On dispersing representations of quivers and their connection with representations of bundles of semichains
In the paper we discuss the notion of “dispersing representation of a quiver” and give, in a natural special case, a criterion for the problem of classifying such representations to be tame. In proving the criterion we essentially use representations of bundles of semichains, introduced about fi...
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Інститут прикладної математики і механіки НАН України
2002
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| Цитувати: | On dispersing representations of quivers and their connection with representations of bundles of semichains / V.M. Bondarenko // Algebra and Discrete Mathematics. — 2002. — Vol. 1, № 1. — С. 19–31. — Бібліогр.: 35 назв. — англ. |
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irk-123456789-1552832019-06-17T01:27:04Z On dispersing representations of quivers and their connection with representations of bundles of semichains Bondarenko, V.M. In the paper we discuss the notion of “dispersing representation of a quiver” and give, in a natural special case, a criterion for the problem of classifying such representations to be tame. In proving the criterion we essentially use representations of bundles of semichains, introduced about fifteen years ago by the author. 2002 Article On dispersing representations of quivers and their connection with representations of bundles of semichains / V.M. Bondarenko // Algebra and Discrete Mathematics. — 2002. — Vol. 1, № 1. — С. 19–31. — Бібліогр.: 35 назв. — англ. 1726-3255 1991 Mathematics Subject Classification 16G20, 16G60, 15A21 http://dspace.nbuv.gov.ua/handle/123456789/155283 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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English |
| description |
In the paper we discuss the notion of “dispersing
representation of a quiver” and give, in a natural special case, a
criterion for the problem of classifying such representations to be
tame. In proving the criterion we essentially use representations of
bundles of semichains, introduced about fifteen years ago by the
author. |
| format |
Article |
| author |
Bondarenko, V.M. |
| spellingShingle |
Bondarenko, V.M. On dispersing representations of quivers and their connection with representations of bundles of semichains Algebra and Discrete Mathematics |
| author_facet |
Bondarenko, V.M. |
| author_sort |
Bondarenko, V.M. |
| title |
On dispersing representations of quivers and their connection with representations of bundles of semichains |
| title_short |
On dispersing representations of quivers and their connection with representations of bundles of semichains |
| title_full |
On dispersing representations of quivers and their connection with representations of bundles of semichains |
| title_fullStr |
On dispersing representations of quivers and their connection with representations of bundles of semichains |
| title_full_unstemmed |
On dispersing representations of quivers and their connection with representations of bundles of semichains |
| title_sort |
on dispersing representations of quivers and their connection with representations of bundles of semichains |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/155283 |
| citation_txt |
On dispersing representations of quivers and their connection with representations of bundles of semichains / V.M. Bondarenko // Algebra and Discrete Mathematics. — 2002. — Vol. 1, № 1. — С. 19–31. — Бібліогр.: 35 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT bondarenkovm ondispersingrepresentationsofquiversandtheirconnectionwithrepresentationsofbundlesofsemichains |
| first_indexed |
2025-07-14T06:17:53Z |
| last_indexed |
2025-07-14T06:17:53Z |
| _version_ |
1837602044847849472 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2002). pp. 19–31
c© Journal ”Algebra and Discrete Mathematics”
On dispersing representations of quivers
and their connection with representations
of bundles of semichains
Vitalij M. Bondarenko
Communicated by V.V. Kirichenko
Dedicated to V. V. Kirichenko on the occasion of his 60th birthday
Abstract. In the paper we discuss the notion of “dispersing
representation of a quiver” and give, in a natural special case, a
criterion for the problem of classifying such representations to be
tame. In proving the criterion we essentially use representations of
bundles of semichains, introduced about fifteen years ago by the
author.
1. Introduction
The classical problems of linear algebra on the reduction of the matrix of
a linear map (by means of elementary row and column transformations)
and the matrix of a linear operator (by means of similarity transforma-
tions) to canonical forms can be generalized in the following two ways:
by considering a greater number of maps or giving more complicated
structure of vector spaces. The first way led finally to the notion of rep-
resentations of a quiver (P. Gabriel). As examples of a generalization of
the second type it may be mentioned the well-known vectorspace problem
[1, p. 82], its natural “two-dimensional” analog [2, 3] and a general ex-
tension of the classical problem on one linear operator [4, 3]. Clearly one
can consider various generalizations of the classical problems combining
two indicated ways. In [3] the author consider a common generalization
1991 Mathematics Subject Classification 16G20, 16G60, 15A21.
Key words and phrases: quiver, bunch of modules, dispersing representation,
bundle of semichaines.
