Homological ideals as integer specializations of some Brauer configuration algebras
UDC 512.5 The homological ideals associated with some Nakayama algebras are characterized and enumerated via integer specializations of some suitable Brauer configuration algebras. In addition, it is shown how the number of these homological ideals can be connected with the&am...
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| author | Fernández Espinosa, Pedro Fernando Moreno Cañadas, Agustín Fernández Espinosa, Pedro Fernando Moreno Cañadas, Agustín |
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UDC 512.5
The homological ideals associated with some Nakayama algebras are characterized and enumerated via integer specializations of some suitable Brauer configuration algebras. In addition, it is shown how the number of these homological ideals can be connected with the  process of categorification of Fibonacci numbers defined by Ringel and Fahr. |
| doi_str_mv | 10.37863/umzh.v74i9.6218 |
| first_indexed | 2026-03-24T03:26:35Z |
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DOI: 10.37863/umzh.v74i9.6218
UDC 512.5
Pedro Fernando Fernández Espinosa (Universidad de Caldas, Colombia),
Agustı́n Moreno Cañadas1(Universidad Nacional de Colombia, Bogotá-Colombia)
HOMOLOGICAL IDEALS AS INTEGER SPECIALIZATIONS
OF SOME BRAUER CONFIGURATION ALGEBRAS
ГОМОЛОГIЧНI IДЕАЛИ, ЯК ЦIЛОЧИСЛОВI СПЕЦIАЛIЗАЦIЇ
ДЕЯКИХ КОНФIГУРАЦIЙНИХ АЛГЕБР БРАУЕРА
The homological ideals associated with some Nakayama algebras are characterized and enumerated via integer specializati-
ons of some suitable Brauer configuration algebras. In addition, it is shown how the number of these homological ideals
can be connected with the process of categorification of Fibonacci numbers defined by Ringel and Fahr.
Охарактеризовано гомологiчнi iдеали, асоцiйованi з деякими алгебрами Накаями, та перераховано їх через цiло-
числовi спецiалiзацiї деяких вiдповiдних конфiгурацiйних алгебр Брауера. Крiм того, показано як кiлькiсть таких
гомологiчних iдеалiв може бути пов’язана з процесом категоризацiї чисел Фiбоначчi, що був визначений Рiнгелем
i Фаром.
1. Introduction. Homological ideals or strong idempotent ideals are ideals of an algebra introduced
by 2, Platzeck and Todorov in [2]. These ideals arise from the research of heredity ideals and
quasi-hereditary algebras. For these ideals the corresponding quotient map induces a full and faithful
functor between derived categories. Recently, homological ideals have been studied in different
contexts, for instance Gatica, Lanzillota and Platzeck and independently Xu and Xi established some
relationships with the so-called finitistic dimension conjecture and the Igusa – Todorov functions [6].
Furthermore, De la Peña and Xi in [9] and Armenta in [1] studied the impact of these ideals in the
context of Hochschild cohomology and one point extensions.
This work deals with the combinatorial properties of homological ideals associated to some
path algebras and their relationships with the novel Brauer Configuration algebras which have been
introduced recently by Green and Schroll in [7]. In particular, we introduce the notion of the
message of a Brauer configuration, such messages enable to compute the number of homological
ideals associated to some Nakayama algebras. Moreover, such number of ideals allow us to obtain
an alternative version of the partition formula for even-index Fibonacci numbers given by Fahr and
Ringel in [3] attaining in this way a new algebraic interpretation for these numbers. Worth noting
that Fahr and Ringel devoted works [3 – 5] to this kind of interpretations also-called categorifications.
This paper is organized as follows. In Section 2, we recall main notation and definitions regarding
homological ideals and Brauer configuration algebras. In particular, we introduce the notion of integer
specialization of a Brauer configuration and the concept of the message of a Brauer configuration. In
Section 3, we give combinatorial conditions to determine whether an idempotent ideal associated to
some Nakayama algebras is homological or not and it is reminded the notion of categorification in
the sense of Fahr and Ringel. We also give the number of such ideals via the integer specialization
of a suitable Brauer configuration algebra and its corresponding message. Moreover, we use the
1 Corresponding author, e-mail: amorenoca@unal.edu.co.
c\bigcirc PEDRO FERNANDO FERNÁNDEZ ESPINOSA, AGUSTIN MORENO CAÑADAS, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9 1201
1202 PEDRO FERNANDO FERNÁNDEZ ESPINOSA, AGUSTIN MORENO CAÑADAS
number of homological ideals to establish a partition formula for even-index Fibonacci numbers.
Some interesting sequences in the On-line Encyclopedia of Integer Sequences [11] arising from these
computations are described as well.
2. Preliminaries. In this section, we recall main definitions and notation to be used throughout
the paper [1, 2, 7, 9, 10].
2.1. Homological ideals. For an algebra A we mean a finite dimensional basic and connected
algebra over an algebraically closed field k. We denote the category of finite dimensional right A-
modules as \mathrm{m}\mathrm{o}\mathrm{d}(A), and the bounded derived category of \mathrm{m}\mathrm{o}\mathrm{d}(A) as Db(A). We will assume that
A is a bounded path algebra of the form kQ/I with Q a finite quiver and I an admissible ideal.
An epimorphism of algebras \phi : A \rightarrow B is called homological epimorphism if it induces a full
and faithful functor
Db(\phi \ast ) : Db(B) \rightarrow Db(A).
Let I be a two sided ideal of A. Since the quotient map \pi : A \rightarrow A/I is an epimorphism then
the induced functor \pi \ast : \mathrm{m}\mathrm{o}\mathrm{d}(A/I) \rightarrow \mathrm{m}\mathrm{o}\mathrm{d}(A) is full and faithful.
A two sided ideal I of A is homological if the quotient map \pi : A \rightarrow A/I is an homological
epimorphism.
The following results characterize homological ideals [2, 9].
Proposition 1. Let I be an ideal of A, then:
1) I is an homological ideal of A if and only if \mathrm{T}\mathrm{o}\mathrm{r}An (I, A/I) = 0 for all n \geq 0; in this case, I
is idempotent;
2) if I is idempotent and A-projective, then I is homological;
3) If I is idempotent, then I is homological if and only if \mathrm{E}\mathrm{x}\mathrm{t}nA(I, A/I) = 0 for all n \geq 0.
We denote the trace of an A-module M in an A-module N as
\mathrm{t}\mathrm{r}M (N) :=
\sum
f\in \mathrm{H}\mathrm{o}\mathrm{m}A(M,N)
\mathrm{I}\mathrm{m}(f) \subset N.
Remark 1. We recall that according to Auslander et al. [2], if P is an A-projective module then
\mathrm{t}\mathrm{r}P (A) is an idempotent ideal of A and one obtains all the idempotent ideals of A this way.
Remark 2. Note that, since the homological ideals are idempotent ideals and the idempotent
ideals are traces of projective modules over A, then there is always a finite number of homological
ideals.
Following the assumption that A is a bounded quiver algebra of the form kQ/I and the number
of vertices of Q are finite for every subset \{ a1, . . . , am\} \subset Q0, we will assume the following
notation for every idempotent ideal generated by the trace of P (a1)\oplus . . .\oplus P (am) in A:
Ia1,...,am = \mathrm{t}\mathrm{r}(P (a1)\oplus ...\oplus P (am))(A). (1)
In this paper, we combine tools developed by Auslander et al. in [2], Xi and De la Peña in [9] and
the integer specializations of some Brauer configuration (see Subsection 2.3) to establish an explicit
formula for the number of homological ideals associated to some Nakayama algebras. This number
allows to establish a partition formula for even-index Fibonacci numbers as Fahr and Ringel define
in [3 – 5].
