On classification of pairs of potent linear operators with the simplest annihilation condition
We study the problem of classifying the pairs of linear operators A,B (acting on the same vector space), when the both operators are potent and AB=0. We describe the finite, tame and wild cases and classify the indecomposable pairs of operators in the first two of them.
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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Інститут прикладної математики і механіки НАН України
2016
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| Цитувати: | On classification of pairs of potent linear operators with the simplest annihilation condition / V.M. Bondarenko, O.M. Tertychna, O.V. Zubaruk // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 18-23. — Бібліогр.: 5 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860241229123420160 |
|---|---|
| author | Bondarenko, V.M. Tertychna, O.M. Zubaruk, O.V. |
| author_facet | Bondarenko, V.M. Tertychna, O.M. Zubaruk, O.V. |
| citation_txt | On classification of pairs of potent linear operators with the simplest annihilation condition / V.M. Bondarenko, O.M. Tertychna, O.V. Zubaruk // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 18-23. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | We study the problem of classifying the pairs of linear operators A,B (acting on the same vector space), when the both operators are potent and AB=0. We describe the finite, tame and wild cases and classify the indecomposable pairs of operators in the first two of them.
|
| first_indexed | 2025-12-07T18:30:00Z |
| format | Article |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 1, pp. 18–23
© Journal “Algebra and Discrete Mathematics”
On classification of pairs of potent linear
operators with the simplest annihilation
condition
Vitaliy M. Bondarenko, Olena M. Tertychna,
Olesya V. Zubaruk
Communicated by V. V. Kirichenko
Abstract. We study the problem of classifying the pairs of
linear operators A,B (acting on the same vector space), when the
both operators are potent and AB = 0. We describe the finite, tame
and wild cases and classify the indecomposable pairs of operators
in the first two of them.
Introduction
Throughout the paper, k is an algebraic closed field of characteristic
char k = 0. All k-vector space are finite-dimensional. Under consideration
maps, morphisms, etc., we keep the right-side notation.
We call a Krull-Schmidt category (i.e. an additive k-category with
local endomorphism algebras for all indecomposable objects) of tame
(respectively, wild) type if so is the problem of classifying its objects up to
isomorphism (see precise general definitions in [1]). For formal reasons
we exclude the categories of finite type (i.e. with finite number of the
isomorphism classes of indecomposable objects) from those of tame type.
In this paper we study the problem of classifying the pairs of anni-
hilating potent linear operators (an operator C is called potent or, more
precisely, s-potent if Cs = C, where s > 1).
2010 MSC: 15A21, 16G20, 16G60.
Key words and phrases: potent operator, quiver, Krull-Schmidt category, func-
tor, canonical form, tame type, wild type, Dynkin graph, extended Dynkin graph.
V. Bondarenko, O. Tertychna, O. Zubaruk 19
Formulate our problem more precisely and in the category language.
Let P(k) denotes the category of pairs of linear operators acting
on the same k-vector space, i.e. the category with objects the triples
U = (U,A,B), consisting of a k-vector space U and linear operators A,B
on U , and with morphisms from U = (U,A,B) to U ′ = (U ′,A′,B′) the
linear maps X : U → U ′ such that AX = XA, BX = XB. Since it
is a Krull-Schmidt category, each object is uniquely determined by its
direct summands. For natural numbers n,m > 1, denote by P◦
k(n,m)
the full subcategory of P(k) consisting of all triples (U,A,B) with A
being n-potent, B being m-potent and AB = 0. Our aim is to describe
the type of every such category and to classify (up to isomorphism) the
indecomposable objects in finite and tame cases.
Theorem 1. A category P◦
k(n,m) is of
• finite type if nm < n+m+ 3,
• tame type if nm = n+m+ 3,
• wild type if nm > n+m+ 3.
With respect to the mentioned classification see section 2.
Note that from Theorems 3.1 and 3.2 of [2] it follows that without
the relation AB = 0 the corresponding overcategory Pk(n,m) is of tame
type if n = m = 2 and of wild type otherwise (see more in 3.6 below).
1. Proof of the theorem
We first establish a connection between the categories P◦
k(n,m) and
the categories of representations of quivers.
Recall the notion of representations of a quiver [3].
Let Q = (Q0, Q1) be a finite quiver (directed graph), where Q0 and
Q1 are the sets of its vertices and arrows, respectively. 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 K-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)
over K will be denoted by repK Q. It is a Krull-Schmidt category.