20 On dispersing representations of quivers
of “mixed” type introducing the notion of “dispersing representation of
a quiver (with relations)”. In terms of these representations one can
formulate many classification problems, among them the problems on
representations of posets [5], bundles of semichaines [6], tangles [7], etc.
(and also all the above mentioned ones). In this paper we study dis-
persing representations of (finite and infinite) quivers without relations.
In considering criterions of tameness we essentially use a main result
on representations of bundles of finitely many semichains [8, 6] and his
extension to the case of infinitely many ones (see the last section).
2. Main notions and examples
Throughout the paper, we will keep the right-side notation. All vector
spaces over a field k will be finite-dimensional; the category of such spaces
will be denoted as usual by mod k. Unless otherwise stated, all quivers
and posets will be finite. The sign
∐
will denote the direct sum of
posets, categories or functors. Singletons will be always identified with
the elements themselves.
We first recall the definition of dispersing representations of a quiver
[3, Section 10].
Let A be a Krull-Schmidt category over a field k. By a (right) module
over A we mean as usual a k-linear functor F : A → mod k. A collection
M = {Mi} of modules Mi : A → mod k, where i run through a set
X, is said to be an X-bunch of modules over A. An X-bunch M is
said to be faithful if AnnM =
⋂
i∈X AnnMi = 0 (AnnMi being the
annihilator of Mi). We call X-bunches M and M ′ of modules over A
and A′, respectively, equivalent if there exists an equivalence F : A → A′
such that, for each i ∈ X, the modules Mi and FM ′
i are isomorphic; in
this case we write M ∼= M ′ or (A,M) ∼= (A′,M ′).
Let Γ = (Γ0,Γ1) be a (not necessarily finite) quiver with the set
of vertices (points) Γ0 and the set of arrows Γ1, and k a field. Fix a
Krull-Schmidt category A over k and a Γ0-bunch M of modules over A.
We call M -dispersing representation of Γ (or dispersing representation
with respect to M , or simply dispersing representation if M is fixed) a
pair U = (M(X), f) formed by the collection of vector spaces M(X) =
{Mi(X)| i ∈ Γ0} for an object X ∈ A and a collection f = {fα|α : i →
j run throughΓ1} of linear maps fα : Mi(X) → Mj(X). A morphism
from U = (M(X), f) to U ′ = (M(X ′), f ′) is determined by a morphism
ϕ : X → X ′ satisfying fαMj(ϕ) = Mi(ϕ)f ′
α for each arrow α : i → j.
The category of M -dispersing representations of Γ is denoted by repMΓ;
by repinv
M Γ we denote the full subcategory of repMΓ consisting of all
objects U = (M(X), f) with invertible linear maps fα (α runs over Γ1).
V. M. Bondarenko 21
If we take A =
∐
i∈Γ0
Ai with Ai = mod k for each i, and Mi =∐
j∈Γ0
Mij with Mij = δij1Aj
: Aj → mod k (δij being the Kroneker
delta), then the case of usual representations of Γ occurs.
Our notion is naturally generalized to the case of quivers with rela-
tions. Moreover, one can take any ring instead of the field k, an arbitrary
category instead of the category A or mod k, etc.
In terms of dispersing representations one can formulate many clas-
sification problems.
Example 2.1. Let Γ be the quiver
1
◦−→
2
◦ and C a finite poset which
is identified with the following category: ObC = C,C(x, y) = {(x|y)} if
x ≤ y and C(x, y) = ∅, otherwise; composition is such that (x|y)(y|z) =
(x|z). Denote by C the category ⊕kC (kC being the linearization of
C and ⊕kC its additive hull) and by N the module over C such that
N(x) = k for each x ∈ C and N(x|y) = 1k. Set A = B
∐
C with
B = mod k, and M1 = 1B
∐
0C , M2 = 0B
∐
N with the identity module
1B : B → mod k and the zero ones 0C : C → mod k, 0B : B → mod k.
