Quivers with potentials for cluster varieties associated to braid semigroups

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by  Efim Abrikosov

QUIVERS WITH POTENTIALS FOR CLUSTER VARIETIES
ASSOCIATED TO BRAID SEMIGROUPS.
EFIM ABRIKOSOV
7
1
0
Abstract. Let C be a simply laced generalized Cartan matrix. Given an element b of the
2
generalized braid semigroup related to C, we construct a collection of mutation-equivalent
n
quivers with potentials. A quiver with potential in such a collection corresponds to an
a
J expression of b in terms of the standard generators. For two expressions that differ by a
braid relation, the corresponding quivers with potentials are related by a mutation.
3
] assTochieatmedaitnoaeplepmliecnattsioonfothfethbirsaridessuelmt iisgraoucponrsetlrautecdtiotno Cof.a family of CY3 A∞-categories
T
In particular, we construct a canonical up to equivalence CY A -category associated to
R quotient of any Double Bruhat cell Gu,v/AdH in a simply lace3d re∞ductive Lie group G.
.
h We describe the full set of parameters these categories depend on by defining a 2-
t dimensional CW-complex and proving that the set of parameters is identified with second
a
m cohomology group of this complex.
[

1
v 1. Introduction.
2
7
Awell-knownconstructionassociatesaKac-Moodyalgebrag(C)toanygeneralizedCartan
6
0 matrix C (see e.g. [Kac90]). If C is n n symmetric matrix it is convenient to encode it by
0 a graph Γ with n vertices, where ×
.
1
0
# of edges between α and β = C ,
7 αβ
−
1
and C is the corresponding entry of the generalized Cartan matrix. In the classical setting
: αβ
v g(C) is a simple Lie algebra or an affine Lie algebra and the graph is the corresponding
i
X Dynkin diagram or its extended version.
r Further, one can define a braid semigroup B . It is generated by the set of generators
a Γ
s α Vertices of Γ subject to braid relations. In the simply laced case, that is, when
α
{ | ∈ }
Γ has at most one edge between any two vertices, these relations take form:
s s s = s s s , if C = 1;
α β α β α β αβ
−
s s = s s , if C = 0.
(cid:40) α β β α αβ
A further quotient by relations s2 = e gives a standard realization of the Weyl group W. We
α
will often use the term “braid semigroup” for the doubled braid semigroup B = B B .
Γ Γ Γ
×
Key words and phrases. Representation Theory, Cluster Varieties, Quivers with Potential.
1
2 EFIM ABRIKOSOV
In [FG05] a cluster variety corresponding to an arbitrary element b B was defined.
b Γ
X ∈
Recallthataclusteralgebra[FZ01]oraclustervarietystructure[FG03], isdescribedbycom-
binatorial data that consists of a collection of seeds related by involutions called mutations.
In simply laced case a seed is just a quiver with variables attached to its vertices.
Thus, for each b B , Fock and Goncharov describe a collection of quivers related
Γ
∈
by mutations. Elements of this collection correspond to reduced presentations for b via
generators. Hence, for every word x = s s ...s , there is a quiver Q(x). If x and y are
i1 i2 ik
related by a single braid relation, then Q(y) is obtained from Q(x) by an explicit mutation.
In particular any pair of elements (u,v) of a classical Weyl group, being lifted to the braid
semigroup, provides a cluster Poisson variety structure on the corresponding double Bruhat
cell of the related simple Lie group Gu,v G [FG05]. The corresponding cluster algebra
⊂
structure was described in [BFZ03].
Main corollary of results of our paper is a CY -categorification of this picture for arbitrary
3
simply laced generalized Cartan matrices. This is done using the construction from [KS08]
that associates to a quiver with potential a CY A -category equipped with a collection of
3 ∞
spherical generators. A potential is a (possibly infinite) linear combination of cyclic paths in
the path algebra of the quiver (see section 2 for more precise definitions). Roughly speaking,
vertices of the quiver correspond to spherical generators of the category, and cycles in the
potential encode higher compositions of Ext1 in the associated CY A -category.
