a, c} ∈ E). The map (σ, v) 7→ σv defines an action of the
group of graph automorphisms on the set of skew-zigzag coefficients.
It is straightforward to verify that σ(Iv) = Iσv. Thus we obtain an isomorphism
σ : Av(Γ)→ Aσv(Γ), [P ] 7→ [σ(P )]. (4.8)
Let
ψ : Av(Γ)→ Au(Γ)
be an isomorphism of graded algebras where v and u are two collections of skew-zigzag coeffi-
cients. Suppose that V = {a1, . . . , an}. Although it follows from more advance concepts, we
will use an elementary approach to prove that vertices must be mapped to vertices. For all
1 ≤ i ≤ n, let ψ([ai]) =
n∑
j=1
αij [aj ], where αij ∈ k for all 1 ≤ i, j ≤ n. For any two vertices
ai, ak ∈ V , 1 ≤ i, k ≤ n, i 6= k, we have
0 = ψ(0) = ψ([ai][ak]) =
n∑
j=1
αij [aj ]
n∑
j=1
αkj [aj ]
=
n∑
j=1
αijαkj [aj ].
Thus, for any j = 1, . . . , n, at least one of αij or αkj is 0. Hence, if αij 6= 0 for some 1 ≤ i, j ≤ n,
then αkj = 0 for all 1 ≤ k ≤ n, k 6= i. Moreover, since ψ is injective, ψ([ai]) 6= 0 for all
i = 1, . . . , n, and so there exists ji ∈ {1, . . . , n} such that αiji 6= 0 and thus αkji = 0 for all
14 C. Couture
1 ≤ k ≤ n, k 6= i. Therefore, for any 1 ≤ i, k ≤ n, i 6= k we have ji 6= jk. As a result, we must
have αij = 0 for all 1 ≤ i, j ≤ n, j 6= ji. Consequently, we must have ψ([ai]) = αiji [aji ] for some
ji ∈ {1, . . . , n}. Furthermore, notice that we have
φ([ai]) = φ([ai][ai]) = φ([ai])φ([ai]) = αijiαiji [aji ] = αijiφ([ai]),
for all 1 ≤ i ≤ n. Hence we must have αiji = 1. Therefore, any isomorphism ψ : Av(Γ)→ Au(Γ)
induces a graph automorphism
σψ : V → V, ai 7→ aji .
Then, we obtain the isomorphism
σ−1
ψ ◦ ψ : Av(Γ)→ Aσ−1u(Γ),
Therefore
ψ = σψ ◦
(
σ−1
ψ ◦ ψ
)
, where σ−1
ψ ◦ ψ([a]) = [a] for all a ∈ V. (4.9)
Lemma 4.10. Let Γ be a connected asymmetric graph with at least three vertices, and let v
and u be two collections of skew-zigzag coefficients. Then Av(Γ) ∼= Au(Γ) as graded algebras if
and only if v ≡ u.
Proof. Let Γ be a connected asymmetric graph with at least three vertices, and v and u be two
collections of skew-zigzag coefficients. Suppose that we have an isomorphism φ : Av(Γ)→ Au(Γ).
Then in particular the map
σφ : V → V, a 7→ s (φ([a])) ,
is a graph automorphism. Since Γ is asymmetric, we must have σφ(a) = a for all a ∈ V . Thus
φ([a]) = [a] for all a ∈ V . Hence v ≡ u by Lemma 4.3.
The reverse implication follows from the definition of the equivalence relation. �
Let Aut(Γ) be the group of graph automorphisms of Γ. We define a group action of Aut(Γ)
on H1(Γ, k∗) by
(σf)(C) = f
(
σ−1(C)
)
, σ ∈ Aut(Γ), f ∈ H1(Γ,k∗), C ∈ C.
For any σ ∈ Aut(Γ) and any collection of skew-zigzag coefficients v we have an isomorphism
given by (4.8) σ : Av(Γ) → Aσv(Γ). It is clear that Av(Γ) 7→ Aσv(Γ) defines a group action of
Aut(Γ) on Σ and that it preserves the equivalence relation ∼. Therefore, we have an induced
action of Aut(Γ) on Σ/∼, given by X 7→ σX for all X ∈ Σ/∼, where σX = {σx |x ∈ X}.
Lemma 4.11. For any graph Γ we have the following isomorphism of sets
(Σ/∼)/Aut(Γ) ∼= Σ/∼=.
Proof. Define
α : Σ/∼ → Σ/∼=, [Av(Γ)]∼ 7→ [Av(Γ)]∼=.
