Example of categorification of a cluster algebra

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by  Laurent Demonet

EXAMPLE OF CATEGORIFICATION OF A CLUSTER ALGEBRA
LAURENT DEMONET
7 Abstract. We present here two detailed examples of additive categorifications of the
1
cluster algebra structure of a coordinate ring of a maximal unipotent subgroup of a
0
simple Lie group. The first one is of simply-laced type (A ) and relies on an article by
2 3
n Geiß, Leclerc and Schr¨oer. The second is of non simply-laced type (C2) and relies on an
article by the author of this note. This is aimed to be accessible, specially for people
a
J who are not familiar with this subject.

6
2
1. Introduction: the total positivity problem

]
T Let N be the subgroup of SL (C) consisting of upper triangular matrices with diagonal
4
R
1. We say that X ∈ N is totally positive if its 12 non-trivial minors are positive real
.
h numbers (a minor is non-trivial if it is not constant on N and not product of other
t
a minors). As a consequence of various results of Fomin and Zelevinsky [3] (see also [1]), in
m
a (very) special case, we get
[

Proposition 1 (Fomin-Zelevinsky). X ∈ N is totally positive if and only if the minors
1
v ∆1(X), ∆12(X), ∆123(X), ∆12(X), ∆2(X), ∆3(X) are positive.
4 34 234 24 4 4
3
6 where ∆(cid:96)1...(cid:96)k(X) is the minor of X with rows (cid:96) , ..., (cid:96) and columns c , ..., c .
5 c1...ck 1 k 1 k
Remark that, as the algebraic variety N has dimension 6, we can not expect to find a
7
0 criterion with less than 6 inequalities to check the total positivity of a matrix.
1. To prove this, just remark that we have the following equality:
0
7 ∆12∆23 = ∆123∆2 +∆3∆12
24 34 234 4 4 34
1
: which immediately implies that ∆1(X), ∆12(X), ∆123(X), ∆12(X), ∆2(X), ∆3(X) are
v 4 34 234 24 4 4
i positive if and only if ∆1(X), ∆12(X), ∆123(X), ∆23(X), ∆2(X), ∆3(X) are positive.
X 4 34 234 34 4 4
Such an equality is called an exchange identity. In Figure 1, we wrote 14 sets of minors
r
a which are related by exchange identities whenever they are linked by an edge. As every
minor appears in this graph, it induces the previous proposition.
These observations lead to the definition of a cluster algebra [4]. A cluster algebra is an
algebra endowed with an additional combinatorial structure. Namely, a (generally infi-
nite)setofdistinguishedelementscalledcluster variables groupedintosubsetsofthesame
cardinality n, called clusters and a finite set {x ,x ,...,x } called the set of coef-
n+1 n+2 m
ficients. For each cluster {x ,x ,...,x }, the extended cluster {x ,...,x ,x ,...,x }
1 2 n 1 n n+1 m
is a transcendence basis of the algebra. Moreover, each cluster {x ,x ,...,x } has n
1 2 n
The paper is in a final form and no version of it will be submitted for publication elsewhere.
1,12,123,
4 34 234
13,1,1
34 2 3
1,12,123,
4 34 234
13,23,1
34 34 3
1,12,123,
4 34 234
13,1,3
34 2 4
1,12,123,
4 34 234
13,23,3
34 34 4
1,12,123,
4 34 234
12,1,3
1,12,123, 24 2 4
4 34 234
23,2,3 1,12,123,
34 4 4 4 34 234
12,2,3
24 4 4 1,12,123,
4 34 234
12,1,12 1,12,123,
24 2 23 4 34 234
1,12,123, 1223,12,13
1,12,123, 4 34 234
4 34 234 12,2,12
23,2,2 24 4 23
34 4 3
1,12,123,
4 34 234
23,2,1
34 3 3
1,12,123,
4 34 234
12,2,2
23 4 3
1,12,123,
4 34 234
12,2,1
23 3 3
Figure 1. Exchange graph of minors
neighbours obtained by replacing one of its elements x by a new one x(cid:48) related by a
k k
relation
x x(cid:48) = M +M
k k 1 2
where M and M are mutually prime monomials in {x ,...,x ,x ,...,x }, given
1 2 1 k−1 k+1 m
by precise combinatorial rules. These replacements, called mutations and denoted by µ
k
are involutive. For precise definitions and details about these constructions, we refer to
[4].