A quiver Q is said to be of finite, tame or wild representation type
over K if the caregory repK Q has respectively finite, tame or wild type.
By results of [3] (respectively, [4] and [5]), a connected quiver is of finite
20 Classification of pairs of potent linear operators
(respectively, tame) representation type if and only if it is a Dynkin
(respectively, extended Dynkin) graph. Note that by a Dynkin graph we
mean a Dynkin diagram with some orientation of edges, and for simplicity
denote it in the same way as the Dynkin diagram (analogously for an
extended Dynkin graph).
Now we proceed to investigate connections between categories of the
forms P◦
k(n,m) and repk Q.
We identify a linear map α of U = U1 ⊕ . . . Up into V = V1 ⊕ . . . Vq
with the matrix (αij)
p
i=1
q
j=1
, where αij : Ui → Vj are the linear maps
induced by α; if p = q and the matrix is diagonal, we write α = ⊕p
i=1
αi.
The identity linear operator on W is denoted by 1W .
For natural numbers n,m > 1, denote by Q(n,m) the quiver with
set of vertices Q0(n,m) = {1, 2 . . . , n+m} and set of arrows Q1(n,m) =
{i → j | j = 1, . . . , n, i = n+ 1, . . . , n+m}. The primitive root of unity
of degree s is denoted by εs.
Define the functor Gnm from repk Q(n − 1,m − 1) to P◦
k(n,m) as
follows. Gnm assigns to each object (V, γ) ∈ repk Q(n − 1,m − 1) the
object (V ⊕,Aγ ,Bγ) ∈ P◦
k(n,m) where V ⊕ = ⊕n+m−2
i=1
Vi, Aγ
ij = εi
n−11Vi
if i = j 6 n − 1 and Aγ
ij = 0 if otherwise, Bγ
n+i−1,n+i−1
= εi
m−11Vn+i−1
if i 6 m − 1, Bγ
n+i−1,j = γij if i 6 m − 1, j 6 n − 1, and Bγ
pq = 0 in all
other cases. Gnm assigns to each morphism λ of repk Q(m− 1, n− 1) the
morphism ⊕n+m−2
i=1
λi of P◦
k(n,m).
Proposition 1. The functor Gnm is full and faithful.
Proof. It is obvious that Gnm is faithful. Prove that it is full. Let δ be a
morphism from (V, γ)Gnm = (V ⊕,Aγ ,Bγ) to (W,σ)Gnm = (W⊕,Aσ,Bσ).
In other words, δ is a linear map of V ⊕ into W⊕ such that Aγδ = δAσ and
Bγδ = δBσ. We consider these equalities as matrix ones (see the definition
of V ⊕), and the induced by them scalar equalities (Aγδ)ij = (δAσ)ij and
(Bγδ)ij = (δBσ)ij denote, respectively, by [a, i, j] and [b, i, j].
Since εn−1, ε
2
n−1, . . . , ε
n−1
n−1 and 0 are pairwise different elements of the
field k, it follows from the equalities [a, i, j] with i, j ∈ {1, . . . , n − 1},
i 6= j, [a, i, j] with i ∈ {1, . . . , n− 1}, j ∈ {n, . . . , n+m− 2} and [a, i, j]
with i ∈ {n, . . . , n+m− 2}, j ∈ {1, . . . , n− 1} that the block (δpq)n−1
p,q=1 of
σ (as a matrix) is diagonal and the blocks (δpq)n−1
p=1
n+m−2
q=n , (δpq)n+m−2
p=n
n−1
q=1
are zero. Then analogously to above, it follows from the equalities [b, i, j]
with i, j ∈ {n, . . . , n+m−2}, i 6= j, that the block (δpq)n+m−2
p,q=n is diagonal.
Thus σ (as a matrix) is diagonal, and it is easy to see that the equalities
[b, i, j] with i ∈ {n, . . . , n + m − 2}, j ∈ {1, . . . , n − 1} means that σ =
V. Bondarenko, O. Tertychna, O. Zubaruk 21
(σ1, . . . , σn+m−2) is a morphism between the objects (V, γ) and (W,σ) of
the category repk Q(n− 1,m− 1). Since σ = σGnm, the fullness of Gnm
is proved.
Proposition 2. Each object of P◦
k(n,m) is isomorphic to an object of
the form RGnm ⊕ (W, 0, 0), where R is an object of repk Q(n− 1,m− 1),
W is a k-vector space of dimension d > 0.