3 ∞
Quivers with potential were introduced in [DWZ07], where authors also described process
of mutation extended to potentials. From the point of view of [KS08] these mutations is a
combinatorial incarnation of certain transformations of collections of spherical generators.
Two quivers with potential are called mutation-equivalent if there is a sequence of mutations
relating them.
TheCY -categorificationofclustervarietiesisinformallydepictedinthefollowingdiagram:
3
CY A -category
Cluster variety 3 ∞
Quivers with potentials
Quivers related by mutations −→
related by mutations
{ }
(cid:26) (cid:27)
We want to stress that the space of all potentials for a given quiver is infinite-dimensional.
So a priori a cluster variety gives rise to an infinite dimensional family of CY A -categories.
3 ∞
Our main result, Theorem 1.1, tells that there is an explicitly described finite dimensional
family of CY A -categories up to equivalences associated to any element b B .
3 ∞ Γ
∈
Theorem 1.1. Let Γ be a graph with at most one edge between any two vertices.
Then any element b of the braid semigroup B (subject to technical restrictions (4.1,4.2)),
Γ
gives rise to a collection of mutation-equivalent quivers with potential, each corresponding to a
reduced expression of b. Furthermore, there is a finite-dimensional family of such collections.
QUIVERS WITH POTENTIALS FOR CLUSTER VARIETIES ASSOCIATED TO BRAID SEMIGROUPS. 3
Here are a few remarks about the theorem. Quivers with potentials associated to bipartite
ribbon graphs were first considered by physicists, see e.g. [FHK+05]. A family of mutation-
equivalent quivers with potentials associated to triangulated surfaces was established by
Labardini-Fragoso in [LF08]. The underlying quivers were considered in [GSV03], [FG03].
Any such quiver corresponds to a triangulation of a surface and if two triangulations are
related by a flip, then two associated quivers differ by mutation. Quivers with potentials
for the moduli space of framed n-dimensional local systems on a decorated surface were
introduced in [Gon16]: from this point of view the n = 2 case is the one studied in [LF08].
The proof of Theorem 1.1 is based on careful study of the class of potentials that we call
primitive, it is preserved by mutations and exhibits particularly nice properties. We denote
a quiver with potential associated to a presentation x of b by (Q(x),W ). We provide a
x
full description of the space of primitive potentials on Q(x) modulo the action of the group
of right-equivalences. This space is identified with the second cohomology group of a 2-
(cid:101)
dimensional CW-complex (x) whose 1-skeleton coincides with the quiver (Proposition 4.8).
(cid:101)
C
If two quivers are related by mutations, then the associated CW-complexes are homotopy
equivalent. Based on these facts the proof of Theorem 1.1 boils down to the check that the
(cid:101)
second cohomology class associated to a primitive potential is preserved under mutations.
The first application of our construction is when Γ is a simply laced Dynkin diagram.
Then we obtain CY -categories related to Double Bruhat cells. More precisely, quivers Q(x)
3
actually describe cluster structure on quotients by the adjoint action of Cartan subgroup:
Gu,v/AdH. The potentials we introduce give rise to certain combinatorially described cat-
(cid:101)
egories. In this case our description of the space of primitive potentials implies that all
of them are right-equivalent, it follows that there is a unique up to equivalence category
associated to a primitive potential for Gu,v/AdH.
The geometric origin of this category is a subject for the future research: we believe
that it should describe a full subcategory of a wrapped Fukaya category of certain non-
compact Calabi-Yau threefold. This program in a different context of quivers with potentials
associated to triangulated surfaces [LF08] was carried out in detail in [BS13],[Smi13]. It was
conjectured in [Gon16] that the CY A -categories associated to the quivers with potentials
3 ∞
forthemodulispaceofframedn-dimensionallocalsystemsondecoratedsurfacesaresimilarly
related to wrapped Fukaya categories of the open CY threefolds studied in [DDP06].