Clearly this is a well-defined surjective map. Moreover, notice that if v and u are two collections
of skew-zigzag coefficients, then we have
α([Av(Γ)]∼) = α([Au(Γ)]∼) ⇐⇒ [Av(Γ)]∼= = [Au(Γ)]∼=
⇐⇒ there exists an isomorphism Φ: Av(Γ)→ Au(Γ).
Skew-Zigzag Algebras 15
Recall from (4.9) that every isomorphism between skew-zigzag algebras can be written as the
composition of a graph automorphism and an isomorphism that fixes the vertices. Thus, there
exists σ ∈ Aut(Γ) and an isomorphism γ that fixes the vertices such that
Φ = σ ◦ γ : Av(Γ)
γ−→ Aσ−1u(Γ)
σ−→ Au(Γ).
Since γ fixes the vertices, we have [Av(Γ)]∼ = [Aσ−1u(Γ)]∼ and so,
σ[Av(Γ)]∼ = σ[Aσ−1u(Γ)]∼ = [Au(Γ)]∼.
Now, let v and u be two collections of skew-zigzag coefficients such that there exists σ ∈ Aut(Γ)
such that σ[Av(Γ)]∼ = [Au(Γ)]∼. Then, since Aσ−1u(Γ) ∈ [Av(Γ)]∼, there exists an isomor-
phism γ that fixes the vertices such that we have the following
Av(Γ)
γ−→ Aσ−1u(Γ)
σ−→ Au(Γ).
Since σ ◦ γ is an isomorphism, we must have [Av(Γ)]∼= = [Au(Γ)]∼= and thus, α([Av(Γ)]∼) =
α([Au(Γ)]∼).
Consequently, for any X1, X2 ∈ Σ/∼, we have
α(X1) = α(X2) ⇐⇒ there exists σ ∈ Aut(Γ) such that σX1 = X2.
Hence, we finally obtain (Σ/∼)/Aut(Γ) ∼= Σ/∼=. �
Theorem 4.12. For any graph Γ we have the following isomorphism of sets
Σ/∼= ∼= H1(Γ,k∗)/Aut(Γ).
Proof. By Theorem 4.8, we know that the map
ψ ◦ φ : Σ/∼ → H1(Γ, k∗), [Av(Γ)]∼ 7→ fv,
is an isomorphism, where the map φ is as in (4.7) and ψ : SZC/≡ → H1(Γ,k∗), [v] 7→ fv. Let
us now prove that it preserves the Aut(Γ)-action, i.e., for all σ ∈ Aut(Γ) and v ∈ SZC, we have
(ψ ◦ φ)(σ[Av(Γ)]∼) = σ(ψ ◦ φ)([Av(Γ)]∼). We have
(ψ ◦ φ)(σ[Av(Γ)]∼) = ψ([σv]) = fσv,
and
σ(ψ ◦ φ)([Av(Γ)]∼) = σ(ψ([v])) = σ(fv).
Let C = (a1 | . . . | an) ∈ C be a cycle. Then,
σ(fv)(C) = fv
(
σ−1(C)
)
= fv
((
σ−1(a1) | . . . |σ−1(an)
))
= v
σ−1(a1)
σ−1(an−1),σ−1(a2)
· · · vσ
−1(an−1)
σ−1(an−2),σ−1(an)
= fσv(C).
Hence σ(fv) = fσv. Consequently, we must have (ψ ◦ φ)(σ[Av(Γ)]∼) = σ(ψ ◦ φ)([Av(Γ)]∼). So
ψ ◦ φ is an Aut(Γ)-set isomorphism and thus,
(Σ/∼)/Aut(Γ) ∼= H1(Γ, k∗)/Aut(Γ).
Lemma 4.11 then yields Σ/∼= ∼= H1(Γ,k∗)/Aut(Γ). �
In [5, Section 4] the authors state the following result without proof.
16 C. Couture
Corollary 4.13. If Γ is a tree then all of its skew-zigzag algebras are isomorphic.
Proof. The follows immediately from Theorem 4.12 and the fact that trees have trivial graph
cohomology. �
Example 4.14. Consider the graph Γ = (V,E) given by V = {a, b, c} and E = {{a, b}, {b, c},
{a, c}}. The associated double graph is
a
b
c
By Proposition 4.4, if u, v ∈ SZC, then Au(Γ) ∼ Av(Γ) if and only if ucycle
P = vcycle
P for every
cycle P . Furthermore, recall that by (4.9), if u, v ∈ SZC are such that Av(Γ) ∼= Au(Γ) via an
isomorphism φ, then φ can be written as the composition of a graph automorphism σ with an
isomorphism γ that fixes the vertices:
σ ◦ γ : Av(Γ)
γ−→ Aσ−1u(Γ)
σ−→ Au(Γ).