In the previous example, the coefficients are ∆1, ∆12 and ∆123 and the cluster variables
4 34 234
are all the other non-trivial minors. The extended clusters are the sets appearing at the
vertices of Figure 1.
–2–
The aim of the following sections is to describe examples of additive categorifications
of cluster algebras. It consists of enhancing the cluster algebra structure with an additive
category, some objects of which reflect the combinatorial structure of the cluster algebra;
moreover, there is an explicit formula, the cluster character associating to these particular
objects elements of the algebra, in a way which is compatible with the combinatorial
structure. The examples we develop here rely on (abelian) module categories. They are
particularcasesofcategorificationsbyexactcategoriesappearingin[6](simply-lacedcase)
and [2] (non simply-laced case). The study of cluster algebras and their categorifications
has been particularly successful these last years. For a survey on categorification by
triangulated categories and a much more complete bibliography, see [7].
2. The preprojective algebra and the cluster character
Let Q be the following quiver (oriented graph):
α (cid:40)(cid:40) (cid:118)(cid:118) β
1 (cid:104)(cid:104) 2 (cid:54)(cid:54) 3
α∗ β∗
Asusual,denotebyCQtheC-algebra,abasisofwhichisformedbythepaths(including
0-length paths supported by each of the three vertices) and the multiplication of which is
defined by concatenation of paths when it is possible and vanishes when paths can not be
composed (we write here the composition from left to right, on the contrary to the usual
composition of maps). Thus, a (right) CQ-module is naturally graded by idempotents
(0-length paths) corresponding to vertices and the action of arrows seen as elements
of the algebra can naturally be identified with linear maps between the corresponding
homogeneous subspaces of the representation. We shall use the following right-hand side
convenient notation:
   
0 0 0
2
−1 1 0 (cid:1)(cid:1) (cid:29)(cid:29)
(cid:43)(cid:43) (cid:116)(cid:116) 1 3
C (cid:105)(cid:105) C2 (cid:52)(cid:52) C2 = (cid:29)(cid:29) (cid:1)(cid:1)
−1 2
(cid:16) (cid:17)  
1 0 1 0
(cid:29)(cid:29)
0 1 3
where each of the digits represents a basis vector of the representation and each arrow a
non-zero scalar (1 when not specified) in the corresponding matrix entry.
Let us now introduce the preprojective algebra of Q:
Definition 2. The preprojective algebra of Q is defined by
CQ
Π =
Q (αα∗,α∗α+β∗β,ββ∗)
the representations of which are seen as particular representations of CQ (in other
words, modΠ is a full subcategory of modCQ).
Q
–3–
Example 3. Among the following representations of CQ, the first one and the second
one are representations of Π :
Q
2
1 2 (cid:1)(cid:1) (cid:29)(cid:29)
(cid:29)(cid:29) (cid:1)(cid:1) (cid:29)(cid:29) (cid:37)(cid:37) 1 3
2 ; 1 3 ; 1(cid:101)(cid:101) 2 ; (cid:29)(cid:29) (cid:1)(cid:1) .
(cid:29)(cid:29) (cid:29)(cid:29) (cid:1)(cid:1) −1 2
3 −1 2 (cid:29)(cid:29)
3
One of the property, which is discussed in many places (for example in [6]), of the
preprojective algebra of Q, fundamental for this categorification, is
Proposition 4. The category modΠ is stably 2-Calabi-Yau. In other words, for every
Q
X,Y ∈ modΠ ,
Q
Ext1(X,Y) (cid:39) Ext1(Y,X)∗
functorially in X and Y, where Ext1(Y,X)∗ is the C-dual of Ext1(Y,X). In particular, it
is a Frobenius category (is has enough projective objects and enough injective objects and
they coincide).