Another application is for Γ = A , n 2, the extended Dynkin diagram of type A.
n
≥
Cluster structure in this case was studied in detail in [FM14]. In this paper, Fock and
Marshakov proved that a class of inte(cid:98)grable systems related to dimers on a torus, introduced
by Goncharov and Kenyon in [GK11], can be realized using cluster varieties associated to
quivers Q(x). Theorem 1.1 in this case gives a 1-parametric family of CY -categories. In
3
fact, the CW-complex that controls the family of potentials is naturally identified with the
torus where dimers are defined. The second cohomology group of the torus is indeed 1-
(cid:101)
dimensional. As explained in [GK11], the cluster integrable systems related to dimers on
4 EFIM ABRIKOSOV
a torus are parametrized by Newton polygons N. It is plausible that our CY -categories
3
describe full subcategories of a 1-parametric family of wrapped Fukaya categories of the
toric Calabi-Yau threefold associated with the cone over the polygon N. CY A -categories
3 ∞
I am very grateful to Alexander Goncharov for introducing me to this subject and for many
useful discussions and remarks.
The structure of the paper is the following: in section 2 we recall precise definitions for
quivers with potentials; in section 3 the construction of quivers Q(x) is given; section 4 is
devoted to the study of primitive potentials on these quivers.
(cid:101)
2. Quivers with potential and their mutations.
2.1. Generalities on quivers with potential. In this section we provide main definitions
related to quivers with potentials and introduce notations.
For us a quiver Q is a finite collection of vertices I connected by oriented edges. In our
approach quivers of the interest will be glued from elementary pieces along subsets of special
vertices called frozen. Arrows between frozen vertices are allowed to have weight 1, so after
2
gluing of two quivers half-arrows either combine to a usual arrow or cancel out depending
on their mutual orientations. More specifically any quiver can be encoded by the following
collection:
Definition 2.1. A quiver Q is given by a tuple (I,I ,E,E ,s,t) where:
0 0
(i) I is the set of vertices;
(ii) I I is the subset of frozen vertices;
0
⊂
(iii) E is the set of arrows;
(iv) E E is the subset of half-arrows;
0
⊂
(v) these sets are related by maps s,t : E I assigning to every arrow its source and
→
target.
We require that half-arrows can be adjacent only to frozen vertices, i.e. restrictions of s and
t to E take values in I .
0 0
Sometimes we use notation Q (resp. Q ) for the set of vertices (resp. arrows) of quiver Q.
0 1
The definition of a potential of a quiver relies on the notion of path algebra R Q that we
(cid:104) (cid:105)
will describe momentarily. Informally, elements of a path algebra R Q can be thought of as
(cid:104) (cid:105)
being (linear combinations of) products of composable arrows. By “composable” we mean
that the source of every successive arrow coincides with the target of the preceding. Such
a product is usually called “a path”. Our convention is to write products from right to left
as for standard composition of maps. Note that source and target maps s,t on Q can be
extended to all paths.
Precisely there is the following definition. Fix a base field k of characteristic zero, and let
R = kQ0 and A = kQ1 be rings of k-valued functions on vertices and arrows respectively.
QUIVERS WITH POTENTIALS FOR CLUSTER VARIETIES ASSOCIATED TO BRAID SEMIGROUPS. 5
Note that R is generated by a collection of idempotents i Q corresponding to δ-functions,
0
∈
and A posesses a natural structure of R-bimodule:
i a j = δ δ a i,j Q , a Q
i,t(e) s(e),j 0 1
· · ∀ ∈ ∈
To obtain the path algebra we take iterated tensor products of A over R.
Definition 2.2. A path algebra of a quiver Q is the graded tensor algebra
∞
R Q = A A... A,
R R
(cid:104) (cid:105) ⊗ ⊗
(cid:77)d=0 d
with the convention that 0-th graded component is R (its elements are sometimes called
(cid:124) (cid:123)(cid:122) (cid:125)
“lazy paths”). Any path algebra has a maximal ideal generated by elements of positive
degree, that is by paths of nonzero length, we denote it by m(Q). Define the completed path
algebra R Q as the completion of R Q with respect to powers of the maximal ideal.