2.2. Brauer configuration algebras. Brauer configuration algebras were introduced by Green
and Schroll in [7] as a generalization of Brauer graph algebras which are biserial algebras of tame
representation type and whose representation theory is encoded by some combinatorial data based on
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
HOMOLOGICAL IDEALS AS INTEGER SPECIALIZATIONS OF SOME BRAUER CONFIGURATION . . . 1203
graphs. Actually, underlying every Brauer graph algebra is a finite graph with acyclic orientation of
the edges at every vertex and a multiplicity function [7]. The construction of a Brauer graph algebra
is a special case of the construction of a Brauer configuration algebra in the sense that every Brauer
graph is a Brauer configuration with the restriction that every polygon is a set with two vertices. In
the sequel, we give precise definitions of a Brauer configuration and a Brauer configuration algebra.
A Brauer configuration \Gamma is a quadruple of the form \Gamma = (\Gamma 0,\Gamma 1, \mu ,\scrO ), where:
(B1) \Gamma 0 is a finite set whose elements are called vertices;
(B2) \Gamma 1 is a finite collection of multisets called polygons; in this case, if V \in \Gamma 1 then the
elements of V are vertices possibly with repetitions, \mathrm{o}\mathrm{c}\mathrm{c}(\alpha , V ) denotes the frequency of the vertex
\alpha in the polygon V and the valency of \alpha denoted \mathrm{v}\mathrm{a}\mathrm{l}(\alpha ) is defined in such a way that
\mathrm{v}\mathrm{a}\mathrm{l}(\alpha ) =
\sum
V \in \Gamma 1
\mathrm{o}\mathrm{c}\mathrm{c}(\alpha , V ); (2)
(B3) \mu is an integer valued function such that \mu : \Gamma 0 \rightarrow \BbbN where \BbbN denotes the set of positive
integers, it is called the multiplicity function;
(B4) \scrO denotes an orientation defined on \Gamma 1 which is a choice, for each vertex \alpha \in \Gamma 0, of a
cyclic ordering of the polygons in which \alpha occurs as a vertex, including repetitions, we denote S\alpha
such collection of polygons; more specifically, if S\alpha =
\bigl\{
V
(\alpha 1)
1 , V
(\alpha 2)
2 , . . . , V
(\alpha t)
t
\bigr\}
is the collection
of polygons where the vertex \alpha occurs with \alpha i = \mathrm{o}\mathrm{c}\mathrm{c}(\alpha , Vi) and V
(\alpha i)
i meaning that S\alpha has \alpha i
copies of Vi, then an orientation \scrO is obtained by endowing a linear order \leq to S\alpha and adding a
relation Vt \leq V1, if V1 = \mathrm{m}\mathrm{i}\mathrm{n}S\alpha and Vt = \mathrm{m}\mathrm{a}\mathrm{x}S\alpha ;
(B5) every vertex in \Gamma 0 is a vertex in at least one polygon in \Gamma 1;
(B6) every polygon has at least two vertices;
(B7) every polygon in \Gamma 1 has at least one vertex \alpha such that \mathrm{v}\mathrm{a}\mathrm{l}(\alpha )\mu (\alpha ) > 1.
The set (S\alpha ,\leq ) is called the successor sequence at the vertex \alpha .
A vertex \alpha \in \Gamma 0 is said to be truncated if \mathrm{v}\mathrm{a}\mathrm{l}(\alpha )\mu (\alpha ) = 1, that is, \alpha is truncated if it occurs
exactly once in exactly one V \in \Gamma 1 and \mu (\alpha ) = 1. A vertex is non-truncated if it is not truncated.
The quiver of a Brauer configuration algebra. The quiver Q\Gamma =
\bigl(
(Q\Gamma )0, (Q\Gamma )1
\bigr)
of a Brauer
configuration algebra is defined in such a way that the vertex set (Q\Gamma )0 = \{ v1, v2, . . . , vm\} of Q\Gamma is
in correspondence with the set of polygons \{ V1, V2, . . . , Vm\} in \Gamma 1, noting that there is one vertex
in (Q\Gamma )0 for every polygon in \Gamma 1.
Arrows in Q\Gamma are defined by the successor sequences. That is, there is an arrow vi
si - \rightarrow vi+1 \in
\in (Q\Gamma )1 provided that Vi \leq Vi+1 in (S\alpha ,\leq ) \cup \{ Vt \leq V1\} for some non-truncated vertex \alpha \in \Gamma 0.
In other words, for each non-truncated vertex \alpha \in \Gamma 0 and each successor V \prime of V at \alpha , there is an
arrow from v to v\prime in Q\Gamma where v and v\prime are the vertices in Q\Gamma associated to the polygons V and
V \prime in \Gamma 1, respectively.
The ideal of relations and definition of a Brauer configuration algebra. Fix a polygon
V \in \Gamma 1 and suppose that \mathrm{o}\mathrm{c}\mathrm{c}(\alpha , V ) = t \geq 1, then there are t indices i1, . . . , it such that V = Vij .
Then the special \alpha -cycles at v are the cycles Ci1 , Ci2 , . . . , Cit , where v is the vertex in the quiver
of Q\Gamma associated to the polygon V. If \alpha occurs only once in V and \mu (\alpha ) = 1, then there is only
one special \alpha -cycle at v.
Let k be a field and \Gamma a Brauer configuration. The Brauer configuration algebra associated to
\Gamma is defined to be the bounded path algebra \Lambda \Gamma = kQ\Gamma /I\Gamma , where Q\Gamma is the quiver associated to \Gamma
and I\Gamma is the ideal in kQ\Gamma generated by the following set of relations \rho \Gamma of type I, II and III.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1204 PEDRO FERNANDO FERNÁNDEZ ESPINOSA, AGUSTIN MORENO CAÑADAS
Fig. 1. The quiver Q\Gamma n defined by the Brauer configuration \Gamma n.
1. Relations of type I. For each polygon V = \{ \alpha 1, . . . , \alpha m\} \in \Gamma 1 and each pair of non-truncated
vertices \alpha i and \alpha j in V, the set of relations \rho \Gamma contains all relations of the form C\mu (\alpha i) - C \prime \mu (\alpha j)
where C is a special \alpha i-cycle and C \prime is a special \alpha j -cycle.
2. Relations of type II. Relations of type II are all paths of the form C\mu (\alpha )a where C is a
special \alpha -cycle and a is the first arrow in C.
3. Relations of type III. These relations are quadratic monomial relations of the form ab in kQ\Gamma
where ab is not a subpath of any special cycle unless a = b and a is a loop associated to a vertex of
valency 1 and \mu (\alpha ) > 1.