Then the category of {M1,M2}-dispersing representations of Γ is in fact
the category of representations of the poset C [5, §4].
A general case of a “decomposable” bunch (as in the example) arise,
in other terms, in studying representations of dyadic sets [9, Section 0].
From the point of view of the author, the most interesting cases occur
when (in contrast to the previous case) a system M of modules is not
“decomposable” or there is a quiver with relations.
Before discussing such examples we give some definitions.
Let S = (A, ∗) be a (not necessarily finite) poset with involution. By
an S-graded vector space over k we mean the direct sum U =
⊕
a∈A Ua of
k-vector spaces Ua such that Ua∗ = Ua for all a ∈ A. A linear map ϕ of
an S-graded space U =
⊕
a∈A Ua into an S-graded space U ′ =
⊕
a∈A U ′
a
will be called an S-map if ϕa∗a∗ = ϕaa for each a ∈ A and ϕbc = 0 for
each b, c ∈ A not satisfying b ≤ c, where ϕxy denotes the linear map of Ux
into U ′
y induced by the map ϕ. The category of S-graded vector spaces
over k (with objects the S-graded spaces and with morphisms the S-
maps) is denoted by modSk 1. Because S = (A, ∗) with trivial involution
is naturally identified with A, these definitions involve the case of usual
posets. For a poset A =
∐n
i=1 Ai, we identify modAk with
∐n
i=1 modAi
k.
Recall that a semichain is by definition a poset A such that every
element of A is comparable with all but at most one elements. Obviously,
any semichain A can be uniquely represented in the form A =
⋃m
i=1 Ai,
1When S is infinite and U ∈ modSk, we have Ua = 0 for all but finitely many
a ∈ A (because we consider only finite-dimensional vector spaces).
22 On dispersing representations of quivers
where each Ai (called a link of A) consist of either one point or two
incomparable points, and A1 < A2 < · · · < Am, where, for subsets X
and Y of a poset, X < Y means that x < y for any x ∈ X, y ∈ Y (if
each Ai consist of one point, the set A is called a chain); the number
m is called the length of A. A semichain A with involution ∗ is called a
∗-semichain if x∗ = x for every x belonging to the union of all two-point
links.
Example 2.2. Let Γ be the quiver with one vertex, one loop ϕ and
one relation f(ϕ) = 0, where f(t) = t2, and let S = (A, ∗) be a poset
with involution. Set A = modSk and denote by M : modSk → mod k
the natural imbedding module. In the case when S is a ∗-semichain, M -
dispersing representations of Γ was classified in [10, §2] (in connection
with classifying the modular representations of quasidihedral groups);
the case of a chain with involution was considered earlier in [11, §1]. The
case, when A is an arbitrary Krull-Schmidt subcategory in mod k and
f(t) an arbitrary polynomial, is considered in [4, 3].
Finally we consider an example which plays a central role in our
consideration.
Example 2.3. Let S = {A1, . . . , An, B1, . . . , Bn} be a family of
pairwise disjoint semichains; set A =
∐n
i=1 Ai and B =
∐n
i=1 Bi. A
bundle of semichains A1, . . . , An, B1, . . . , Bn is a pair S = (S, ∗), where
∗ is an involution on the set S0 = A
∐
B such that x∗ = x for each x
from the union of all two-point links (of the given semichains).
Let S = (S, ∗) be a bundle of semichaines A1, . . . , An, B1, . . . , Bn. A
representation of the bundle S = (S, ∗) over a field k is a triple (U, V, ϕ),
where
(1) U = {U1, . . . , Un} and V = {V1, . . . , Vn} are collections of k-
spaces such that Ui ∈ modAi
k, Vi ∈ modBi
k (i = 1, . . . , n) , and the
A
∐
B-graded space (
⊕n
i=1 Ui) ⊕ (
⊕n
i=1 Vi) belong to the subcategory
mod(A
∐
B,∗) k of modA
∐
B k;
(2) ϕ = {ϕ1, . . . , ϕn} is a collection of linear maps ϕi ∈ Homk(Ui, Vi),
i = 1, . . . , n.