(cid:104)(cid:104) (cid:105)(cid:105) (cid:104) (cid:105)
One of the advantages of using completed path algebras is that that R-homomorphisms of
completed algebras have a very simple form. Let Q and Q(cid:48) be two quivers with the same set
of vertices and with arrow spaces A and A(cid:48). Then structure of these maps is explained by
the following proposition which is a little reformulation of Proposition 2.4 from [DWZ07]:
Proposition 2.3. Any pair (ϕ(1),ϕ(2)) of R-bimodule homomorphisms ϕ(1) : A A(cid:48) and
→
ϕ(2) : A m(Q(cid:48))2 gives rise to a unique homomorphism of completed path algebras ϕ :
→
R Q R Q(cid:48) such that ϕ = id and ϕ = (ϕ(1),ϕ(2)). Furthermore, ϕ is isomorphism
R A
(cid:104)(cid:104) (cid:105)(cid:105) → (cid:104)(cid:104) (cid:105)(cid:105) | |
if and only if ϕ(1) is a R-bimodule isomorphism.
Thus,automorphismsofpathalgebrascanbeunderstoodascertainlower-triangularmatrices
with respect to the grading.
Main objects of our study are potentials on quivers. They may be understood as (possibly
infinite) linear combinations of closed paths considered up to cyclic order of arrows:
Definition 2.4. Let R Q be the linear subspace of the completed path algebra gener-
cyc
ated by cyclic paths, i.e(cid:104).(cid:104)pr(cid:105)o(cid:105)ducts of the form n a such that t(a ) = s(a ). A potential
i=1 i 1 n
on Q is an element of R Q considered up to cyclic shift:
cyc
(cid:104)(cid:104) (cid:105)(cid:105) (cid:81)
a a ...a a a ...a
1 2 n n 1 n−1
←→
Remark: ItmightbehelpfultodefinedualpathalgebraR Q∗ whichisobtainedbyreplacing
(cid:104) (cid:105)
vector spaces of arrows by their dual vector spaces. Then functionals on this algebra are
isomorphic to R Q . Potentials can be viewed as functionals on R Q∗ /[R Q∗ ,R Q∗ ], so
(cid:104)(cid:104) (cid:105)(cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
it becomes clear why they are considered up to cyclic shifts.
Thus a quiver with potential (QP for short) is just a pair (Q,W) where Q is a quiver and
W is an element of R Q considered up to cyclic shifts. It immediately follows from
cyc
(cid:104)(cid:104) (cid:105)(cid:105)
definitions that R-homomorphisms of completed path algebras induce maps of QPs.
6 EFIM ABRIKOSOV
i i i
d d d
b ¯b ¯b
a k e l a [bc] k e¯ l k e¯ l
c c¯ c¯
[ec] [ec]
j j j
Figure 1. (Q,W = abc) Figure 2. (µkQ,µkW) Figure 3. (µkQ,µkW = c¯e¯[ec])
Definition 2.5. Let (Q,W) and (Q(cid:48),W(cid:48)) be t(cid:101)wo qu(cid:101)ivers with potentials on the same set
of vertices. We say that they are right-equivalent if there is an R-algebra isomorphism
ϕ : R Q R Q(cid:48) such that ϕ(W) = W(cid:48).
(cid:104)(cid:104) (cid:105)(cid:105) → (cid:104)(cid:104) (cid:105)(cid:105)
Given two quivers with potential on the same set of vertices one can form their direct sum
(Q,W) (Q(cid:48),W(cid:48)) in an intuitive sense: this is new quiver with potential where all arrow
⊕
spaces are direct sums of arrow spaces of the summands and the potential is the sum of their
potentials.
Definition 2.6. A quiver with potential is called trivial if it is right-equivalent to the direct
sum (Q ,W ) where each summand has exactly two arrows a and b forming a 2-cycle and
i i i i
the potential is the product of these arrows W = a b .
i i i
(cid:76)
If degree 2 component of the potential of a quiver is zero then it is called reduced.
Next we quote “Splitting Theorem” from [DWZ07] which is essential for mutations:
Theorem 2.7. Every quiver with potential (Q,W) is right-equivalent to direct sum of reduced
and trivial quivers with potential (Q ,W ) (Q ,W ). Moreover, right-equivalence
red red triv triv
⊕
class of each of the summands is determined by right-equivalence class of (Q,W).
It is often useful to replace a quiver with potential by its reduced part, we will call this
procedure reduction.