As an example for n \geq 4 fixed, we consider a Brauer configuration \Gamma n = (\Gamma 0,\Gamma 1, \mu ,\scrO ) such
that:
1) \Gamma 0 = \{ n - k - 1 \in \BbbN | 2 \leq k \leq n - 1\} \cup \{ n - 2\} ,
2) \Gamma 1 = \{ Uk = \{ n - 2, n - k - 1\} | 2 \leq k \leq n - 1\} ,
3) the orientation \scrO is defined in such a way that
(a) vertex n - 2 has associated the successor sequence U2 < U3 < . . . < Un - 1, in this case,
\mathrm{v}\mathrm{a}\mathrm{l}(n - 2) = n - 2,
(b) if 2 \leq k \leq n - 1, then at vertex n - k - 1, it holds that the corresponding successor sequence
consists only of Uk, and for each k, \mathrm{v}\mathrm{a}\mathrm{l}(n - k - 1) = 1,
4) \mu (n - 2) = 1,
5) \mu (n - k - 1) = n - 2, 2 \leq k \leq n - 1.
The ideal I\Gamma n of the corresponding Brauer configuration algebra \Lambda \Gamma n is generated by the follo-
wing relations (see Fig. 1), for which it is assumed the following notation for the special cycles:
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HOMOLOGICAL IDEALS AS INTEGER SPECIALIZATIONS OF SOME BRAUER CONFIGURATION . . . 1205
CUk
n - 2 =
\left\{
an - 2
1 an - 2
2 . . . an - 2
k - 1 , if k = 2,
an - 2
k - 1a
n - 2
k . . . an - 2
k - 2 , otherwise,
CUk
n - k - 1 = an - k - 1
1 .
(3)
1. ahi a
s
r, if h \not = s, for all possible values of i and r unless for the loops associated to the vertices
n - k - 1.
2. CUk
n - 2 -
\Bigl(
CUk
n - k - 1
\Bigr) n - 2
for all possible values of k.
3. CUk
n - 2a with a being the first arrow of CUk
n - 2 for all k.
4.
\Bigl(
CUk
n - k - 1
\Bigr) n - 2
a\prime with a\prime being the first arrow of CUk
n - k - 1 for all k.
Figure 1 shows the quiver Q\Gamma n associated to this configuration.
The following results describe the structure of a Brauer configuration algebra [7].
Theorem 1. Let \Lambda be a Brauer configuration algebra with Brauer configuration \Gamma .
1. There is a bijective correspondence between the set of projective indecomposable \Lambda -modules
and the polygons in \Gamma .
2. If P is a projective indecomposable \Lambda -module corresponding to a polygon V in \Gamma . Then
\mathrm{r}\mathrm{a}\mathrm{d}P is a sum of r indecomposable uniserial modules, where r is the number of (non-truncated)
vertices of V and where the intersection of any two of the uniserial modules is a simple \Lambda -module.
3. A Brauer configuration algebra is a multiserial algebra.
4. The number of summands in the heart of an indecomposable projective \Lambda -module P such that
\mathrm{r}\mathrm{a}\mathrm{d}2 P \not = 0 equals the number of non-truncated vertices of the polygons in \Gamma corresponding to P
counting repetitions.
5. If \Lambda \prime is a Brauer configuration algebra obtained from \Lambda by removing a truncated vertex of a
polygon in \Gamma 1 with d \geq 3 vertices then \Lambda is isomorphic to \Lambda \prime .
Proposition 2. Let \Lambda be a Brauer configuration algebra associated to the Brauer configuration
\Lambda and \scrC = \{ C1, . . . , Ct\} be a full set of equivalence class representatives of special cycles. Assume
that for i = 1, . . . , t, Ci is a special \alpha i-cycle, where \alpha i is a non-truncated vertex in \Gamma . Then
\mathrm{d}\mathrm{i}\mathrm{m}k \Lambda = 2| Q0| +
\sum
Ci\in \scrC
| Ci| (ni| Ci| - 1),
where | Q0| denotes the number of vertices of Q, | Ci| denotes the number of arrows in the \alpha i-cycle
Ci and ni = \mu (\alpha i).
Proposition 3. Let \Lambda be the Brauer configuration algebra associated to a connected Brauer
configuration \Gamma . The algebra \Lambda has a length grading induced from the path algebra kQ if and only
if there is an N \in \BbbZ >0 such that for each non-truncated vertex \alpha \in \Gamma 0 \mathrm{v}\mathrm{a}\mathrm{l}(\alpha )\mu (\alpha ) = N.
Sierra [10] proved the following result regarding the center of a Brauer configuration algebra.
Theorem 2. Let \Gamma be a reduced (i.e., without truncated vertices) and connected Brauer confi-
guration and let Q be its induced quiver and \Lambda be the induced Brauer configuration algebra such
that \mathrm{r}\mathrm{a}\mathrm{d}2 \Lambda \not = 0. Then the dimension of the center of \Lambda denoted \mathrm{d}\mathrm{i}\mathrm{m}k Z(\Lambda ) is given by the formula
\mathrm{d}\mathrm{i}\mathrm{m}k Z(\Lambda ) = 1 +
\sum
\alpha \in \Gamma 0
\mu (\alpha ) + | \Gamma 1| - | \Gamma 0| +\#(\mathrm{L}\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{s}Q) - | C\Gamma | , (4)
where C\Gamma = \{ \alpha \in \Gamma 0 | \mathrm{v}\mathrm{a}\mathrm{l}(\alpha ) = 1 and \mu (\alpha ) > 1\} .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1206 PEDRO FERNANDO FERNÁNDEZ ESPINOSA, AGUSTIN MORENO CAÑADAS
As an example the following is the numerology associated to the algebra \Lambda \Gamma n = kQ\Gamma n/I\Gamma n with
Q\Gamma n as shown in Fig. 1 and special cycles given in (3)
\bigl(
| r(Q\Gamma n)| is the number of indecomposable
projective modules, | Ci| = \mathrm{v}\mathrm{a}\mathrm{l}(i)
\bigr)
:
| r(Q\Gamma n)| = n - 2,
| Cn - 2| = n - 2, | Cn - k - 1| = 1,
\sum
\alpha \in \Gamma 0
\sum
X\in \Gamma 1
\mathrm{o}\mathrm{c}\mathrm{c}(\alpha ,X) = n - 1, the number of special cycles,
\mathrm{d}\mathrm{i}\mathrm{m}k \Lambda \Gamma n = 2(n - 2) + (n - 2)(n - 3) + (n - 3)(n - 2) = 2(n - 2)2,
\mathrm{d}\mathrm{i}\mathrm{m}k Z(\Lambda \Gamma n) = 1 + 1 + (n - 2)2 + (n - 2) - (n - 1) + (n - 2) - (n - 2) = n2 - 4n+ 5.
Remark 3. \Lambda \Gamma n is a Brauer graph algebra and according to Proposition 3, the Brauer configura-
tion algebra \Lambda \Gamma n with quiver Q\Gamma n shown in Fig. 1 has a length grading induced by the path algebra
kQ\Gamma n , provided that for any \alpha \in \Gamma 0 it holds that \mu (\alpha ) \mathrm{v}\mathrm{a}\mathrm{l}(\alpha ) = n - 2.
2.3. Message of a Brauer configuration. The concept of the message of a Brauer configuration
is helpful to categorify some integer sequences in the sense of Fahr and Ringel (see Subsection 3.1
of the present document, [3, 4]).
Let \Gamma = \{ \Gamma 0,\Gamma 1, \mu ,\scrO \} be a Brauer configuration and let U \in \Gamma 1 be a polygon such that
U =
\Bigl\{
\alpha f1
1 , \alpha f2
2 , . . . , \alpha fn
n
\Bigr\}
, where fi = \mathrm{o}\mathrm{c}\mathrm{c}(\alpha i, U). The term
w(U) = \alpha f1
1 \alpha f2
2 . . . \alpha fn
n (5)
is said to be the word associated to U . The sum
M(\Gamma ) =
\sum
U\in \Gamma 1
w(U) (6)
is said to be the message of the Brauer configuration \Gamma .
