On representations of the semigroups S(I,J) with acyclic quiver
In this paper we study representations of semigroups (over a field k) generated by idempotents with partial null multiplication in the case when the corresponding quiver has not oriented cycles.
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| Дата: | 2009 |
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Інститут математики НАН України
2009
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| Цитувати: | On representations of the semigroups S(I,J) with acyclic quiver / V.M. Bondarenko, O.M. Tertychna // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 478-483. — Бібліогр.: 3 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-63262010-02-24T12:01:04Z On representations of the semigroups S(I,J) with acyclic quiver Bondarenko, V.M. Tertychna, O.M. Геометрія, топологія та їх застосування In this paper we study representations of semigroups (over a field k) generated by idempotents with partial null multiplication in the case when the corresponding quiver has not oriented cycles. 2009 Article On representations of the semigroups S(I,J) with acyclic quiver / V.M. Bondarenko, O.M. Tertychna // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 478-483. — Бібліогр.: 3 назв. — англ. 1815-2910 http://dspace.nbuv.gov.ua/handle/123456789/6326 512.5+512.6 en Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Геометрія, топологія та їх застосування Геометрія, топологія та їх застосування |
| spellingShingle |
Геометрія, топологія та їх застосування Геометрія, топологія та їх застосування Bondarenko, V.M. Tertychna, O.M. On representations of the semigroups S(I,J) with acyclic quiver |
| description |
In this paper we study representations of semigroups (over a field k) generated by idempotents with partial null multiplication in the case when the corresponding quiver has not oriented cycles. |
| format |
Article |
| author |
Bondarenko, V.M. Tertychna, O.M. |
| author_facet |
Bondarenko, V.M. Tertychna, O.M. |
| author_sort |
Bondarenko, V.M. |
| title |
On representations of the semigroups S(I,J) with acyclic quiver |
| title_short |
On representations of the semigroups S(I,J) with acyclic quiver |
| title_full |
On representations of the semigroups S(I,J) with acyclic quiver |
| title_fullStr |
On representations of the semigroups S(I,J) with acyclic quiver |
| title_full_unstemmed |
On representations of the semigroups S(I,J) with acyclic quiver |
| title_sort |
on representations of the semigroups s(i,j) with acyclic quiver |
| publisher |
Інститут математики НАН України |
| publishDate |
2009 |
| topic_facet |
Геометрія, топологія та їх застосування |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/6326 |
| citation_txt |
On representations of the semigroups S(I,J) with acyclic quiver / V.M. Bondarenko, O.M. Tertychna // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 478-483. — Бібліогр.: 3 назв. — англ. |
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2025-07-02T09:15:19Z |
| last_indexed |
2025-07-02T09:15:19Z |
| _version_ |
1836526044748709888 |
| fulltext |
Çáiðíèê ïðàöüIí-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè2009, ò.6, �2, 478-483ÓÄÊ 512.5+512.6V. M. BondarenkoInstitute of Mathemati
s, NAS, KyivE-mail: vit-bond�imath.kiev.uaO. M. Terty
hnaKyiv National Taras Shev
henko UniversityE-mail: olena-terty
hna�mail.ruOn representations of the semigroups
S(I, J)S(I, J)S(I, J) with a
y
li
quiver
In this paper we study representations of semigroups (over a field k) gen-
erated by idempotents with partial null multiplication in the case when
the corresponding quiver has not oriented cycles.
We will use the definitions, notions and conventions of [1]. The
main of them will be repeated.
Let I be a finite set without 0 and J a subset of I × I without
elements of the form (i, i). We define S(I, J) to be the semigroup
with generators ei, where i runs through all elements of I ∪ 0, and
the following relations:
1) e0 = 0;
2) e2i = ei for every i ∈ I;
3) eiej = 0 for every 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.
c© V. M. Bondarenko, O. M. Tertychna, 2009
On representations of the semigroups ... 479
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 . We also
put
Y = {(a, b) ∈ (X ×X) \ Y | a 6= b}.
Throughout, k denotes a field.
Let S be a semigroup. A matrix representation of S (of de-
gree 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).