Let us now define the three following one-parameter subgroups of N:
     
1 t 0 0 1 0 0 0 1 0 0 0
0 1 0 0 0 1 t 0 0 1 0 0
x (t) =   x (t) =   x (t) =  .
1 0 0 1 0 2 0 0 1 0 3 0 0 1 t
0 0 0 1 0 0 0 1 0 0 0 1
For X ∈ modΠ and any sequence of vertices a ,a ,...,a of Q, we denote by
Q 1 2 n
(cid:26) (cid:27)
X
i
Φ = 0 = X ⊂ X ⊂ ··· ⊂ X ⊂ X = X|∀i ∈ {1,2,...,n}, (cid:39) S
X,a1a2...an 0 1 n−1 n X ai
i−1
the variety of composition series of X of type a a ...a (S is the simple module, of
1 2 n ai
dimension 1, supported at vertex a ). This is a closed algebraic subvariety of the product
i
of Grassmannians
Gr (X)×Gr (X)×···×Gr (X).
1 2 n
We denote by χ the Euler characteristic. Using results of Lusztig and Kashiwara-Saito,
Geiß-Leclerc-Schro¨er proved the following result:
Theorem 5 ([6]). Let X ∈ modΠ . There is a unique ϕ ∈ C[N] such that
Q X
(cid:88) (cid:16) (cid:17) ti1ti2...ti6
ϕ (x (t )x (t )...x (t )) = χ Φ 1 2 6
X a1 1 a2 2 a6 6 X,ai1ai2...ai6 i !i !...i !
i1,i2,...,i6∈N 1 2 6 1 2 6
for every word a a a a a a representing the longest element of S (aik is the repetition
1 2 3 4 5 6 4 k
i times of a ).
k k
The map ϕ : modΠ → C[N] is called a cluster character.
Q
Remark 6. (1) Theuniquenessintheprevioustheoremiseasybecauseitiswellknown
that
x (t )x (t )...x (t )
a1 1 a2 2 a6 6
runs over a dense subset of N ;
–4–
1 2 2 3
X ∈ modΠQ S1 S2 S3 (cid:29)(cid:29) (cid:1)(cid:1) (cid:29)(cid:29) (cid:1)(cid:1)
2 1 3 2
ϕ ∈ C[N] ∆1 ∆2 ∆3 ∆12 ∆1 ∆23 ∆2
X 2 3 4 23 3 34 4
1 2 3
2 1 3 (cid:29)(cid:29) (cid:1)(cid:1) (cid:29)(cid:29) (cid:1)(cid:1)
X ∈ modΠQ (cid:1)(cid:1) (cid:29)(cid:29) (cid:29)(cid:29) (cid:1)(cid:1) 2 1 3 2
1 3 2 (cid:29)(cid:29) (cid:29)(cid:29) (cid:1)(cid:1) (cid:1)(cid:1)
3 2 1
ϕ ∈ C[N] ∆13 ∆12 ∆123 ∆12 ∆1
X 34 24 234 34 4
Figure 2. Cluster character
(2) the existence is much harder and strongly relies on the construction of semi-
canonical bases by Lusztig [8]. In particular, the fact that it does not depend
on the choice of a a a a a a is not clear a priori (see the following examples).
1 2 3 4 5 6
Example 7. We suppose that a a a a a a = 213213. Then
1 2 3 4 5 6
 
1 t +t t t t t t
2 5 2 4 2 4 6
0 1 t +t t t +t t +t t 
x (t )x (t )x (t )x (t )x (t )x (t ) =  1 4 1 3 1 6 4 6.
a1 1 a2 2 a3 3 a4 4 a5 5 a6 6 0 0 1 t3 +t6 
0 0 0 1
• The module S has only one composition series, of type 1. Therefore Φ (S ) is
1 1 1
one point and Φ (S ) = ∅ for any other a. Identifying the two members in the
a 1
formula of the previous theorem,
ϕ (x (t )x (t )x (t )x (t )x (t )x (t )) = t +t = ∆1.
