On connections between representations of semigroups S(I,J) and representations of quivers
In this paper we study connections between representations of one natural class of semigroups and representations of quivers.
Збережено в:
| Дата: | 2009 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2009
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/6324 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On connections between representations of semigroups S(I,J) and representations of quivers / V.M. Bondarenko // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 470-473. — Бібліогр.: 2 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859726662886752256 |
|---|---|
| author | Bondarenko, V.M. |
| author_facet | Bondarenko, V.M. |
| citation_txt | On connections between representations of semigroups S(I,J) and representations of quivers / V.M. Bondarenko // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 470-473. — Бібліогр.: 2 назв. — англ. |
| collection | DSpace DC |
| description | In this paper we study connections between representations of one natural class of semigroups and representations of quivers.
|
| first_indexed | 2025-12-01T11:18:36Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöüIí-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè2009, ò.6, �2, 470-473ÓÄÊ 512.5+512.6V. M. BondarenkoInstitute of Mathemati
s, NAS, KyivE-mail: vit-bond�imath.kiev.uaOn
onne
tions betweenrepresentations of semigroups S(I, J)S(I, J)S(I, J)and representations of quivers
In this paper we study connections between representations of one natural
class of semigroups and representations of quivers.
Let I be a finite set without 0 and let J be a subset of the
cartesian product I×I without diagonal elements (i, i). We define
S(I, J) to be the semigroup with generators ei, where i ∈ I ∪ 0,
and the following relations:
1) e0 = 0;
2) e2i = ei for any i ∈ I;
3) eiej = 0 for any pair (i, j) ∈ J .
The set of the semigroups of the form S(I, J) is denoted by I.
We call S(I, J) ∈ I a semigroup generated by idempotents with
partial null multiplication.
Throughout, k denotes a field. All vector space over k are finite-
dimensional. Under consideration maps, morphisms, etc., we keep
the right-side notation.
Let S be a semigroup and let Mn(k) denotes the algebra of all
n×n matrices with entries in k. A matrix representation of S (of
degree n) over k is a homomorphism T from S to the multiplicative
semigroup of Mn(k). If there is an identity (resp. zero) element
a ∈ S, we assume that the matrix T (a) is identity (resp. zero).
c© V. M. Bondarenko, 2009
On connections between representations ... 471
Since Mn(k) can be considered as the algebra of all linear transfor-
mations of any fixed n-dimensional vector space, we can consider
representations of the semigroup S in terms of vector spaces and
linear transformations. Thus, a representation of S over k is a
homomorphism ϕ from S to the multiplicative semigroup of the
algebra EndkU with U being a finite-dimensional vector space.
Two representation ϕ : S → EndkU and ϕ′ : S → EndkU
′ are
called equivalent if there is a linear map σ : U → U ′ such that
ϕσ = ϕ′.
A representation ϕ : S → EndkU of S is also denoted by (U,ϕ).
By the dimension of (U,ϕ) one means the dimension of U . The
representations of S form a category which will be denoted by
repk S (it has as morphisms from (U,ϕ) to (U ′, ϕ) the maps σ
such that ϕσ = ϕ′).
In this paper we study connections between representations of
the semigroups S(I, J) and representations of quivers.
Recall the notion of representations of a quiver [1].
Let Q = (Q0, Q1) be a (finite) quiver, where Q0 is the set of its
vertices and Q1 is the set of its arrows α : x→ y.
A representation of the quiver Q = (Q0, Q1) over a field k is a
pair R = (V, γ) formed by a collection V = {Vx |x ∈ Q0} of vector
spaces Vx and a collection
γ = {γα |α : x→ y runs throughQ1}
of linear maps γα : Vx → Vy. A morphism from R = (V, γ)
to R′ = (V ′, γ ′) is given by a collection λ = {λx |x ∈ Q0} of
linear maps λx : Vx → V ′
x, such that γαλy = λxγ
′
α for any arrow
α : x→ y.
The category of representations of Q = (Q0, Q1) will be denoted
by repkQ.
We identify a linear map α of
U = U1 ⊕ . . .⊕ Up
into
V = V1 ⊕ . . .⊕ Vq
472 V. M. Bondarenko
with the matrix (αij), i = 1, . . . , p, j = 1, . . . , q, where
αij : Ui → Vj
are the linear maps induced by α (then the sum and the compo-
sition of maps are given by the matrix rules).