An integer specialization of a Brauer configuration \Gamma is a Brauer configuration \Gamma e = (\Gamma e
0,\Gamma
e
1,
\mu e,\scrO e) endowed with a preserving orientation map e : \Gamma 0 \rightarrow \BbbN such that
\Gamma e
0 = \mathrm{I}\mathrm{m}\mathrm{g} e \subset \BbbN ,
\Gamma e
1 = e(\Gamma 1), if H \in \Gamma 1 then e(H) = \{ e(\alpha i) | \alpha i \in H\} \in e(\Gamma 1), (7)
\mu e(e(\alpha )) = \mu (\alpha ) for any \alpha \in \Gamma 0.
Besides e(U) \preceq e(V ) in \Gamma e
1 provided that U \preceq V in \Gamma 1.
Let we(U) = (e(\alpha 1))
f1(e(\alpha 2))
f2 . . . (e(\alpha n))
fn denote the specialization under e of a word
w(U). In such a case, M(\Gamma e) =
\sum
U\in \Gamma e
1
we(U) is the specialized message of the Brauer configu-
ration \Gamma with the usual integer sum and product (in general with the sum and product associated to
\mathrm{I}\mathrm{m}\mathrm{g} e).
Example 1. For the Brauer configuration \Gamma n whose associated quiver is shown in Fig. 1, we
define the specialization e(\alpha ) = 2\alpha , \alpha \in \Gamma 0, with the concatenation in each word given by the
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
HOMOLOGICAL IDEALS AS INTEGER SPECIALIZATIONS OF SOME BRAUER CONFIGURATION . . . 1207
difference of the specializations of the vertices belonging to a determined polygon, in such a case for
n fixed, we have
w(Uk) = (n - 2)(n - k - 1) for 2 \leq k \leq n - 1,
we(Uk) = 2n - 2 - 2n - k - 1 for 2 \leq k \leq n - 1, (8)
M(\Gamma e
n) =
\sum
Uk\in \Gamma 1
we(Uk) =
n - 1\sum
k=1
2n - 2 - 2n - k - 1.
3. Homological ideals associated to Nakayama algebras. In this section, we prove some
combinatorial conditions which allow to establish whether an idempotent ideal in some Nakayama
algebras is homological or not. We also give the number of homological ideals associated to these al-
gebras via the integer specialization of the Brauer configuration \Gamma n defined in Example 1. Moreover,
we use the number of homological ideals to establish a partition formula for even-index Fibonacci
numbers.
Let Q be either a linearly oriented quiver with underlying graph \BbbA n or a cycle \widetilde \BbbA n with cyclic
orientation. That is, Q is one of the following quivers (see Fig. 2):
8 Pedro Fernando Fernández Espinosa et al
3 Homological Ideals Associated to Nakayama
Algebras
In this section, we prove some combinatorial conditions which allow to establish
whether an idempotent ideal in some Nakayama algebras is homological or not. We
also give the number of homological ideals associated to these algebras via the integer
specialization of the Brauer configuration �n defined in Example 9. Moreover, we
use the number of homological ideals to establish a partition formula for even-index
Fibonacci numbers.
Let Q be either a linearly oriented quiver with underlying graph An or a cycle fAn with
cyclic orientation. That is, Q is one of the following quivers
1
2
3
4
5
.
.
.
n � 1
n
•
1
•
2
· · · •
n � 1
•
n
or
Figure 2: Quiver fAn with cyclic orientation and Dynkin diagram An linearly
oriented.
A quotient A of kQ by an admissible ideal I is called a Nakayama algebra [8].
In this work, for n � 3 fixed, we consider the algebras AR(i,j,k)
= kQ/I where Q is
a Dynkin diagram of type An linearly oriented and I is an admissible ideal generated
by one relation R(i,j,k) of length k starting at a vertex i and ending at a vertex j of
the given quiver, 1 i < j n. The following picture shows the general structure of
quivers Q which we are focused in this paper.
An = 1 ! · · · ! i ! i + 1 ! · · · ! i + k = j ! j + 1 ! · · · ! n � 1 ! n.
The following Lemmas 10-17 allow to determine which idempotent ideals of an algebra
AR(i,j,k)
are also homological ideals. In this case, Lemmas 10 and 11 regard the case
whenever the idempotent ideal is generated by the trace of just one projective module
associated to a vertex of the quiver.
Lemma 10. Every idempotent ideal Ir of an algebra AR(i,j,k)
(see (1)) with j r or
r i is homological.
Proof. For r i, we have the following cases:
1. trP (r)(P (t)) = 0 if t > r.
2. trP (r)(P (t)) = P (r) if t r, where P (r) denotes the k-th projective module.
If r � j, we consider the following cases:
1. trP (r)(P (t)) = P (r) if i < t r, where P (r) denotes the k-th projective module.
2. trP (r)(P (t)) = 0.
Fig. 2. Quiver \widetilde \BbbA n with cyclic orientation and Dynkin diagram \BbbA n linearly oriented.
A quotient A of kQ by an admissible ideal I is called a Nakayama algebra [8].
In this work, for n \geq 3 fixed, we consider the algebras AR(i,j,k)
= kQ/I, where Q is a Dynkin
diagram of type \BbbA n linearly oriented and I is an admissible ideal generated by one relation R(i,j,k)
of length k starting at a vertex i and ending at a vertex j of the given quiver, 1 \leq i < j \leq n. The
following picture shows the general structure of quivers Q which we are focused in this paper:
\BbbA n = 1 \rightarrow . . . \rightarrow i \rightarrow i+ 1 \rightarrow . . . \rightarrow i+ k = j \rightarrow j + 1 \rightarrow . . . \rightarrow n - 1 \rightarrow n.
The following lemmas allow to determine which idempotent ideals of an algebra AR(i,j,k)
are also
homological ideals. In this case, Lemmas 1 and 2 regard the case whenever the idempotent ideal is
generated by the trace of just one projective module associated to a vertex of the quiver.
Lemma 1. Every idempotent ideal Ir of an algebra AR(i,j,k)
(see (1)) with j \leq r or r \leq i is
homological.
Proof. For r \leq i, we have the following cases:
1) \mathrm{t}\mathrm{r}P (r)(P (t)) = 0 if t > r,
2) \mathrm{t}\mathrm{r}P (r)(P (t)) = P (r) if t \leq r, where P (r) denotes the kth projective module.
If r \geq j, we consider the following cases:
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1208 PEDRO FERNANDO FERNÁNDEZ ESPINOSA, AGUSTIN MORENO CAÑADAS
1) \mathrm{t}\mathrm{r}P (r)(P (t)) = P (r) if i < t \leq r, where P (r) denotes the kth projective module,
2) \mathrm{t}\mathrm{r}P (r)(P (t)) = 0.
In all cases \mathrm{t}\mathrm{r}P (r)(AR(i,j,k)
) = P (r)l for some l \in \BbbN . The result follows as a consequence of
Proposition 1, item 2. We are done.
Lemma 2. Every idempotent ideal It of an algebra AR(i,j,k)
with i + 1 \leq t \leq j - 1 is not
homological.