In terms of vector spaces and linear transformations, a represen-
tation of S over k is a homomorphism ϕ from S to the multi-
plicative 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 representations of semigroups S(I, J)
over k in the case when the quiver Q(I, J) is acyclic, i.e. has not
oriented cycles.
Recall the notion of representations of a quiver [2].
Let Q = (Q0, Q1) be a finite quiver with the setQ0 of its vertices
and the set Q1 of its arrows α : x→ y.
A representation of the quiver Q = (Q0, Q1) over a field k is
given by 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 a collection λ = {λx |x ∈ Q0} of linear maps
480 V. M. Bondarenko, O.M. Tertychna
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.
A linear map α of U = U1 ⊕ . . . Up into V = V1 ⊕ . . . Vq is
identified with the matrix (αij), i = 1, . . . , p, j = 1, . . . , q, where
αij : Ui → Vj are the induced linear maps.
Let S = S(I, J) with I = {1, 2, . . . ,m}. Define the functor
F = F (I, J) : repkQ(I, J) → repk S(I, J)
as follows. F = F (I, J) assigns to an 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 a morphism λ
of repkQ(I, J) the morphism ⊕i∈Iλi of repk S(I, J).
In [1] the first author proved that the functor F = F (I, J) is
full and faithful (see the only theorem).
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 tame and wild
classification problems are given in [3].
In this paper we prove the following theorem.
Theorem 1. Let S(I, J) be a semigroup from I such that the
quiver Q(I, J) is acyclic.
Then each object of the category repk S(I, J) is isomorphic to
an object of the form XF (I, J) ⊕ (W, 0), where X ∈ repkQ(I, J)
and W is a vector space of dimension d ≥ 0.
Proof. For simplicity, the quiver Q(I, J) is designated by
Q = (Q0, Q1).
We use the induction on m; the case m = 0, 1 are trivial.
Now let m > 1 and R = (U,ϕ) be a representation of S(I, J).
Fix s ∈ Q0 such that there is no arrow i→ s; we will assume (with-
out loss of generality) that s = m. Consider the subsemigroup S′
On representations of the semigroups ... 481
of S generated by ei, i ∈ I ′ ∪ 0, where I ′ = {1, . . . ,m− 1}. Then
S′ = S(I ′, J ′) with
J ′ = {(i, j) ∈ I × I | i, j ∈ I ′},
and Q′ = Q(I ′, J ′) is the full subquiver of Q with vertex set Q′
0 =
I ′.
Denote by R′ = (U,ϕ′) the restriction of R to S′ (ϕ′(x) = ϕ(x)
for any x ∈ S′). It follows by induction that
R′ ∼= R′ = X ′F (I ′, J ′) ⊕ (W ′, 0),
where X ′ is a representation of the quiver Q(I ′, J ′).
Let R′ = (U,ϕ′) and X ′ = (V ′, γ′) with V ′ = {V ′
i | i ∈ Q′
0} and
γ′ = {γ′α |α : i→ j runs throughQ′
1}.
Since R′ ∼= R′, there exists a linear map
σ : U → U = V ′
1 ⊕ V ′
2 ⊕ . . .⊕ V ′
m−1 ⊕W ′
such that ϕ′σ = ϕ′. Then the representation R = (U,ϕ) is equiv-
alent to the representation R = (U,ϕ), where ϕ(ei) = ϕ′(ei) for
any i = 1, . . . ,m − 1 and ϕ(em) = ϕ(em)σ (since, for i 6= m,
ϕ′(ei) = ϕ′(ei)σ = ϕ(ei)σ, and so ϕ(x) = ϕ(x)σ for each x ∈ S).
We consider the representation R = (U,ϕ) in more detail. Put
Vm = W ′ and consider ϕ as a matrix, taking into account that
U = V1 ⊕ V2 ⊕ . . .⊕ Vm−1 ⊕ Vm.
For (p, q) ∈ J , we denote by [p, q, i, j] the scalar equality
[ϕ(ep)ϕ(eq)]ij = 0,
induced by the (matrix) equality ϕ(ep)ϕ(eq) = 0 (the last equation
holds since epeq = 0 in S(I, J)). It follows from [m, q, i, q] (for any
fixed q 6= m) that (ϕ(em))iq = 0, and consequently (ϕ(em))ij = 0
for any (i, j) ∈ I × I ′.