S1 a1 1 a2 2 a3 3 a4 4 a5 5 a6 6 2 5 2
• The module
2
(cid:1)(cid:1) (cid:29)(cid:29)
P = 1 3
2
(cid:29)(cid:29) (cid:1)(cid:1)
2
has two composition series, of type 2312 and 2132. Therefore,
ϕ (x (t )x (t )x (t )x (t )x (t )x (t )) = t t t t = ∆12.
P2 a1 1 a2 2 a3 3 a4 4 a5 5 a6 6 1 2 3 4 34
Remark that, in this case, the only composition series which is playing a role
is 2132, even if the situation is symmetric. This justify the second part of the
previous remark.
The other indecomposable representations of Π and their cluster character values are
Q
collected in Figure 2.
Two important properties of this cluster character were proved by Geiß-Leclerc-Schro¨er
(see for example [6]):
Proposition 8. Let X,Y ∈ modΠ .
Q
(1) ϕ = ϕ ϕ .
X⊕Y X Y
–5–
(2) Suppose that dimExt1(X,Y) = 1 (and therefore dimExt1(Y,X) = 1) and let
0 → X → T → Y → 0 and 0 → Y → T → X → 0
a b
be two (unique up to isomorphism) non-split short exact sequences. Then
ϕ ϕ = ϕ +ϕ .
X Y Ta Tb
3. Minimal approximations
This section recall the definition and elementary properties of approximations. It is
there for the sake of ease. In what follows, modΠ can be replaced by any additive
Q
Hom-finite category over a field.
Definition 9. Let X and T be two objects of modΠ . A left add(T)-approximation of
Q
X is a morphism f : X → T(cid:48) such that
• T(cid:48) ∈ add(T) (which means that every indecomposable summand of T(cid:48) is an inde-
composable summand of T) ;
• every morphism g : X → T factors through f.
If, moreover, there is no strict direct summand T(cid:48)(cid:48) of T(cid:48) and left add(T)-approximation
f(cid:48) : X → T(cid:48)(cid:48), then f is said to be a minimal left add(T)-approximation.
In the same way, we can define
Definition 10. Let X and T be two objects in modΠ . A right add(T)-approximation
Q
of X is a morphism f : T(cid:48) → X such that
• T(cid:48) ∈ add(T) ;
• every morphism g : T → X factors through f.
If, moreover, thereisnostrictdirectsummandT(cid:48)(cid:48) ofT(cid:48) andrightadd(T)-approximation
f(cid:48) : T(cid:48)(cid:48) → X, then f is said to be a minimal right add(T)-approximation.
Now, a classical proposition which permits to explicitly compute approximations:
Proposition 11. Let X and T (cid:39) Ti1⊕Ti2⊕···⊕Tin be two objects in modΠ (the T ’s
1 2 n Q i
are non-isomorphic indecomposable). For i,j ∈ {1,...,n}, we denote by I the subvector
ij
space of Hom(T ,T ) consisting of the non-invertible morphisms (I = Hom(T ,T ) if
i j ij i j
i (cid:54)= j). Thus, for j ∈ {1,...,n}, we obtain a linear map
(cid:77) I ⊗Hom(X,T ) −ϕ→j Hom(X,T )
ij i j
i∈{1,...,n}
(g,f) (cid:55)→ g ◦f.
Let B be a basis of cokerϕ lifted to Hom(X,T ). Then the morphism
j j j
X −(−f)−j∈−{−1,−...−,n−},−f∈−B→j (cid:77) T#Bj
j
j∈{1,...,n}
is a minimal left add(T)-approximation of X. Moreover, any minimal left add(T)-
approximation of X is isomorphic to it.