A morphism from
(U, V, ϕ) = ({U1, . . . , Un}, {V1, . . . , Vn}, {ϕ1, . . . , ϕn})
to
(U ′, V ′, ϕ′) = ({U ′
1, . . . , U
′
n}, {V
′
1 , . . . , V ′
n}, {ϕ
′
1, . . . , ϕ
′
n})
is determined by a pair (α, β) formed by a collection α = {α1, . . . , αn}
of Ai-maps αi: Ui → U ′
i and a collection β = {β1, . . . , βn} of Bi-maps
βi: Vi → V ′
i (i = 1, . . . , n) such that
V. M. Bondarenko 23
(3) the A
∐
B-map (
⊕n
i=1 αi) ⊕ (
⊕n
i=1 βi ) of (
⊕n
i=1 Ui) ⊕ (
⊕n
i=1 Vi)
into (
⊕n
i=1 U ′
i) ⊕ (
⊕n
i=1 V ′
i ) belong to the subcategory mod(A
∐
B,∗) k;
(4) ϕiβi = αiϕ
′
i for each i = 1, . . . , n.
The category of representations of the bundle of semichains S = (S, ∗)
is denoted by Bk (S) = Bk (S, ∗) = Bk (A1, . . . , An, B1, . . . , Bn, ∗).
The definition of representations of bundles of semichaines can be
easily rewrited in terms of dispersing representations.
Denote by Λ(n) the quiver with the set of vertices
Λ0(n) = {1−, . . . , n−, 1+, . . . , n+}
and the arrows (i−, i+) : i− → i+ for i = 1, . . . , n. In our new terms, a
representation of the bundle S = (S, ∗) is a P -dispersing representation
of Λ(n) with the category S = K(S) = mod(A
∐
B,∗)k (as A) and the
modules Pi = Pi(S) : S → mod k (i run through Λ0(n)) to be the
composition of the natural embedding of S in S0 = modA
∐
Bk and the
projection of S0 onto modAi
k (resp. modBi
k) for i = j− (resp. i = j+).
Obviously, the category Bk (S) is isomorphic to the category repP Λ(n)
with P = {Pi|i ∈ Λ0(n)}.
The representations of a bundle of semichains (and the notion of
“bundle of semichains” itself) were introduced in [6, §1] (for the first
time, in [8]). In these papers the author give (in terms of matrices) a
complete classification of the indecomposable representations of an arbi-
trary bundle of semichains; the classifying is obtained in the explicit and
invariant (without “trace” of the method of solution) form.
In special case, when there is only two semichaines, representations
of bundles arose under consideration a problem of I. M. Gelfand [12]2, in
the classification of the modular representations of quasidihedral groups
[13, 10] (see also [6, §2]) and in studying numerous other problems: in
studying representations of different classes of quivers with relations and
algebras (see e.g. [14, 15, 16, 17, 18]), in the classification of faithful
posets of infinite (non-polynomial, in other terminology) growth [19],
under consideration representations of posets with involution [20] and
equivalence relation [21]. In studying representations of posets with non-
singularity conditions [22, 23] there arose representations of bundles of
four semichaines. For an arbitrary (even) number of semichaines, repre-
sentations of bundles arose first in solving the Gelfand problem and its
generalizations [6, §3]. Recently the main classification theorem of [6, §1]
is used in solving various classification problems of representation theory,
2The tameness of the problem under consideration also follows from properties of
an algorithm described in [12, §2]), but an inductive answer indicated there (for two
semichaines, if one use our terminology) is false.
24 On dispersing representations of quivers
topology and algebraic geometry (see e.g. [24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34].
The main reason of wide application of representations of bundles
of semichains is that, for many classification problems, “most” of tame
cases are reduced to them (such a reduction is today the only method
of solving many problems of infinite growth, as for representations of
quasidihedral groups [10] or partially ordered sets [19]). As will be seen
below, this is also true for dispersing representations of quiver.
3. Main result
We assume from now on that k is an algebraic closed field. For a Krull-
Schmidt category A (over k), we denote by A0 a fixed full subcategory
of A formed by chosen representatives of all isomorphism classes of inde-
composables; we will assume throughout this section that |ObA0| < ∞
(for the case |ObA0| = ∞ see the next section).