Proof. Let T = (U,A,B) be an objects of the category P◦
k(n,m). Since
the roots εn−1, . . . , ε
n−1
n−1 and 0 of the polynomial xn − x are pairwise
different, we can assume (by the theorem on the Jordan canonical form)
that U = U1⊕. . .⊕Un−1⊕U0 with Us = Ker(A−εs
n−11U ) and U0 = Ker A;
then A = A1 ⊕ . . . ⊕ An−1 ⊕ A0 with As : Us → Us to be the scalar
operator εs
n−11Us
and A0 : U0 → U0 to be zero (here s = 1, . . . , n − 1).
From AB = 0 it follows that U1 ⊕ . . .⊕ Un−1 ∈ Ker B, and consequently
we have (since Bm = B) that the operator B0 : U0 → U0, induced
by B, satisfies the equality Bm
0 = B0. Then, analogously as above, U0 =
Un⊕. . .⊕Un+m−2⊕W with Un+s−1 = Ker(B0−εs
m−11U0
), s = 1, . . . ,m−1,
and W = Ker B0. Besides, it follows from Bm = B that W0 ∈ Ker B.
Thus, U = U1 ⊕ . . .⊕ Un+m−2 ⊕W and now the operators A,B are
uniquely defined by the maps Bij : Ui → Uj with i and j running from
n to n+m− 2 and from 1 to n− 1, respectively. The representation R
of the quiver Q(n− 1,m− 1), corresponding to these maps, satisfies the
required condition, i. e. T = RGnm ⊕ (W, 0, 0).
Denote by P̂◦
k(n,m) the full subcategory of P◦
k(n,m) consisting of all
objects that have no objects (W, 0, 0), with W 6= 0, as direct summands.
We have as an immediate consequence of Propositions 1 and 2 the
following statement.
Theorem 2. The functor Gnm, viewed as a functor from the category
repk Q(n− 1,m− 1) to the category P̂◦
k(n,m), is an equivalence of cate-
gories.
Using this theorem it is easy to show by the standard method that
the types of categories P◦
k(n,m) and repk Q(n− 1,m− 1) coincide.
Now Theorem 1 follows from the simple facts that Q = Q(n−1,m−1)
is a Dynkin graph iff either n = 2,m = 2, 3, 4, or vice versa,n = 2, 3, 4,m =
2 (then Q = A2, A3, D4, respectively), and an extended Dynkin graph
iff either n = 2,m = 5, or vice versa, n = 5,m = 2 (then Q = D̃4), or
n = m = 3 (then Q = Ã3).
22 Classification of pairs of potent linear operators
2. The classification of the indecomposable pairs
of annihilating potent operators
The functor Gnm allows to obtain a classification of indecomposable
objects (up to isomorphism) of any category P◦
k(n,m) of finite and tame
types (see Theorem 2). To do this, it is need to take representatives of the
classes of isomorphic indecomposable objects (one from each class) of the
category repk Q with Q = Q(n− 1,m− 1) and apply to them the functor
Gnm (as a result we get all representatives of the classes of isomorphic
indecomposable objects of P◦
k(n,m), except (k, 0, 0)). Such (of the most
simple form) representatives are well-known: see [3] for Q = A2, A3, D4
(our cases of finite type) and [4, 5] for Q = Ã3, D̃4 (our cases of tame
type).
3. Remarks
3.1. All the above results are true if k is any field of characteristic 0
and εn, εm ∈ k.
3.2. All the above results are true if k is an algebraic closed field of
characteristic p 6= 0, which does not divide nm.
3.3. All the above results are true if k is as in 3.2, but does not
necessarily algebraically closed, and εn, εm ∈ k.
3.4. All the above results are true if k is an algebraic closed field of any
characteristic and A,B satisfy, respectively, polynomials ϕ(x) and ψ(x)
of degrees n and m without multiple roots such that ϕ(0) = 0, ψ(0) = 0
(without the last condition the problem is trivial).
3.5. Theorem 1 is true if k is any field of any characteristic and A,B
satisfy, respectively, any fixed separable polynomials ϕ(x) and ψ(x) of
degrees n and m such that ϕ(0) = 0, ψ(0) = 0 (see the definitions in [1]).
3.6. Classifying the pairs of idempotent operators. As the first
author pointed out, the following classification of the pairs of idempotent
operators (the objects of Pk(2, 2)) follows from [2, Section 3] and [4].
One will adhere to the matrix language. The field k is assumed to
be any algebraic closed (otherwise, it is necessary to replace the below
Jordan blocks in 1) by indecomposable Frobenius companion ones).