Thus, if P is a cycle, then vcycle
P = ucycle
σ−1(P )
. Conversely, if there exists σ ∈ Aut(Γ), such that
vcycle
P = ucycle
σ−1(P )
for every cycle P , then we have Av(Γ) ∼= Aσ−1u(Γ) by Proposition 4.4 and
Aσ−1u(Γ) ∼= Au(Γ) by (4.8). Thus, Av(Γ) ∼= Au(Γ). Consequently, Av(Γ) ∼= Au(Γ) if and only if
there exists some σ ∈ Aut(Γ) such that vcycle
P = ucycle
σ−1(P )
for all cycles P . Therefore, in order to
compute Σ/∼ and Σ/∼=, it suffices to consider the cycle products of u and v (u, v ∈ SZC) along
every cycle.
It is clear that the C = (a | b | c | a) yields a basis for the cycle space C. So, if u, v ∈ SZC,
then Au(Γ) ∼ Av(Γ) if and only if ucycle
C = vcycle
C and Au(Γ) ∼= Av(Γ) ⇐⇒ ucycle
C = vcycle
C or
ucycle
C∗ = vcycle
C .
Let k = C and xRy ⇐⇒ x = y or x−1 = y (x, y ∈ C∗). Consider the maps
φ : Σ/∼ → C∗, [Av(Γ)]∼ 7→ vcycle
C , ψ : Σ/∼= → C∗/R, [Av(Γ)]∼= 7→ vcycle
C .
These maps are well-defined and injective by the above remark. Surjectivity follows from the
fact that if x ∈ C∗, then the following is a collection of skew-zigzag coefficients:
vab,b = vac,c = vba,a = vbc,c = vca,a = vcb,b = vab,c = vac,b = vba,c = vbc,a = 1,
vca,b = x, vcb,a = x−1.
Thus, Σ/∼ ∼= C∗, whereas Σ/∼= ∼= C∗/R. Note that, C∗ � C∗/R since, in C∗/R, every element
is its own inverse which is not true in C∗.
5 Other constructions of some skew-zigzag algebras
We conclude with a discussion of other constructions of certain skew-zigzag algebras that have
appeared in the literature [2, 3].
Let Γ = (V,E) be a connected graph. We call Ω = (εa,b ∈ k∗ | a, b ∈ V ) a collection of
orientation coefficients if, for any pair of vertices a, b ∈ V , we have εa,b = 0 if {a, b} /∈ E and
εa,b = −εb,a if {a, b} ∈ E. If Γ has at least 3 vertices, we define the algebra BΓ
Ω to be the quotient
algebra of the path algebra of DΓ by the two sided ideal IΩ generated by elements of the form
Skew-Zigzag Algebras 17
• (a | b | c) for {a, b}, {b, c} ∈ E and a 6= c, and
• εa,b(a | b | a)− εa,c(a | c | a) such that a is connected to both b and c.
Lemma 5.1. Let Γ be a connected graph. For any collection of orientation coefficients, Ω, there
exists a collection of skew-zigzag coefficients, v, such that BΓ
Ω = Av(Γ).
Proof. Let Ω = (εa,b) be a collection of orientation coefficients and set vab,c = εa,c/εa,b for all
{a, b}, {a, c} ∈ E. For any {a, b}, {a, c}, {a, d} ∈ E we have
vab,b =
εa,b
εa,b
= 1, vab,cv
a
c,b =
εa,c
εa,b
εa,b
εa,c
= 1, and vab,cv
a
c,dv
a
d,b =
εa,c
εa,b
εa,d
εa,c
εa,b
εa,d
= 1.
Thus the set v = (vab,c | {a, b}, {a, c} ∈ E) is a collection of skew-zigzag coefficients. It is clear
that IΩ = Iv. Thus, we have BΓ
ε = Av(Γ). �
Definition 5.2 (orientation). Let Γ = (V,E) be a connected graph and let DΓ = (V,E′) be its
double graph. A set ε ⊆ E′ is said to be an orientation of DΓ if, for every {a, b} ∈ E, exactly
one directed edge in DΓ between a and b is in ε.
In [3, Section 2.1, p. 109] the authors fix an orientation ε of DΓ. Then, they define orientation
coefficients Ω = (εa,b | a, b ∈ V ) as follows:
εa,b =
1 if (a | b) ∈ ε,
−1 if (b | a) ∈ ε,
0 if a and a are not connected.