Let N be a module over A, and define
supp0N = {X ∈ ObA0|N(X) 6= 0};
for X,Y ∈ ObA0, set N(X,Y ) = N(A0(X,Y )). We call N satu-
rated if dimk N(X,X) = dimk N(X)(dimk N(X) − 1)/2 + 1 for any
X ∈ supp0N and dimk N(X,Y ) + dimk N(Y,X) is equal to 0 or to
dimk N(X) dimk N(Y ) for any distinct X,Y ∈ supp0N (i.e. nonzero
dimk N(X,X) and dimk N(X,Y ) + dimk N(Y,X) take the greatest pos-
sible values). For X ∈ ObA0, we will denote by NX the submodule of N
generated by N(X). Let L(N) denote the lattice of submodules in N or-
dered by inclusion. We call N lattice-chained (resp. lattice-semichained)
if L(N) is a chain (resp. a semichaine), and chained (resp. semichained)
if in addition it is saturated. Finally, we say that a submodule N ′ of N
is singular if it is comparable (in L(N)) to each submodule of N .
Let Γ = (Γ0,Γ1) be a quiver. For an arrow α, denote by s(α) and
e(α) its starting point and its endpoint, respectively. By w−(i) (resp.
w+(i)), where i ∈ Γ0, denote the number of arrows α with s(α) = i
(resp. e(α) = i); put w(i) = w−(i) + w+(i). A vertex i is said to be
trivial if w(i) = 0, outer if w(i) = 1 and inner if w(i) > 1. The sets
of all trivial, outer and inner vertices are denoted by Γ0
0, Γ1
0 and Γ2
0,
respectively. Let M = {Mi} be a fixed Γ0-bunch of A-modules. We call
Mi isolated if supp0Mi∩supp0Mj = ∅ for any j 6= i. An isolated chained
module Mi 6= 0 with dimk Mi(X) ≤ 1 for any object X ∈ A0 is said to
be elementary.
We call Γ M -tame (resp. M -wild) if so is the problem of classifying
the objects of the category repMΓ [35]; a quiver of M -finite (M -infinite)
V. M. Bondarenko 25
type is defined similarly. Further, we call Γ M -inv-wild if the problem of
classifying the object of the category repinv
M Γ is wild. In considering these
problems, it is obviously sufficient to confine oneself to quivers without
trivial vertices.
Our main result is the following theorem.
Theorem 3.1. Let Γ be a finite (not necessarily connected) quiver with-
out trivial vertices and M = {Mi} a Γ0-bunch of nonzero A-modules
without elementary ones for outer vertices. Then Γ is M -tame if and
only if the following conditions hold:
(1) w(i) ≤ 2 for any i ∈ Γ0;
(2) the module Mi is semichained for each i ∈ Γ1
0 and is simple and
isolated for each i ∈ Γ2
0;
(3)
∑
i∈Γ1
0
dimk Mi(X) ≤ 2 for each object X ∈ A0; moreover, when
dimk Mj(X) = dimk Ms(X) = 1 for j 6= s, the submodules MX
j ⊆ Mj
and MX
s ⊆ Ms are both singular.
Otherwise, the quiver Γ is M -inv-wild.
Note that in all cases Γ is of M -infinite type.
Sketch of proof. We may assume Γ1
0 = Γ0, because otherwise one can
take the new quiver
→
Γ with
→
Γ0= {α−, α+|α ∈ Γ1},
→
Γ1= {α : α− →
α+|α ∈ Γ1} and the
→
Γ0-bunch of A-modules
→
M with
→
Mα−= Ms(α),
→
Mα+= Me(α) (taking into account that Γ is M -tame iff
→
Γ is
→
M -tame).
Then (1)–(3) imply that (A/AnnM,M) ∼= (K(S), P (S)) for a bundle of
semichaines S = (S, ∗) with S = {Aα, Bα|α ∈ Γ1}, and it follows from
[6, §1] that Γ is M -tame (of M -infinite type). The proof of the fact that
Γ is M -wild if the condition (1), (2) or (3) does not hold is divided into
several steps.