Proof. Consider Lt = \mathrm{t}\mathrm{r}P (t) P (i) = P (i)/S(i) \oplus . . . \oplus S(t - 1), where S(k) denote the
kth simple module. Also, note that there are not morphisms from P (t) to P (j) if t \not = j which
means that \mathrm{E}\mathrm{x}\mathrm{t}1AR(i,j,k)
(Lt, P (j)) is a direct summand of \mathrm{E}\mathrm{x}\mathrm{t}1AR(i,j,k)
(It, AR(i,j,k)
/It), provided that
Lt is a direct summand of It and P (j) is a direct summand of AR(i,j,k)
/It. Applying the functor
\mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
( - , P (j)) to a projective resolution of Lt with the form
0 \rightarrow P (j) \rightarrow P (t) \rightarrow Lt \rightarrow 0,
it is obtained the exact sequence
0 \rightarrow \mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
(P (t), P (j)) \rightarrow \mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
(P (j), P (j)) \rightarrow 0.
Thus, \mathrm{E}\mathrm{x}\mathrm{t}1AR(i,j,k)
(Lt, P (n)) \sim = k and \mathrm{E}\mathrm{x}\mathrm{t}1AR(i,j,k)
(Ii, AR(i,j,k)
/Ii) \not = 0. Then the idempotent ideal It
is not an homological ideal as a consequence of Proposition 1, item 3.
Lemma 3. If each idempotent ideal I\alpha w of an algebra AR(i,j,k)
is not homological, then every
idempotent ideal of the form I\alpha 1,...,\alpha l
is not homological for 2 \leq l \leq k - 1.
Proof. For l fixed, we start by computing I\alpha 1,...,\alpha l
,
I\alpha 1,...,\alpha l
= \mathrm{t}\mathrm{r}P (\alpha 1)\oplus ...\oplus P (\alpha l)(AR(i,j,k)
) =
l\sum
w=1
\mathrm{t}\mathrm{r}P (\alpha w)(AR(i,j,k)
).
In accordance with the hypothesis \alpha w \in [i+ 1, j - 1] and taking into account that
\mathrm{t}\mathrm{r}P (\alpha w)(AR(i,j,k)
) = L\alpha w\underbrace{} \underbrace{}
i times
\oplus P (\alpha w)\underbrace{} \underbrace{}
\alpha w - i times
\oplus 0\underbrace{} \underbrace{}
n - \alpha w times
, (9)
\mathrm{t}\mathrm{r}P (\alpha 1)\oplus ...\oplus P (\alpha l)(AR(i,j,k)
) = L\alpha 1\underbrace{} \underbrace{}
i times
\oplus
l\bigoplus
w=1
P (\alpha w)\oplus 0\underbrace{} \underbrace{}
n - i - l times
(10)
it holds that according to the identity (10), P (j) is a direct summand of AR(i,j,k)
/I\alpha 1...\alpha l
and L\alpha 1
has the following projective resolution:
0 \rightarrow P (j) \rightarrow P (\alpha 1) \rightarrow L\alpha 1 \rightarrow 0.
Applying the functor \mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
( - , P (j)), we have that \mathrm{E}\mathrm{x}\mathrm{t}1AR(i,j,k)
(L\alpha 1 , P (j)) \not = 0 and by Propo-
sition 1, item 3, we conclude that the idempotent ideal I\alpha 1...\alpha l
is not an homological ideal.
Lemma 4. For l fixed, if each idempotent ideal I\alpha w of an algebra AR(i,j,k)
with 1 \leq w \leq l is
homological, then every idempotent ideal of the form I\alpha 1,...,\alpha l
is also homological.
Proof. It suffices to observe that \mathrm{t}\mathrm{r}P (\alpha w)(AR(i,j,k)
) = P (\alpha w)
l for some l \in \BbbN .
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HOMOLOGICAL IDEALS AS INTEGER SPECIALIZATIONS OF SOME BRAUER CONFIGURATION . . . 1209
Lemma 5. Every ideal Ii,t or It,j of an algebra AR(i,j,k)
is homological.
Proof. In accordance with the previous lemma we can conclude that if It is homological then
the result holds. If it is not the case then we consider the following cases:
1. For It non homological we can compute Ii,t = \mathrm{t}\mathrm{r}P (i)\oplus P (t)(AR(i,j,k)
) (see identity (9)), since
r \leq i, then \mathrm{t}\mathrm{r}P (i) P (r) = P (i). Therefore ideal Ii,t is projective and idempotent. Thus, for Proposi-
tion 1, item 2, we conclude that ideal Ii,t is homological.
2. We start by computing It,j as follows:
It,j = \mathrm{t}\mathrm{r}P (t)\oplus P (j)(AR(i,j,k)
) = Lt\underbrace{} \underbrace{}
i times
\oplus P (t)\underbrace{} \underbrace{}
t - i times
\oplus P (j)\underbrace{} \underbrace{}
j - t times
\oplus 0\underbrace{} \underbrace{}
n - j times
,
AR(i,j,k)
/It,j is given by
AR(i,j,k)
/It,j =
P (1)\oplus P (2)\oplus . . .\oplus P (i)\oplus . . .\oplus P (t)\oplus . . .\oplus P (j)\oplus . . .\oplus P (n)
Lt \oplus . . .\oplus Lt \oplus P (t)\oplus . . .\oplus P (t)\oplus P (j)\oplus . . .\oplus P (j)\oplus 0\oplus . . .\oplus 0
.
In order to compute \mathrm{E}\mathrm{x}\mathrm{t}1AR(i,j,k)
= (It,j , AR(i,j,k)
/It,j) we consider the projective resolution of Lt
0 \rightarrow P (j) \rightarrow P (t) \rightarrow Lt \rightarrow 0.
Applying the functor \mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
( - , P (j)), we obtain
0 \rightarrow \mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
(P (t), AR(i,j,k)
/It,j) \rightarrow \mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
(P (j), AR(i,j,k)
/It,j) \rightarrow 0.
Taking into account that
\left\{
\mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
\biggl(
P (t),
P (z)
Lt
\biggr)
= 0 if 1 \leq z \leq i,
\mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
\biggl(
P (t),
P (y)
P (t)
\biggr)
= 0 if i+ 1 \leq y \leq t - 1,
\mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
\biggl(
P (t),
P (v)
P (j)
\biggr)
= 0 if t+ 1 \leq v \leq j - 1,
\mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
(P (t), P (u)) = 0 if j + 1 \leq u \leq n,
\left\{
\mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
\biggl(
P (j),
P (z)
Lt
\biggr)
= 0 if 1 \leq z \leq i,
\mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
\biggl(
P (j),
P (y)
P (t)
\biggr)
= 0 if i+ 1 \leq y \leq t - 1,
\mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
\biggl(
P (j),
P (v)
P (j)
\biggr)
= 0 if t+ 1 \leq v \leq j - 1,
\mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
(P (j), P (u)) = 0 if j + 1 \leq u \leq n.
We conclude that \mathrm{E}\mathrm{x}\mathrm{t}nAR(i,j,k)
(It,j , AR(i,j,k)
/It,j) = 0 and that the idempotent ideal It,j is an homo-
logical ideal as a consequence of Proposition 1, item 3.
Remark 4. If the non homological ideal It has the form It1,...,tn the previous Lemma 5 also
holds.