(5.1)
Notice that εa,b = −εb,a. Thus, by Lemma 5.1, we have BΓ
Ω = Av(Γ), where
v =
(
vab,c =
εa,c
εa,b
| {a, b}, {a, c} ∈ E
)
.
However, the following proposition shows that the converse of Lemma 5.1 is false. Thus, the
alternate definition of skew-zigzag algebras in terms of orientation coefficients is more restrictive.
Proposition 5.3. If Γ is not a bipartite graph, then A(Γ) is not isomorphic to BΓ
Ω for any
collection of orientation coefficients Ω.
Proof. Suppose that Γ is not a bipartite graph and that there exists an isomorphism
φ : A(Γ)→ BΓ
Ω
for some collection of orientation coefficients Ω. For all a ∈ V let φ([a]) = [xa]. So, for all
{a, b} ∈ E we have φ([a | b]) = αa,b[xa |xb] for some αa,b ∈ k. Consequently, for any {a, b},
{a, c} ∈ E we have
αa,bαb,a[xa |xb |xa] = φ([a | b | a]) = φ([a | c | a]) = αa,cαc,a[xa |xc |xa].
Therefore,
αa,cαc,a
αa,bαb,a
=
εxa,xc
εxa,xb
. (5.2)
By [4, Proposition 1.6.1], since Γ is not bipartite, it contains a cycle of odd length. Thus, there
is a cycle C = (a1 | . . . | an) in DΓ with n even. So, (5.2) yields
1 =
αa1,a2αa2,a1
αa1,an−1αan−1,a1
αa2,a3αa3,a2
αa2,a1αa1,a2
· · ·
αan−1,a1αa1,an−1
αan−1,an−2αan−2,an−1
=
εxa1 ,xa2
εxa1 ,xan−1
εxa2 ,xa3
εxa2 ,xa1
· · ·
εxan−1 ,xa1
εxan−1 ,xan−2
. (5.3)
18 C. Couture
Since {ai, ai+1} ∈ E for all 1 ≤ i ≤ n− 1, εxaj ,xak 6= 0 for 1 ≤ j, k,≤ n− 1, j 6= k. Moreover, we
know that εa,b = −εb,a for all a, b ∈ V . Consequently, (5.3) yields
1 = (−1)n−1 = −1.
This contradiction implies that Au(Γ) � BΓ
Ω for any Ω. �
Suppose (εa,b) is a collection of orientation coefficients. For x ∈ V , define Vx and yx as in
Proposition 3.7. Notice that if we modify the set J in (3.2) by setting
J ′ := {[a], [b | c], εx,yx [x | yx |x] | a, x ∈ V, b, c ∈ V such that {b, c} ∈ E},
then J ′ is independent of the choice of yx for every x ∈ V . Indeed, for any y, z ∈ Vx, we have
εx,y[x | y |x] = εx,z[x | z |x].
In [2, Section 6.1, p. 2516], the authors fix an orientation ε of Γ and define coefficients εa,b
(a, b ∈ V ) as in (5.1). They then define a diagrammatic algebra using these coefficients. Their
algebra is in fact isomorphic to the algebra BΓ
Ω, for Ω = (εa,b | a, b ∈ V ), via the map
A→ A(Γ),
y
x
7→
{
[x | y] x 6= y,
εx,yx [x | yx |x] if x = y,
x 7→ [x].
So, by Lemma 5.1, A is isomorphic to Av(Γ) where v =
(
vab,c =
εa,c
εa,b
| {a, b}, {a, c} ∈ E
)
. However,
if Γ is not bipartite, then this is not isomorphic to the zigzag algebra by Proposition 5.3.
Note on the LATEXversion. For the interested reader, the tex file of this paper includes
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Acknowledgements
This work was completed under the supervision of Professor Alistair Savage. The author would
like to thank Professor Savage immensely for his patience and guidance throughout this paper as
well as the opportunity to write this paper. The author would also like to thank the University
of Ottawa and the Work-Study Program for their support. Finally, the author would like to
thank the referees for their useful comments and for providing a reference for Proposition 3.9.
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http://dx.doi.org/10.1007/978-4-431-54177-6
http://dx.doi.org/10.1007/978-3-319-07968-4
1 Introduction
2 Graph theory background
3 Skew-zigzag algebras
3.1 Definitions
3.2 Bases
3.3 Skew-zigzag algebras as Frobenius algebras
4 Moduli spaces of skew-zigzag algebras
5 Other constructions of some skew-zigzag algebras
References
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