Step 1. Let S = {A1, . . . , An, B1, . . . , Bn} be a family of pairwise
disjoint posets. We call ∗-bundle (or involution bundle) of these posets a
pair S = (S, ∗), where ∗ is an involution on S0 = A
∐
B (A =
∐n
i=1 Ai,
B =
∐n
i=1 Bi). S is said to be nodal if x∗ 6= x implies that x is comparable
to any element of his poset. Nonempty Ai or Bi is said to be elementary
if it is a chain with all elements being involutory to themselves. We
say “bundle of semichaines” instead “nodal ∗-bundle of semichaines”.
Representations of a ∗-bundle S are defined in the same way as those of
a bundle of semichains.
We have the following statement: a ∗-bundle S of nonempty and
nonelementary posets is wild if (a) there is a poset Ai or Bi which is not
semichained, or (b) the bundle is not nodal.
Present the idea of the proof. For x, y ∈ S0, we write x ∼∗ y iff x = y
or x∗ = y, and x — y iff, for some i, x ∈ Ai, y ∈ Bi or x ∈ Bi, y ∈ Ai;
26 On dispersing representations of quivers
put r∗(x) = |{y|y ∼∗ x}|, and, for X ⊆ S0, r∗(X) = maxx∈X r∗(x).
The notation X — Y for subsets X,Y of S0 means that x — y for any
x ∈ X, y ∈ Y . A chain {1 < 2 < . . . < p} is denoted by 〈p〉 and a poset
〈i〉
∐
. . .
∐
〈j〉 by 〈i, . . . , j〉.
It is proved that (a) or (b) holds iff there is an “alternating” chain
f = {C — x1 ∼∗ x2 — · · · — x2m−1 ∼∗ x2m — D} (m ≥ 0) with
C,D ⊂ S0 such that (c) C ∼= 〈1, 1〉 and r∗(C) = 2, or C ∼= 〈1, 2〉 and
r∗(C) = 1, or C ∼= 〈1, 1, 1〉 and r∗(C1) = 1; (d) D ∼= 〈1, 1〉 and r∗(D) = 1,
or D = {xi} with 1 ≤ i < 2m; (e) xi 6= xj for any i 6= j (for m = 0,
f = {C — D} with D to be of the first form). The main stage of the
proof is to describe all minimal ∗-bundles with a chain f of the above
type and construct for each such ∗-bundle a k〈x, y〉-representation from
the definition of wildness.
Step 2. One can introduce an ∼-bundle of posets and its representa-
tions (and define elementary posets, etc.) in the same way as those in
the case of an involution ∗, replacing everywhere (in particular, in the
definition of modSk) ∗, or equivalently the equivalence relation ∼∗, by
an arbitrary equivalence relation ∼. It is proved that an ∼-bundle S of
nonempty and nonelementary posets is wild if r∼(S0) > 2. The idea of
the proof is similar to that in Step 1. The differences are only (besides
the taking ∼ instead of ∼∗) that, in (c), C is only of the form {y} with
r∼(y) > 2, that, in (d), D can be (in addition) of this form, and that, in
(e), in addition r∼(xi) = 2 for any i.
Step 3. Keeping the notation of Step 1, we call (∗, ◦)-bundle (or
biinvolution bundle) of the given posets a triple S = (S, ∗, ◦), where ∗ and
◦ are, respectively, involutions on S0 and S2
0 satisfying (x, y)◦ = (y, x)◦ for
any x, y, (x, y)◦ = (x, y) for incomparable x, y and the natural conditions
1)–4) of [5, 4.11] if x ≤ y. Its representations are defined similar to that
for a ∗-bundle (for a poset A, (A, ∗, ◦)-graded spaces are (A, ∗)-graded
ones; by (A, ∗, ◦)-maps one must mean (A, ∗)-maps ϕ such that ϕab = ϕcd
whenever (a, b)◦ = (c, d)). It is proved that an (∗, ◦)-bundle of nonempty
and nonelementary (respect to ∗) posets is wild if ◦ is nontrivial. The
idea of the proof is similar to that in Step 2. The difference is only that
the role of x with r∼(x) = 1, 2 is played by x with r◦(x) = 1, 2, where
r◦(x) = 1 if (x, y)◦ = (x, y) for any y and r◦(x) = 2 otherwise.