2.2. Mutation of quivers with potential. Throughoutthissectionweassumethatquiver
Q has no loops and there are no half-arrows. It is known that to every vertex of Q there
corresponds a transformation of the quiver called a mutation. Remarkably this process can
be defined for quivers with potentials too. We briefly recall this story in this section.
Let k be any nonfrozen vertex of (Q,W) that does not belong to a 2-cycle. Then we are
going to produce another quiver with potential denoted by (µ Q,µ W) that is the mutation
k k
QUIVERS WITH POTENTIALS FOR CLUSTER VARIETIES ASSOCIATED TO BRAID SEMIGROUPS. 7
of (Q,W) at vertex k. To do so we first construct premutated quiver (µ Q,µ W) possibly
k k
having 2-cycles and then we reduce it using Theorem 2.7.
Define arrow spaces of the premutated quiver as follows:
(cid:101) (cid:101)
A A A if i,j =k;
i,j i,k k,j
(2.1) µ A = ⊕ ⊗ (cid:54)
k i,j A∗ otherwise.
(cid:40) j,i
Here A = i A j denote(cid:101)s the vector space of arrows from vertex j to i. One can interpret
i,j
· ·
this operation on the level of quivers in simple combinatorial terms: in the first case we add
new shortcut arrow called [ab] for every 2-step path ab in Q passing through vertex k, in the
second case we reverse all arrows adjacent to vertex k. Let is indicate this by putting bars
on the top, e.g. a¯ stands for reversed arrow a.
Now we describe the effect of mutation on the potential. Denote by [W] the potential for
the quiver µ Q obtained by replacing all occurrences of 2-step paths through k by corre-
k
sponding shortcuts. Then by definition:
(cid:101)
¯
(2.2) µ W = [W]+ ba¯[ab]
k
a,b: s(a)=t(b)=k
(cid:88)
The sum over 2-step paths thr(cid:101)ough k corresponds to triangular terms that are formed by new
¯
shortcut arrows [ab] and reversed arrows a¯, b. Note that these terms are canonical elements
in A∗ A∗ A A . Finally, as mentioned above µ (Q,W) is obtained by
k,s(b) ⊗ t(a),k ⊗ t(a),k ⊗ k,s(b) k
removing the trivial part of (µ Q,µ W).
k k
Example: In the example in the sequence of Figures 1-3 we take potential of the original
quiver shown on the left to be W = abc. After mutation at vertex k, two new shortcut
(cid:101) (cid:101)
arrows appear in µ Q (they are shown in green). Furthermore, all arrows attached to k are
k
reversed. We have
¯
(cid:101) µkW = [W]+c¯b[bc]+c¯e¯[ec],
since [W] = a[bc] and there are two additional triangles corresponding to each shortcut.
Finally, to reduce premutated qu(cid:101)iver we have to decompose it to the direct sum of its trivial
and reduced parts. That can be done as follows: apply right-equivalence ϕ that sends
¯
a a c¯b and leaves other arrows unchanged, then ϕ(µ W) = a[bc] + c¯e¯[ec]. Hence the
k
(cid:55)→ −
2-cycle a[bc] can be split out and it is deleted after reduction. The result of the mutation is
shown in Figure 3, corresponding potential is µ W = c¯e¯[ec].
k (cid:101)
The following two theorems from [DWZ07] justify that mutation is well-defined on right-
equivalence classes of QPs and that mutation acts there as an involution.
Theorem2.8. Theright-equivalenceclassof(µ Q,µ W)isdeterminedbytheright-equivalence
k k
class of (Q,W).
8 EFIM ABRIKOSOV
Theorem 2.9. The correspondence µ : (Q,W) (µ Q,µ W) acts as an involution on
k k k
−→
the set of right-equivalence classes of reduced quivers with potentials.
3. Quivers with potentials from words of the free semigroup M .
Γ
In this section we describe the class of quivers of our interest. Particular examples of
such quivers correspond to cluster structures on Poisson leaves of Poisson-Lie groups. Our
quivers are naturally obtained by gluing certain elementary pieces, this approach called
amalgamation was developed by V. Fock and A. Goncharov in [FG05].