–6–
The previous proposition has a dual version which permits to compute minimal right
approximations. In practice, this computation relies on searching morphisms up to fac-
torization through other objects. There is an explicit example of computation in Example
19.
4. Maximal rigid objects and their mutations
Let us introduce the objects the combinatorics of which will play the role of the cluster
algebra structure.
Definition 12. Let X ∈ modΠ .
Q
• The module X is said to be rigid if it has no self-extension, (i.e., Ext1(X,X) = 0).
• The module X is said to be basic maximal rigid if it is basic (i.e., it does not
have two isomorphic indecomposable summands), rigid, and maximal for these
two properties.
Remark 13. A basic maximal rigid Π -module contains Π as a direct summand (because
Q Q
Π is both projective and injective and therefore has no extension with any module).
Q
Example 14. The object
1 2 3
1 3 3 1 (cid:29)(cid:29) (cid:1)(cid:1) (cid:29)(cid:29) (cid:1)(cid:1)
(cid:29)(cid:29) (cid:1)(cid:1) ⊕ (cid:1)(cid:1) ⊕ (cid:29)(cid:29) ⊕ 2 ⊕ 1 3 ⊕ 2 ,
2 2 2 (cid:29)(cid:29) (cid:29)(cid:29) (cid:1)(cid:1) (cid:1)(cid:1)
3 2 1
thelastthreesummandsofwhicharetheindecomposableprojective-injectiveΠ -modules,
Q
is basic maximal rigid. It is easy to check that it is basic and rigid, but more difficult to
prove that it is maximal for these properties (see [6] for more details).
Remark 15. We can prove that all basic maximal rigid objects have the same number of
indecomposable summands (six in the example we are talking about).
The following result permits to define a mutation on basic maximal rigid objects. Con-
sideredasanoperationonisomorphismclassesofbasicmaximalrigidobjects, theinduced
combinatorial structure will correspond to the one of a cluster algebra.
Theorem 16 ([6]). Let T (cid:39) T ⊕ T ⊕ T ⊕ P ⊕ P ⊕ P ∈ modΠ be basic maximal
1 2 3 1 2 3 Q
rigid such that P , P and P are the indecomposable projective Π -modules and T , T
1 2 3 Q 1 2
and T are indecomposable non-projective Π -modules. Then, for i ∈ {1,2,3}, there are
3 Q
two (unique) short exact sequences
0 → T →−f T −→f(cid:48) T∗ → 0 and 0 → T∗ →−g T −→g(cid:48) T → 0
i a i i b i
such that
(1) f and g are minimal left add(T/T )-approximations ;
i
(2) f(cid:48) and g(cid:48) are minimal right add(T/T )-approximations ;
i
(3) T∗ is indecomposable and non-projective ;
i
(4) dimExt1(T ,T∗) = dimExt1(T∗,T ) = 1 and the two short exact sequences do not
i i i i
split ;
(5) µ (T) = T/T ⊕T∗ is basic maximal rigid ;
i i i
–7–
(6) T and T do not have common summands.
a b
Remark 17. In the previous theorem, the existence and uniqueness, regarding the first
two conditions, are automatic, except the fact that the extremities of the two short exact
sequences coincide up to order. This fact strongly relies on the stably 2-Calabi-Yau
property. It implies that µ is involutive.
i
Definition 18. In the previous theorem, µ is called the mutation in direction i. The
i
short exact sequences appearing are called exchange sequences.
Example 19. Let
1 2 3
1 3 3 1 (cid:29)(cid:29) (cid:1)(cid:1) (cid:29)(cid:29) (cid:1)(cid:1)
T = (cid:29)(cid:29) (cid:1)(cid:1) ⊕ (cid:1)(cid:1) ⊕ (cid:29)(cid:29) ⊕ 2 ⊕ 1 3 ⊕ 2 .