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1210 PEDRO FERNANDO FERNÁNDEZ ESPINOSA, AGUSTIN MORENO CAÑADAS
Lemma 6. For 1 \leq h \leq i - 1, 1 \leq l \leq k - 1 and 1 \leq m \leq n - j fixed, every idempotent ideal
of the form Iz1,...,zh,t1,...,tl,y1,...,ym of an algebra AR(i,j,k)
, where za \in [1, i - 1], tb \in [i + 1, j - 1],
yc \in [j + 1, n], is not homological.
Proof. For h, l and m fixed, we start by computing Iz1,...,zh,t1,...,tl,y1,...,ym ,
Iz1,...,zh,t1,...,tl,y1,...,ym = \mathrm{t}\mathrm{r}P (z1)\oplus ...\oplus P (zh)\oplus P (t1)\oplus ...\oplus P (tl)\oplus P (y1),\oplus ...\oplus P (ym)(AR(i,j,k)
) =
=
h\sum
a=1
\mathrm{t}\mathrm{r}P (za)(AR(i,j,k)
)
\underbrace{} \underbrace{}
(1)
+
l\sum
b=1
\mathrm{t}\mathrm{r}P (tb)(AR(i,j,k)
)
\underbrace{} \underbrace{}
(2)
+
m\sum
c=1
\mathrm{t}\mathrm{r}P (yc)(AR(i,j,k)
)
\underbrace{} \underbrace{}
(3)
. (11)
The traces (1) – (3) can be written as follows:
h\sum
a=1
\mathrm{t}\mathrm{r}P (za)(AR(i,j,k)
) =
h\bigoplus
a=1
P (za)\oplus 0\oplus . . .\oplus 0,
l\sum
b=1
\mathrm{t}\mathrm{r}P (tb)(AR(i,j,k)
) = Lt1\underbrace{} \underbrace{}
i times
\oplus
l\bigoplus
b=1
P (tb)\oplus 0\underbrace{} \underbrace{}
n - i - l times
, (12)
m\sum
c=1
\mathrm{t}\mathrm{r}P (yc)(AR(i,j,k)
) = 0\underbrace{} \underbrace{}
i times
\oplus P (y1)\underbrace{} \underbrace{}
j - i times
\oplus
m\bigoplus
c=1
P (yc)\oplus 0\underbrace{} \underbrace{}
n - m - j times
.
Thus, the ideal Iz1,...,zh,t1,...,tl,y1,...,ym has the following form:
h\bigoplus
a=1
P (za)\oplus Lt1\underbrace{} \underbrace{}
i - h times
\oplus
l\bigoplus
b=1
P (tb)\oplus P (y1)\underbrace{} \underbrace{}
j - i - l times
\oplus
m\bigoplus
c=1
P (yc)\oplus 0\underbrace{} \underbrace{}
n - m - j times
. (13)
In accordance with (13) we have that
P (j)
P (y1)
is a direct summand of the quotient
AR(i,j,k)
/Iz1,...,zh,t1,...,tl,y1,...,ym and Lt1 has the following projective resolution:
0 \rightarrow P (j) \rightarrow P (t1) \rightarrow Lt1 \rightarrow 0. (14)
Applying the functor \mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
\biggl(
- ,
P (j)
P (y1)
\biggr)
to the resolution (14), we obtain the exact sequence
0 \rightarrow \mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
\biggl(
P (t),
P (j)
P (y1)
\biggr)
\rightarrow \mathrm{H}\mathrm{o}\mathrm{m}AR(i,j,k)
\biggl(
P (j),
P (j)
P (y1)
\biggr)
\rightarrow 0.
Then \mathrm{E}\mathrm{x}\mathrm{t}1AR(i,j,k)
\biggl(
Lt,
P (j)
P (y1)
\biggr)
\sim = k and
\mathrm{E}\mathrm{x}\mathrm{t}1AR(i,j,k)
(Iz1,...,zh,t1,...,tl,y1,...,ym , AR(i,j,k)
/Iz1,...,zh,t1,...,tl,y1,...,ym) \not = 0
by Proposition 1, item 3, we conclude that the idempotent ideal Iz1,...,zh,t1,...,tl,y1,...,ym is not an
homological ideal.
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HOMOLOGICAL IDEALS AS INTEGER SPECIALIZATIONS OF SOME BRAUER CONFIGURATION . . . 1211
Lemma 7. For 1 \leq h \leq i - 1, 1 \leq l \leq k - 1 and 1 \leq m \leq n - j fixed, the idempotent ideals
Iz1,...,zh,t1,...,tl and It1,...,tl,y1,...,ym of an algebra AR(i,j,k)
, where za \in [1, i - 1], tb \in [i + 1, j - 1],
yc \in [j + 1, n], are not homological.
Proof. It is enough to consider in (11) the trace
\sum h
a=1
\mathrm{t}\mathrm{r}P (za)(AR(i,j,k)
) = 0 or the trace
\sum m
c=1
\mathrm{t}\mathrm{r}P (yc)(AR(i,j,k)
) = 0.
3.1. On the number of homological ideals associated to some Nakayama algebras. The
following results allow us to compute the number of homological and non homological ideals in
a bounded algebra AR(i,j,k)
by using the integer specialization e of the Brauer configuration \Gamma n
introduced in Example 1.
Theorem 3. For n \geq 4 fixed and 2 \leq k \leq n - 1, the number | \BbbN \BbbH \BbbI kn| of non homological ideals
of an algebra AR(i,j,k)
is given by the identity | \BbbN \BbbH \BbbI kn| = we(Uk).
Proof. We note that according to Lemmas 2 and 3 there are 2k - 1 - 1 non homological ideals
associated only to the vertices inside the relation R(i,j,k), by Lemma 6 there are additional 2n - k - 1
non homological ideals arising from the combination of vertices which are inside and outside of the
relation. The theorem follows taking into account the product rule and Example 1.
Corollary 1. For n \geq 4 fixed and 2 \leq k \leq n - 1, the number of homological ideals | \BbbH \BbbI kn| of
an algebra AR(i,j,k)
is given by the identity | \BbbH \BbbI kn| = 2n - we(Uk) = 3 \cdot 2n - 2 + 2n - k - 1.
Proof. Since there are 2n idempotent ideals in AR(i,j,k)
, then the result holds as a consequence
of Theorem 3.
The formula obtained in Theorem 3 induces the following triangle:
Non homological triangle \BbbN \BbbH \BbbI \BbbT
n
k
2 3 4 5 6 7 8 . . .
3 1 – – – – – – –
4 2 3 – – – – – –
5 4 6 7 – – – – –
6 8 12 14 15 – – – –
7 16 24 28 30 31 – – –
...
...
...
...
...
...
...
...
...
Entries | \BbbN \BbbH \BbbI kn| of triangle \BbbN \BbbH \BbbI \BbbT can be calculated inductively as follows: we start by defining
| \BbbN \BbbH \BbbI 2n| = 2n - 3 for all n \geq 3. Now, we assume that | \BbbN \BbbH \BbbI kn| = 0 with k \leq 1 and for the sake
of clarity we denote the specialization under e of a word w(Uk) of the polygon Uk in the Brauer
configuration \Gamma n as we(Un
k ) (see Example 1). Then, for k \geq 3,
we(Uk) = we(Un
k ) =
\bigl(
we(Un
k - 1) + we(Un - 1
k - 1 )
\bigr)
- we(Un - 1
k - 2 )
or, equivalently,
| \BbbN \BbbH \BbbI kn| = (| \BbbN \BbbH \BbbI k - 1
n | + | \BbbN \BbbH \BbbI k - 1
n - 1| ) - | \BbbN \BbbH \BbbI k - 2
n - 1| .