Let Jm(λ) denotes the (upper) m × m Jordan block with diagonal
entries λ, Em the m×m identity matrix. Define 0Em (respectively, 0Em)
as Em with added null first column (respectively, last row). For an m×m
matrix X, put X+ = X, X− = Em − X, and for a pair of m × m
matrices P = (X,Y ) and µ, ν ∈ {+,−}, put Pµν = (Xµ, Y ν). Finally,
V. Bondarenko, O. Tertychna, O. Zubaruk 23
for matrices A,B with the same number of rows, introduce the squared
matrices
F [A,B] =
(
A B
0 0
)
, S[A,B] =
(
0 0
A B
)
.
Theorem 3. The set of all pairs of matrices over k of the forms
1) P = (F [En, En], S[Jn(λ), En]), λ ∈ k \ 0,
2) Pµν for P = (F [En, En], S[Jn(0), En]) and µ, ν ∈ {+,−},
3) Pµν for P = (F [En, 0En−1], S[0En−1, En−1]) and µ, ν ∈ {+,−},
where n runs through the natural numbers, is a complete set of pairwise
nonsimilar indecomposable pairs of idempotent matrices over k.
Note that this classification implies those of the pairs of involutory
matrices (the representations of the infinite dihedral group) if char k 6= 2.
References
[1] Yu. A. Drozd, Tame and wild matrix problems, Lecture Notes in Math. 832 (1980),
pp. 242-258.
[2] V. M. Bondarenko, Linear operators on vector spaces graded by posets with
involution: tame and wild cases. Proc. of Institute of Math. of NAS of Ukraine
(Mathematics and its Applications), 63, Kiev, 2006, 168 pp.
[3] P. Gabriel, Unzerlegbare Darstellungen, Manuscripts Math., V.6, 1972, pp. 71-103.
[4] L. A. Nazarova, Representations of quivers of infinite type, Izv. Akad. Nauk SSSR
Ser. Mat. 37 (1973), pp. 752-791 (in Russian).
[5] P. Donovan, M.-R. Freislich, The representation theory of finite graphs and asso-
ciated algebras, Carleton Math. Lecture Notes, No. 5. Carleton University, Ottawa,
Ont., 1973, 83 pp.
Contact information
V. M. Bondarenko Institute of Mathematics, NAS, Kyiv, Ukraine
E-Mail(s): vitalij.bond@gmail.com
O. M. Tertychna Vadim Hetman Kyiv National Economic Uni-
versity, Kiev, Ukraine
E-Mail(s): olena-tertychna@mail.ru
O. V. Zubaruk Kyiv National Taras Shevchenko University,
Volodymyrska str., 64, 01033 Kyiv, Ukraine
E-Mail(s): Sambrinka@ukr.net
Received by the editors: 18.02.2016.
|
| id | nasplib_isofts_kiev_ua-123456789-155201 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T18:30:00Z |
| publishDate | 2016 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Bondarenko, V.M. Tertychna, O.M. Zubaruk, O.V. 2019-06-16T10:49:33Z 2019-06-16T10:49:33Z 2016 On classification of pairs of potent linear operators with the simplest annihilation condition / V.M. Bondarenko, O.M. Tertychna, O.V. Zubaruk // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 18-23. — Бібліогр.: 5 назв. — англ. 1726-3255 2010 MSC:15A21, 16G20, 16G60. https://nasplib.isofts.kiev.ua/handle/123456789/155201 We study the problem of classifying the pairs of linear operators A,B (acting on the same vector space), when the both operators are potent and AB=0. We describe the finite, tame and wild cases and classify the indecomposable pairs of operators in the first two of them. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On classification of pairs of potent linear operators with the simplest annihilation condition Article published earlier |
| spellingShingle | On classification of pairs of potent linear operators with the simplest annihilation condition Bondarenko, V.M. Tertychna, O.M. Zubaruk, O.V. |
| title | On classification of pairs of potent linear operators with the simplest annihilation condition |
| title_full | On classification of pairs of potent linear operators with the simplest annihilation condition |
| title_fullStr | On classification of pairs of potent linear operators with the simplest annihilation condition |
| title_full_unstemmed | On classification of pairs of potent linear operators with the simplest annihilation condition |
| title_short | On classification of pairs of potent linear operators with the simplest annihilation condition |
| title_sort | on classification of pairs of potent linear operators with the simplest annihilation condition |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155201 |
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