We first consider two special cases:
a) ϕmm = 0;
b) ϕmm = 1 = 1Vm
.
482 V. M. Bondarenko, O.M. Tertychna
In case a) (ϕ)2 = ϕ implies ϕ = 0 and so
R = XF (I, J) ⊕ (W, 0)
withX = (V, γ), where V = {V ′
1 . . . , V
′
m−1, 0}, γα = γ′α for α ∈ Q′
1,
γα = 0 for α /∈ Q′
1 and W = W ′.
In case b) an equality [p,m, p,m] for (p,m) /∈ J implies
(ϕ)pm = 0
and so
R = XF (I, J) ⊕ (W, 0)
withX = (V, γ), where V = {V ′
1 . . . , V
′
m−1, 0}, γα = γ′α for α ∈ Q′
1,
γα = 0 for α /∈ Q′
1 and W = W ′.
Now we consider the general case. Since (ϕmm)2 = ϕmm, there
is an invertible map ν = (ν1, ν2) : Vm →W1 ⊕W2 such that
ϕmm(ν1, ν2) = (ν1, ν2)
(
1 0
0 0
)
,
where 1 = 1W1 . Then the representation R′ = (U,ϕ′) is isomor-
phic to the the representation R̂′ = (Û , ϕ̂′), where
Û = Û1 ⊕ Û2 ⊕ . . . ⊕ Ûm+1
with Ûi = Vi for i = 1, . . . m − 1, Ûm = W1, Ûm+1 = W2,
and ϕ̂′(ei) = ϕ′(ei) for i = 1, . . . m − 1, (ϕ̂′(em))ij = (ϕ′(em))ij
for (i, j) ∈ I ′ × I ′, (ϕ̂′(em))ij = 0 for i = m,m + 1, j ∈ I ′,
(ϕ̂′(em))m,mj = 1 = 1W1 , (ϕ̂′(em))m,m+1 = 0, (ϕ̂′(em))m+1,m = 0,
(ϕ̂′(em))m+1,m+1 = 0 (for instance, one can take the isomorphism
β : R̂′ → R′ with ϕ̂′(ei) = µ−1R′µ, where µ = 1U1⊕. . .⊕1Um−1⊕ν.
From (ϕ̂′(ei))2 = ϕ̂′(ei) it follows that ϕ̂′(em))i,m+1 = 0 for
any i ∈ I ′ (see the partial case a)); (then ϕ̂′(em))i,m+1 = 0 for
any i = 1, . . . ,m + 1). From the scalar equalities [p,m, p,m] for
(p,m) /∈ J implies (ϕ̂)pm = 0 (see the partial case b)). Thus,
R = (Û , ϕ̂) ∼= R = (U,ϕ)
On representations of the semigroups ... 483
has the form XF (I, J) ⊕ (W, 0), where X = (V, γ) with V =
{Ûi | i ∈ Q0}, γ = {γα |α : i → j runs throughQ1} with γα = γ′α
for α ∈ Q′
1, γα = ϕ̂(em)ij for α /∈ Q′
1 (then j = m), and
W = Ŵm+1,
as claimed. �
Let rep◦
k S(I, J) denotes the full subcategory of the category
repk S(I, J) consisting of all objects that have no objects (W, 0),
with W 6= 0, as direct summands.
Then from the Theorem of [1] and Theorem 1 it follows the
following statement.
Theorem 2. Let S(I, J) be as in Theorem 1. Then the functor
F = F (I, J), viewed as a functor from repkQ(I, J) to rep◦
k S(I, J),
is an equivalence of categories.
From this theorem it follows that a semigroup S(I, J) and the
quiver Q(I, J have the same representation type (in the case when
Q(I, J is acyclic).
References
[1] V. M. Bondarenko. On connections between representations of semigroups
S(I, J)S(I, J)S(I, J) and representations of quivers// This book
[2] P. Gabriel. Unzerlegbare Darstellungen // Manuscripts Math. – 1972. –
6. – pp. 71-103,309.
[3] Yu. A. Drozd. Tame and wild matrix problems // Lecture Notes in Math.
– 1980. – 832. – pp. 242-258.
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