These arguments prove the following proposition.
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1212 PEDRO FERNANDO FERNÁNDEZ ESPINOSA, AGUSTIN MORENO CAÑADAS
Proposition 4. M(\Gamma e
n) equals the sum of the elements in the nth row of the non homological
triangle \BbbN \BbbH \BbbI \BbbT (see Example 1).
Remark 5. The integer sequence generated by M(\Gamma e
n) =
\sum n - 1
k=1
2n - 2 - 2n - k - 1 = \{ 1, 5, 17, 49,
129, 321, 769, 1793, 4097, 9217, . . .\} is encoded A000337 in the OEIS. Elements of the sequence
A000337 also correspond to the sums of the elements of the rows of the Reinhard Zumkeller triangle.
Remark 6. The sum of entries in the diagonals of the non homological triangle is the sequence
A274868 in the OEIS, and it is related with the number of set partitions of [n] into exactly four
blocks such that all odd elements are in blocks with an odd index, whereas all even elements are in
blocks with an even index.
Similarly, for the homological ideals Corollary 1 induces the following triangle:
Homological triangle \BbbH \BbbI \BbbT
n
k
2 3 4 5 6 7 8 . . .
3 7 – – – – – – –
4 14 13 – – – – – –
5 28 26 25 – – – – –
6 56 52 50 49 – – – –
7 112 104 100 98 97 – – –
...
...
...
...
...
...
...
...
...
The elements of the homological triangle are closely related with the research of categorification
of integer sequences. Particularly, these numbers deal with the work of Fahr and Ringel regarding
categorification of Fibonacci numbers. In Subsection 3.2, we reconstruct the partition formula for
even-index Fibonacci numbers given in [3, 5] by using the number of homological ideals of some
Nakayama algebras.
3.2. Categorification of integer sequences. In this subsection, we give some relationships
between the number of homological ideals of an algebra AR(i,j,k)
and the partition formula given by
Fahr and Ringel for even-index Fibonacci numbers in [3].
According to Fahr and Ringel [4] a categorification of a sequence of numbers means to consider
instead of these numbers suitable objects in a category (for instance, representation of quivers)
so that the numbers in question occur as invariants of the objects, equality of numbers may be
visualized by isomorphisms of objects functional relations by functorial ties. The notion of this
kind of categorification arose from the use of suitable arrays of numbers to obtain integer partitions
of dimensions of indecomposable preprojective modules over the 3-Kronecker algebra (see Fig. 3
where it is shown the 3-Kronecker quiver and a piece of the oriented 3-regular tree or universal
covering (T,E,\Omega t) as described by Fahr and Ringel in [3]). Firstly, they noted that the vector
dimension of these kind of modules consists of even-index Fibonacci numbers (denoted fi and such
that fi = fi - 1 + fi - 2 for i \geq 2, f0 = 0, f1 = 1) then they used results from the universal covering
theory developed by Gabriel and his students to identify such Fibonacci numbers with dimensions of
representations of the corresponding universal covering.
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HOMOLOGICAL IDEALS AS INTEGER SPECIALIZATIONS OF SOME BRAUER CONFIGURATION . . . 1213
1 2
• •
• •• •
• •
• •
• •• •
• •
Fig. 3. The 3-Kronecker quiver and an illustration of its
corresponding universal covering.
Homological Ideals as ... 15
Finally, we recall that Ringel and Fahr interpreted weights wi,i�j as distances in a
3-regular tree (T, E) (with T a vertex set and E a set of edges) from a fixed point
x0 2 T to any point y 2 T . They define sets Tr whose points have distance r to x0 , in
such a case T0 = {x0}, T1 are the neighbors of x0 and so on (note that |Tr| = 3(2r�1)
if r � 1). A given vertex y is said to be even or odd according to this parity [3].
Any vertex y 2 T yields a suitable reflection �y on the set of functions T ! Z with finite
support, denoted Z[T ], and some reflection products denoted �0 and �1 according to
the parity of y are introduced in [3]. Then some maps at : N0 ! Z 2 Z[T ] are defined
in such a way that if a0 is the characteristic function of T0 then a0(x) = 0 unless
x = x0 in which case a0(x0) = 1, and at = (�0�1)
ta0, for t � 0, with at[r] = at(x),
for r 2 N0 and x 2 Tr, these maps at give the values di,j of the array (see Figure 4).
The following table is an example of such array with n = 7. Rows are giving by the
values of t, Pt is a notation for a 3-Kronecker preprojective module with dimension
vector [f2t+2 f2t] (see [5]).
According to the present discussion the identity (18) adopts one of the following forms
defined by Ringel and Fahr in [3]:
f4t =
X
r odd
|Tr| · at[r] = 3
X
m�1
22m · at[2m + 1],
f4t+2 =
X
r even
|Tr| · at[r] = at[0] + 3
X
m�1
22m�1 · at[2m].
(19)
0
1
2
3
4
5
6
7
...
f2
f4
f6
f8
f10
f12
f14
f16
...
...
...
...
...
...
1
2
7
29
1
3
12
53
1
4
18
1
5
25
1
6
1
7
1
1
P1
P2
P3
P4
P5
P6
P7
at[0]· · ·
Figure 4: The even-index Fibonacci partition triangle [5].
For example for t = 3 and t = 4, we compute f8 and f10 as follows;
21 = f8 = 0 + 3(3 · 20) + 0 + 1(3 · 22),
55 = f10 = 1 · 7 + 0 + 4(3 · 21) + 0 + 1(3 · 23).
(20)
Fig. 4. The even-index Fibonacci partition triangle [5].
First of all note that the road to a categorification of the Fibonacci numbers has several stops some
of them dealing with diagonal (lower) arrays of numbers of the form D = (di,j) with 0 \leq j \leq i \leq n
(columns numbered from right to the left, see Fig. 4) for some n \geq 0 fixed and such that
di,i = 1 for all i \geq 0,
di,j = 0 for all j > i,
d2k+i,i - 1 = 0, if i \geq 1, k \geq 0,
d2k,0 = 3d2k - 1,1 - d2(k - 1),0, k \geq 1,
di+1,j - 1 = 2di,j + di,j - 2 - di - 1,j - 1, i, j \geq 2.
In addition, if i \geq 4, then the following identity (hook rule) holds:
i - 2\sum
k=0
di+k,i - k + d2i - 2,0 = d2i - 1,1.
Note that to each entry di,i - j it is possible to assign a weight wi,i - j by using the numbers in the
homological triangle \BbbH \BbbI \BbbT as follows:
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1214 PEDRO FERNANDO FERNÁNDEZ ESPINOSA, AGUSTIN MORENO CAÑADAS
wi,i - j =
\left\{
| \BbbH \BbbI k2s+2| - 22\cdot s - k+1, if is even, i is odd and i \not = j + 1,
| \BbbH \BbbI k2s+1| - 22\cdot s - k, if j is even, i is even,
3, if i odd, j even and i = j + 1,
1, if i = j = 2h for some h \geq 0,
0, if j is odd, i \not = j,
where s =
\biggl\lfloor
i - j
2
\biggr\rfloor
and \lfloor x\rfloor is the greatest integer number less than x. If we consider the multipli-
cation of the entry di,i - j with its corresponding weight wi,i - j , we can define a partition formula for
even-index Fibonacci numbers in the following form:
f2i+2 =
i\sum
j=0
(wi,i - j)(di,i - j). (15)
Finally, we recall that Fahr and Ringel interpreted weights wi,i - j as distances in a 3-regular tree
(T,E) (with T a vertex set and E a set of edges) from a fixed point x0 \in T to any point y \in T.