2 2 2 (cid:29)(cid:29) (cid:29)(cid:29) (cid:1)(cid:1) (cid:1)(cid:1)
3 2 1
(cid:18) (cid:19)
3 3
Using Proposition 11, we get a left add T/ (cid:1)(cid:1) -approximation of (cid:1)(cid:1) :
2 2
3 1 3
(cid:1)(cid:1) → (cid:29)(cid:29) (cid:1)(cid:1)
2 2
and computing the cokernel, we get the exchange sequence:
3 1 3
0 → (cid:1)(cid:1) → (cid:29)(cid:29) (cid:1)(cid:1) → S1 → 0
2 2
so that
1 2 3
1 3 1 (cid:29)(cid:29) (cid:1)(cid:1) (cid:29)(cid:29) (cid:1)(cid:1)
µ2(T) = (cid:29)(cid:29) (cid:1)(cid:1) ⊕S1 ⊕ (cid:29)(cid:29) ⊕ 2 ⊕ 1 3 ⊕ 2 .
2 2 (cid:29)(cid:29) (cid:29)(cid:29) (cid:1)(cid:1) (cid:1)(cid:1)
3 2 1
Doing mutation in the reverse direction:
3
(cid:1)(cid:1) 3
0 → S1 → 2 → (cid:1)(cid:1) → 0.
(cid:1)(cid:1) 2
1
Let us now compute µ µ (T) with its two exchange sequences:
1 2
2
1 3 (cid:1)(cid:1) (cid:29)(cid:29) 2
0 → (cid:29)(cid:29) (cid:1)(cid:1) → S1 ⊕ 1 3 → (cid:1)(cid:1) → 0
2 (cid:29)(cid:29) (cid:1)(cid:1) 1
2
3
2 1 (cid:1)(cid:1) 1 3
0 → (cid:1)(cid:1) → (cid:29)(cid:29) ⊕ 2 → (cid:29)(cid:29) (cid:1)(cid:1) → 0
1 2 (cid:1)(cid:1) 2
1
1 2 3
2 1 (cid:29)(cid:29) (cid:1)(cid:1) (cid:29)(cid:29) (cid:1)(cid:1)
µ1µ2(T) = (cid:1)(cid:1) ⊕S1 ⊕ (cid:29)(cid:29) ⊕ 2 ⊕ 1 3 ⊕ 2 .
1 2 (cid:29)(cid:29) (cid:29)(cid:29) (cid:1)(cid:1) (cid:1)(cid:1)
3 2 1
–8–
Computing inductively all the mutations, we obtain the exchange graph of maximal
rigid objects of Π (Figure 3).
Q
Then, using Proposition 8 and Theorem 16 together with other technical results, we
get the following proposition:
Proposition 20 ([6]). If we project the mutation of maximal rigid objects to C[N] through
the cluster character ϕ, we get a cluster algebra structure on C[N] (in the sense of [4]).
Moreover, this structure is the one proposed combinatorially in [1]. Under this projection,
we get the correspondence:
{non projective indecomposable objects} ↔ {cluster variables}
{projective indecomposable objects} ↔ {coefficients}
{basic maximal rigid objects} ↔ {extended clusters}
Example 21. Taking the notation of Example 19 and looking at Figure 2, we get:
∆1∆2 = ϕ ϕ = ϕ +ϕ = ∆12 +∆1
2 4 S1 3 1 3 3 24 4
(cid:1)(cid:1) (cid:29)(cid:29) (cid:1)(cid:1) (cid:1)(cid:1)
2 2 2
(cid:1)(cid:1)
1
and
∆12∆1 = ϕ ϕ = ϕ +ϕ
24 3 1 3 2 2 3
(cid:29)(cid:29) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:29)(cid:29) 1 (cid:1)(cid:1)
2 1 S1⊕1 3 (cid:29)(cid:29) ⊕ 2
(cid:29)(cid:29) (cid:1)(cid:1) 2 (cid:1)(cid:1)
2 1
= ϕ ϕ +ϕ ϕ = ∆1∆12 +∆12∆1.