3.1. Amalgamation of quivers. Here we review amalgamation technique from [FG05].
Let (Qp) = (Ip,Ip,Ep,Ep,sp,tp) be a collection of quivers labelled by a finite set
p∈P 0 0 p∈P
P. In order to amalgamate them we need to fix gluing data that consists of a set IP and
P
a collection of maps ιp : Ip (cid:44) IP that cover IP. If some k IP is covered by more than one
→ ∈
ιr then all its preimages are required to be frozen.
Assume that all quivers have no 2-cycles, then arrows in Qp can be uniquely described by
skew-symmetric adjacency matrix whose entries are half-integers εr , i,j Ip equal to the
i,j ∈
number of arrows from j to i minus the number of arrows from i to j in Qp (half-arrows are
taken with weight 1).
2
Definition 3.1. The amalgamationofthecollection(Qp) correspondingtogluingdata
p∈P
is the new quiver QP with the set of vertices IP and the adjacency matrix given by
P
εP = εp
i,j i(cid:48),j(cid:48)
where the sum is taken over the set of pairs (cid:88)(i(cid:48),j(cid:48)) i(cid:48), j(cid:48) Ip, ιp(i(cid:48)) = i, ιp(j(cid:48)) = j .
{ | ∈ }
In other words, we take all arrows whose endpoints are mapped to vertices i and j in IPand
compute the resulting number of arrows taken with appropriate signs and weights. The
set of frozen vertices IP IP is defined as the union of images of frozen vertices. After
0 ⊂
amalgamation it may happen that there are frozen vertices with no adjacent half-arrows. If
this is the case we can defrost them, i.e. exclude them from IP. Since it is usually evident
0
from the context what arrows can be defrosted we usually omit these details.
δ 3.2. Construction of quivers Q(x). In this section
a class of quivers generalizing [FG05] is defined. Let
Γ be a graph with no loops and at most one edge con-
α α necting any two vertices, denote its set of vertices by
1 2
Π. In geometric applications related to simply laced
Poisson-Lie groups and loop groups this graph is just
the corresponding Dynkin diagram or its extended
γ
version and elements of Π are identified with simple
roots. WedefineafreesemigroupM andconstructa
Γ
β
Figure 4. Elementary quiver
QUIVERS WITH POTENTIALS FOR CLUSTER VARIETIES ASSOCIATED TO BRAID SEMIGROUPS. 9
collection of quivers with potential (Q(x),W(x)) as-
sociated to every element x M . These quivers
Γ
∈
admit a number of mutations that preserve the collection. Such mutations are related to
braid relations exchanging generators of the free semigroup.
The semigroup M is generated by letters s ,s α Π . Denote by C the associated
Γ α α¯
{ | ∈ }
generalized Cartan matrix. Its entries are defined as follows:
C = 1, if α and β are connected by an edge in Γ;
αβ
−
(3.1) C = 2, if α = β;
 αβ
Cαβ = 0, otherwise.
Braid semigroup BΓ is the quotient of Mγ by relations of three types:
(i) s s = s s , s s = s s if C = 0
α β β α α¯ β¯ β¯ α¯ αβ
(ii) s s s = s s s , s s s = s s s if C = 1
α β α β α β α¯ β¯ α¯ β¯ α¯ β¯ αβ (3.2)
−
(iii) s s = s s
α β¯ β¯ α
In [FG05] a quiver Q(x) was associated to any word x M (where D is arbitrary Dynkin
D
∈
diagram). Moreover, it is shown there that if two words differ by relation of the first type
then corresponding quivers are either identical or differ by explicit mutation. It follows from
this construction that one can associate cluster variety to any element of the braid semigroup
B . Our main goal is to extend this construction to quivers with potentials.
D
Take an element x = s s ...s M (subindices i can be α or α¯). To construct
i1 i2 ik ∈ Γ p
a general quiver Q(x) we first associate a quiver Q(s ) to every generator s M , such
α α Γ
∈
quiverswillbecalledelementary. AfterthatwedescribeQ(x)asamalgamationofelementary
quivers (Q(s )) .
ip p=1...k
For any α Π the set of vertices of elementary quiver Q(s ) is identified with I =
α α
∈
(Π α) α(cid:48) α(cid:48)(cid:48). There are no arrows between i,j I unless one of the vertices is α(cid:48)
α
− ∪ ∪ ∈
or α(cid:48)(cid:48), in the latter case the corresponding entry in the adjacency matrix ε(α) is defined
by formulas below (see Fig. 4). Elementary quiver Q is obtained by reversing all arrows
sα¯
in Q .
sα
C C
αβ αβ
(3.3) ε(α) = 1, ε(α) = , ε(α) = , β (Π α).