They define sets Tr whose points have distance r to x0, in such a case T0 = \{ x0\} , T1 are the
neighbors of x0 and so on (note that | Tr| = 3(2r - 1) if r \geq 1). A given vertex y is said to be even
or odd according to this parity [3].
Any vertex y \in T yields a suitable reflection \sigma y on the set of functions T \rightarrow \BbbZ with finite
support, denoted \BbbZ [T ], and some reflection products denoted \Phi 0 and \Phi 1 according to the parity of
y are introduced in [3]. Then some maps at : \BbbN 0 \rightarrow \BbbZ \in \BbbZ [T ] are defined in such a way that if a0
is the characteristic function of T0 then a0(x) = 0 unless x = x0 in which case a0(x0) = 1, and
at = (\Phi 0\Phi 1)
ta0, for t \geq 0, with at[r] = at(x), for r \in \BbbN 0 and x \in Tr, these maps at give the
values di,j of the array (see Fig. 4). The following table is an example of such array with n = 7.
Rows are giving by the values of t, Pt is a notation for a 3-Kronecker preprojective module with
dimension vector [f2t+2 f2t] (see [5]).
According to the present discussion the identity (15) adopts one of the following forms defined
by Fahr and Ringel in [3]:
f4t =
\sum
r odd
| Tr| at[r] = 3
\sum
m\geq 1
22m \cdot at[2m+ 1],
f4t+2 =
\sum
r even
| Tr| at[r] = at[0] + 3
\sum
m\geq 1
22m - 1 \cdot at[2m].
(16)
For example, for t = 3 and t = 4, we compute f8 and f10 as follows:
21 = f8 = 0 + 3(3 \cdot 20) + 0 + 1(3 \cdot 22),
55 = f10 = 1 \cdot 7 + 0 + 4(3 \cdot 21) + 0 + 1(3 \cdot 23).
Sequences at[0] = d2i,0 and at[1] = d2i+1,1 are encoded respectively as A132262 and A110122 in
the OEIS. Actually, sequence at[0] had not been registered in the OEIS before the publication of
Fahr and Ringel.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
HOMOLOGICAL IDEALS AS INTEGER SPECIALIZATIONS OF SOME BRAUER CONFIGURATION . . . 1215
The following result giving a relationship between the number of homological ideals and Fi-
bonacci numbers is a direct consequence of identities (15) and (16).
Theorem 4.
2t\sum
j=0
(w2t,2t - j)(d2t,2t - j) =
\sum
r even
| Tr| at[r], t \geq 0,
2t - 1\sum
j=0
(w2t - 1,2t - 1 - j)(d2t - 1,2t - 1 - j) =
\sum
r odd
| Tr| at[r], t \geq 1.
References
1. M. Armenta, Homological ideals of finite dimensional algebras, UNAM, Mexico (2016).
2. M. Auslander, M. I. Platzeck, G. Todorov, Homological theory of idempotent ideals, Trans. Amer. Math. Soc., 332,
№ 2, 667 – 692 (1992).
3. P. Fahr, C. M. Ringel, A partition formula for Fibonacci numbers, J. Integer Seq., 11, Article 08.14 (2008).
4. P. Fahr, C. M. Ringel, Categorification of the Fibonacci numbers using representations of quivers, J. Integer Seq.,
15, Article 12.2.1 (2012).
5. P. Fahr, C. M. Ringel, The Fibonacci triangles, Adv. Math., 230, 2513 – 2535 (2012).
6. M. Lanzilotta, M. A. Gatica, M. I. Platzeck, Idempotent ideals and the Igusa – Todorov functions, Algebras and
Represent. Theory, 20, 275 – 287 (2017).
7. E. L. Green, S. Schroll, Brauer configuration algebras: a generalization of Brauer graph algebras, Bull. Sci. Math.,
141, 539 – 572 (2017).
8. D. Happel, D. Zacharia, Algebras of finite global dimension, Algebras, Quivers and Representations, Abel Symp., 8,
Springer, Heidelberg (2013).
9. J. A. De la Peña, Changchang Xi, Hochschild cohomology of algebras with homological ideals, Tsukuba J. Math.,
30, № 1, 61 – 79 (2006).
10. A. Sierra, The dimension of the center of a Brauer configuration algebra, J. Algebra, 510, 289 – 318 (2018).
11. N. J. A. Sloane, The on-line encyclopedia of integer sequences, The OEIS Foundation; https://oeis.org.
Received 08.07.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
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| id | umjimathkievua-article-6218 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:35Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/06/139fc3bea3240d6b563d9412196dd006.pdf |
| spelling | umjimathkievua-article-62182023-01-07T13:45:33Z Homological ideals as integer specializations of some Brauer configuration algebras Homological ideals as integer specializations of some Brauer configuration algebras Fernández Espinosa, Pedro Fernando Moreno Cañadas, Agustín Fernández Espinosa, Pedro Fernando Moreno Cañadas, Agustín Brauer configuration algebra Categorification Homological ideal Integer specialization 16G20 16G30 16G60 UDC 512.5 The homological ideals associated with some Nakayama algebras are characterized and enumerated via integer specializations of some suitable Brauer configuration algebras.&nbsp;In addition, it is shown how the number of these homological ideals can be connected with the&nbsp; process of categorification of Fibonacci numbers defined by Ringel and Fahr. УДК 512.5 Гомологічні ідеали, як цілочислові спеціалізації деяких конфігураційних алгебр Брауера Охарактеризовано гомологічні ідеали, асоційовані з деякими алгебрами Накаями, та перераховано їх через цілочислові спеціалізації деяких відповідних конфігураційних алгебр Брауера. Крім того, показано як кількість таких гомологічних ідеалів може бути пов'язана з процесом категоризації чисел Фібоначчі, що був визначений Рінгелем і Фаром. Institute of Mathematics, NAS of Ukraine 2022-11-08 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6218 10.37863/umzh.v74i9.6218 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 9 (2022); 1201 - 1215 Український математичний журнал; Том 74 № 9 (2022); 1201 - 1215 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6218/9297 Copyright (c) 2022 Agustín Moreno Cañadas, Fernández Espinosa |
| spellingShingle | Fernández Espinosa, Pedro Fernando Moreno Cañadas, Agustín Fernández Espinosa, Pedro Fernando Moreno Cañadas, Agustín Homological ideals as integer specializations of some Brauer configuration algebras |
| title | Homological ideals as integer specializations of some Brauer configuration algebras |
| title_alt | Homological ideals as integer specializations of some Brauer configuration algebras |
| title_full | Homological ideals as integer specializations of some Brauer configuration algebras |
| title_fullStr | Homological ideals as integer specializations of some Brauer configuration algebras |
| title_full_unstemmed | Homological ideals as integer specializations of some Brauer configuration algebras |
| title_short | Homological ideals as integer specializations of some Brauer configuration algebras |
| title_sort | homological ideals as integer specializations of some brauer configuration algebras |
| topic_facet | Brauer configuration algebra Categorification Homological ideal Integer specialization 16G20 16G30 16G60 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6218 |
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