S1 2 1 3 2 34 23 4
(cid:1)(cid:1) (cid:29)(cid:29) (cid:29)(cid:29) (cid:1)(cid:1)
1 3 2 2
(cid:29)(cid:29) (cid:1)(cid:1) (cid:1)(cid:1)
2 1
which can be easily checked by hand. These are part of the equalities which appear in
the proof of Proposition 1.
5. From simply-laced case to general one
Define the following symplectic form:
 
0 0 0 1
 0 0 −1 0
Ψ =  
 0 1 0 0
−1 0 0 0
and the subgroup
N(cid:48) = {M ∈ N|tMΨM = Ψ} or, equivalently N(cid:48) = NZ/2Z
where Z/2Z = (cid:104)g(cid:105) acts on N by M (cid:55)→ Ψ−1(tM−1)Ψ. The group N(cid:48) is a maximal
unipotent subgroup of a symplectic group of type C .
2
–9–
2 2
(cid:0)(cid:0) (cid:30)(cid:30) ⊕S1⊕ (cid:0)(cid:0)
1 3 1
2 2 2
(cid:0)(cid:0) (cid:30)(cid:30) ⊕ (cid:30)(cid:30) ⊕ (cid:0)(cid:0)
1 3 3 1
2
(cid:0)(cid:0) (cid:30)(cid:30) ⊕S1⊕S3
1 3
2 2
(cid:0)(cid:0) (cid:30)(cid:30) ⊕ (cid:30)(cid:30) ⊕S3
1 3 3
1 3
(cid:30)(cid:30) (cid:0)(cid:0) ⊕S1⊕S3
2
2 3
(cid:30)(cid:30) ⊕ (cid:0)(cid:0) ⊕S3
3 2 1 (cid:30)(cid:30) (cid:0)(cid:0) 3⊕ (cid:0)(cid:0) 3⊕S3
2 2
1 3 1
(cid:30)(cid:30) (cid:0)(cid:0) ⊕S1⊕ (cid:30)(cid:30)
2 2 1 2
(cid:30)(cid:30) ⊕S1⊕ (cid:0)(cid:0)
2 1
1 3 3 1
2 3 (cid:30)(cid:30) (cid:0)(cid:0) ⊕ (cid:0)(cid:0) ⊕ (cid:30)(cid:30)
(cid:30)(cid:30) ⊕ (cid:0)(cid:0) ⊕S2 2 2 2
3 2
2 2
(cid:30)(cid:30) ⊕S2⊕ (cid:0)(cid:0)
3 1
1 3
(cid:30)(cid:30) ⊕ (cid:0)(cid:0) ⊕S2
2 2
1 2
(cid:30)(cid:30) ⊕S2⊕ (cid:0)(cid:0)
2 1
Figure 3. Exchange graph of maximal rigid objects (up to projective summands)
The only non-trivial action of Z/2Z on Q induces an action on Π and therefore on
Q
modΠ . Denote by π : C[N] → C[N(cid:48)] the canonical projection. We can now formulate
Q
the following result:
Theorem 22 ([2]). (1) If T is a Z/2Z-stable basic maximal rigid Π -module, then
Q
µ µ (T) = µ µ (T). Moreover, µ µ (T) and µ (T) are also Z/2Z-stable.
1 3 3 1 1 3 2
(2) If X ∈ modΠ , then π(ϕ ) = π(ϕ ).
Q X gX
(3) If we denote µ¯ = µ and µ¯ = µ µ = µ µ , acting on the set of Z/2Z-stable
2 2 1 1 3 3 1
maximal rigid Π -modules, µ¯ induces through π ◦ ϕ the structure of a cluster
Q
algebra on C[N(cid:48)], the clusters of which are projections of the Z/2Z-stable ones of
C[N].
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