α(cid:48)α(cid:48)(cid:48) α(cid:48)β α(cid:48)(cid:48)β
− 2 − 2 ∈ −
In order to specify the gluing data for the collection (Q(s )) we define the set of
ij j=1...k
vertices of amalgamated quiver and introduce suitable notation on the way. Let n be
α
the number of letters s and s in the expression for x. Then the set of vertices of the
α α¯
amalgamated quiver Q(x) is by definition identified with the set of pairs
10 EFIM ABRIKOSOV
α3 α3 α3 α3
0 1 0 1
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
α2
α2 1 α2 α2 α22 α2
0 (cid:0) (cid:1) 2 0 2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) α2 (cid:0) (cid:1) (cid:0) (cid:1)
1
(cid:0) (cid:1)
α1 α1 α1 α1 α1 α1 α1
0 1 2 3 0 1 2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Figure 5. Q for x = s s s s s s Figure 6. Q for y = s s s s s s
x α1 α2 α3 α1 α2 α1 y α1 α2 α3 α2 α1 α2
α
(3.4) I = α Π,0 r n ,
x α
r ∈ ≤ ≤
(cid:26)(cid:18) (cid:19) (cid:27)
(cid:12)
where α is just our a notation for the(cid:12) corresponding pair. It is convenient to use the
i (cid:12)
following terminology:
(cid:0) (cid:1)
Definition 3.2. Decoration on the set of vertices of Q(x) or Q(x) is the map from I to
x
Π that sends vertex α to α Π. We say that a vertex of the quiver is decorated by α if its
r ∈
image under the decoration map is α. The set of all vertices decorated by one α Π form
(cid:101)
(cid:0) (cid:1) ∈
a perch, and the number of vertices on α-perch in Q(x) is n .
α
TodescribeembeddingsofvertexsetsofelementaryquiversQ(s )toI inthegluingdata,
ij x
we introduce additional indexing on letters of x. Write(cid:101)s(r) (resp. s(r)) for r-th occurrence of
α α¯
s or s in x. For instance, the word s s s s s s acquires indices s(1)s(2)s(3)s(1)s(4)s(1)s(2),
α α¯ α α α¯ β¯ α β α α α¯ β¯ α γ β
here letters s ,s are numbered from 1 to 4; letters s ,s are numbered from 1 to 2; and
α α¯ β β¯
there is a single letter s .
γ
Now consider some quiverQ(s(r)), as mentioned above itsvertex set is(Π α ) α(cid:48),α(cid:48)(cid:48) .
α
Corresponding embedding I I maps α(cid:48) and α(cid:48)(cid:48) to α , α . A−n{y o}th∪e{r verte}x
s(αr) −→ x r−1 r
β = α(cid:48),α(cid:48)(cid:48) in Q(s(r)) is mapped to β , where j is the number of letters s or s to the left of
(cid:54) α j (cid:0) (cid:1) (cid:0) (cid:1) β β¯
s(r), or j = 0 if there are no such letters. The procedure for elementary quivers for opposite
α (cid:0) (cid:1)
letters s goes along same lines.
α¯
Example: In Figures 5,6 graph Γ is Dynkin diagram A , and x,y M are two lifts
3 ∈ A3
of the longest element of the Weyl group: x = s s s s s s , y = s s s s s s .
α1 α2 α3 α1 α2 α1 α1 α2 α3 α2 α1 α2
Informally, toobtainQ(x)orQ(y)onecanimaginethatallelementaryquiversQ areplaced
si
on parallel perches labelled by elements of Γ in the order of their occurrence in the word
(see Fig. 7,8). We say that perches labelled by α,β Γ are neighbors if C = 1. After
αβ
∈ −