Step 4. Identifying the modules Mi (i ∈ Γ0) with their images in
mod k, it is proved (with the help of not very complicated arguments)
that the general case is reduced to the cases of Step 1–3.
It follows from the above that our main result can be reformulated
in the following way.
Theorem 3.2. Let Γ and M be as in Theorem 3.1. Then Γ is M -tame
V. M. Bondarenko 27
if and only if there is a bundle of semichaines S = (S, ∗) with S =
{Aα, Bα|α ∈ Γ1} such that (A/AnnM,
→
M) ∼= (K(S), P (S)). Otherwise,
Γ is M -inv-wild.
For ∼-bundles of posets (which include ∗-bundles), we classify tame
cases in the general situation.
4. Extensions of the main result
4.1. The main result for |ObA0| = ∞|ObA0| = ∞|ObA0| = ∞.
Let A be a Krull-Schmidt category over a field k, with |ObA0| = ∞. The
definitions of various types of A-modules, which we gave for |ObA0| < ∞
(at the beginning of Section 2), can be directly transferred to this case.
It is easy to see that a module N is chained (resp. semichained) if and
only if so is N |⊕B (the restriction of N on ⊕B), for any B to be a full
subcategory of A0 with finite many objects (because an infinite poset is
a chaine if and only if all its finite subposets are chaines, and the same
is true for semichaines). Using these facts, one can easily prove that the
main result of this section is also true for |ObA0| = ∞.
4.2. The main result for infinite quivers.
Our main result remains also true for an infinite quiver Γ, and the proof
of this fact can be carried out in the same way as that for finite quivers;
moreover, in view of what we said in the preceding section, it suffices
to consider the case when |ObA0| < ∞. But here we already need to
know that the problem of classifying the representations of a bundle of
semichains is tame when the number of ones is infinite (because |ObA0| <
∞, all the semichaines can be assumed to be finite). The intuition tell
us that this fact is true and that the representations of such bundle can
be classified analogously to that for finitely many semichaines. In this
subsection we clarify an explicit solution of this problem.
Let S be S = {Ai, Bi| i ∈ I} be a family of pairwise disjoint (finite)
semichains, where I is some set. Put A =
∐
i∈I Ai, B =
∐
i∈I Bi,
S0 = A
∐
B. A bundle of semichains Ai, Bi, where i runs through ∈
I, is defined similar to that for finitely many semichaines: it is a pair
S = (S, ∗) with ∗ to be is an involution on S0 such that x∗ = x for each
x belonging to the union of all two-point links. In the new situation,
the category Bk (S) of representations of the bundle S are defined in the
same way as that for finitely many semichaines, and it is a Krull-Schmidt
category too.
28 On dispersing representations of quivers
It is easy to see that a faithful bundle of infinitely many semichaines
has only countable many ones, and hence we can confine oneself to the
countable case3. As usual, Z denotes the integer numbers and N the
natural ones.
Thus, let S = {Ai, Bi| i ∈ N} be a family of pairwise disjoint (finite)
semichains and S = (S, ∗) a bundle of semichains A1, A2, . . . , B1, B2, . . .;
recall that A =
∐
i∈N
Ai, B =
∐
i∈N
Bi and S0 = A
∐
B. If R = (U, V, ϕ)
is a representation of S with the dimension-function d : S0 → N ∪ 0
(sending x ∈ Ai to dimk(Ui)x and y ∈ Bi to dimk(Vi)y), then the set of
all elements x ∈ S0 such that d(x) 6= 0 will be called the support of R.
The indecomposable representations with finite supports (or equiv-
alently, with finitely many semichains) were classified in [8, 6]. Here
we classify the indecomposable representations (of a bundle of countable
many semichains) with infinite supports.
Let, for a semichaine X, L(X) denotes the set of its links (which is
ordered in a natural way). Put L(A) = ∪i∈NL(Ai), L(B) = ∪i∈NL(Bi),
and denote by L(S), or simply L, the union of the sets L(A) and L(B).
It is convenient for us to denote elements of L by lower case letters and
to identify the one-points links with the points themselves. The number
of points of a link x ∈ L is denoted by l(x).