For a finite set X and Y ⊆ X ×X, we denote by Q(X,Y ) the
quiver with vertex set X and arrows a→ b, (a, b) ∈ Y .
Let S = S(I, J), where, as before, I = {1, 2, . . . ,m} and
J = {(i, j) ∈ (I × I) \ J | i 6= j}.
Define the functor F from repkQ(I, J) to repk S(I, J) as follows.
F = F (I, J) assigns to each object (V, γ) ∈ repkQ(I, J) the object
(V ′, γ ′) ∈ repk S(I, J), where V ′ = ⊕i∈IVi, (γ ′(ei))jj = 1Vj if
i = j, (γ ′(ei))ij = γij if (i, j) ∈ J , and (γ ′(ei))js = 0 in all other
cases. F assigns to each morphism λ of repkQ(I, J) the morphism
⊕i∈Iλi of repk S(I, J).
Theorem 1. The functor
F = F (I, J) : repkQ(I, J) → repk S(I, J)
is full and faithful.
Proof. It is obvious that the functor F is faithful. It remains to
prove that it is full. Let δ be a morphism from (V, γ)F = (V ′, γ ′)
to (W,σ)F = (W ′, σ′). In other words, δ is a linear map of V ′
into W ′ such that γ ′(es)δ = δσ′(es) for s = 1, . . . ,m. We will
consider these equalities as matrix ones (taking into account that
V ′ = ⊕i∈IVi and W ′ = ⊕i∈IWi) and denote by [s, i, j] the scalar
equality (γ ′(es)δ)ij = (δσ′(es))ij , induced by the (matrix) equality
γ ′(es)δ = δσ′(es).
From an equation [j, i, j] with j 6= i it follows that δij = 0, and
consequently δ is a diagonal matrix: δ = δ11 ⊕ δ22 ⊕ · · · ⊕ δmm.
Further, if α : i → j is an arrow of the quiver Q(I, J), then from
the equation [i, i, j] we have that γαδjj = δiiσα. Consequently, a
collection δ = {δss | s = 1, . . . ,m} is a morphism from (V, γ) to
On connections between representations ... 473
(W,σ). Since δ = δ11 ⊕ δ22 ⊕ · · · ⊕ δmm, we have that δ = λF ,
where λ = δ, as claimed. �
From this theorem it follows that a semigroup S(I, J) is wild if
so is the quiver Q(I, J) (the general definitions of wild classifica-
tion problems are given in [2].
References
[1] P. Gabriel. Unzerlegbare Darstellungen // Manuscripts Math. – 1972. –
6. – pp. 71-103,309.
[2] Yu. A. Drozd. Tame and wild matrix problems // Lecture Notes in Math.
– 1980. – 832. – pp. 242-258.
|
| id | nasplib_isofts_kiev_ua-123456789-6324 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-2910 |
| language | English |
| last_indexed | 2025-12-01T11:18:36Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bondarenko, V.M. 2010-02-23T14:46:42Z 2010-02-23T14:46:42Z 2009 On connections between representations of semigroups S(I,J) and representations of quivers / V.M. Bondarenko // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 470-473. — Бібліогр.: 2 назв. — англ. 1815-2910 https://nasplib.isofts.kiev.ua/handle/123456789/6324 512.5+512.6 In this paper we study connections between representations of one natural class of semigroups and representations of quivers. en Інститут математики НАН України Геометрія, топологія та їх застосування On connections between representations of semigroups S(I,J) and representations of quivers Article published earlier |
| spellingShingle | On connections between representations of semigroups S(I,J) and representations of quivers Bondarenko, V.M. Геометрія, топологія та їх застосування |
| title | On connections between representations of semigroups S(I,J) and representations of quivers |
| title_full | On connections between representations of semigroups S(I,J) and representations of quivers |
| title_fullStr | On connections between representations of semigroups S(I,J) and representations of quivers |
| title_full_unstemmed | On connections between representations of semigroups S(I,J) and representations of quivers |
| title_short | On connections between representations of semigroups S(I,J) and representations of quivers |
| title_sort | on connections between representations of semigroups s(i,j) and representations of quivers |
| topic | Геометрія, топологія та їх застосування |
| topic_facet | Геометрія, топологія та їх застосування |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/6324 |
| work_keys_str_mv | AT bondarenkovm onconnectionsbetweenrepresentationsofsemigroupssijandrepresentationsofquivers |