Define two symmetric binary relations, α and β, on the set L by
putting xαy if and only if x 6= y, l(x) = l(y) = 1 and x∗ = y, or x = y
and l(x) = 2; xβy if and only if either x ∈ L(Ai), y ∈ L(Bi) or x ∈ L(Bi),
y ∈ L(Ai) for some i ∈ N.
We now introduce the notion of L-chains of type (0,+∞), (−∞, 0)
and (−∞,+∞).
Throughout, all graphs are nonoriented. For a graph C, we denote by
C0 and C1 the sets of its vertices and edges, respectively. Let C+∞ be the
graph with C+∞
0 = N and C+∞
1 = {(i, i + 1)| i ∈ N}, C−∞ be the graph
with C−∞
0 = {−n|n ∈ N} and C−∞
1 = {(−i − 1,−i)| i ∈ N}, and C∞ be
the graph with C∞
0 = Z and C∞
1 = {(i, i+1)| i ∈ Z}. A countable L-chain
is a function g, defined on a graph C ∈ {C+∞, C−∞, C∞}, that associates
to each j ∈ C0 an element g(j) ∈ L and to each edge (j, j + 1) ∈ C1 a
relation g(j, j + 1) ∈ {α, β} subject to the following conditions: (a) g(j)
and g(j + 1) satisfy the relation g(j, j + 1); (b) g(j − 1, j) 6= g(j, j + 1);
(c) for each x ∈ L, the set g−1(x) = {j ∈ C0| g(j) = x} is finite. An
isomorphism of L-chains g and g′, defined on C and C ′, respectively, is
an isomorphism τ : C → C ′ such that g = τg′.
3A representation (U, V, ϕ) of a bundle S is called faithful if (Ui)x, (Vi)y 6= 0 for
any i ∈ I, x ∈ Ai and y ∈ Bi; the bundle S is called faithful if it has a faithful
indecomposable representation.
V. M. Bondarenko 29
A countable L-chain defined on C = C+∞, C−∞, C∞ will be called
an L-chain of type (0,+∞), (−∞, 0) and (−∞,+∞), respectively.
A countable L-chain g is called admissible if xαy for distinct elements
x, y ∈ L and g(j) = x imply the existence of an edge ρ containing
the vertex j and satisfying g(ρ) = α (an L-chain of type (−∞,+∞) is
always admissible), and symmetric if there exist a vertex i such that
g(i− s) = g(i + s) for any s ∈ N (an L-chain of type (0,+∞) or (−∞, 0)
is always nonsymmetric). The vertex 1 (resp. −1) of an L-chain of type
(0,+∞) (resp. (−∞, 0)) is called double if g(1, 2) = β and g(1)αg(1) in
L (resp. g(−2,−1) = β and g(−1)αg(−1) in L). We write d(g) = 1 if
the vertex 1 (resp. -1) is double and d(g) = 0, otherwise; for an L-chain
of type (−∞,+∞), we put d(g) = 0.
Denote by G1(L) the set of admissible nonsymmetric (countable) L-
chains. To an g ∈ G1(L), we associate the representation U1(g) if d(g) =
0, and the representations U1(g), U2(g) if d(g) = 1. These representations
are defined in the same way as those in [8, 6] for finite many semichains
(in these paper we used the language of matrices, but all the results and
proofs can be easily rewrited in terms of vector spaces and linear maps).
The representations Ui(g) of the bundle S = (S, ∗) are all indecom-
posable. Moreover, the following statement holds.
Theorem 4.1. Let S = (S, ∗) be a bundle of countable many semichains.
Choose one representative in each isomorphism class of L-chains of type
(0,+∞), (−∞, 0) and (−∞,+∞) belonging to G1(L). Then the set of
representations of the form Ui(g) associated to the chosen L-chains is a
complete set of pairwise nonisomorphic indecomposable representations
with infinite support.
The idea of the proof is similar to that in [8] for finite many semi-
chaines.
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Contact information
V. M. Bondarenko Institute of Mathematics, Tereshchenkivska
3, 01601 Kyiv, Ukraine
E-Mail: vit-bond@imath.kiev.ua
Received by the editors: 23.09.2002.
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