On Frobenius full matrix algebras with structure systems
Let n ≥ 2 be an integer. In [5] and [6], an n × n A-full matrix algebra over a field K is defined to be the set Mn(K) of all square n × n matrices with coefficients in K equipped with a multiplication defined by a structure system A, that is, an n-tuple of n × n matrices with certain properties....
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irk-123456789-1573562019-06-21T01:27:44Z On Frobenius full matrix algebras with structure systems Fujita, H. Sakai, Y. Simson, D. Let n ≥ 2 be an integer. In [5] and [6], an n × n A-full matrix algebra over a field K is defined to be the set Mn(K) of all square n × n matrices with coefficients in K equipped with a multiplication defined by a structure system A, that is, an n-tuple of n × n matrices with certain properties. In [5] and [6], mainly A-full matrix algebras having (0, 1)-structure systems are studied, that is, the structure systems A such that all entries are 0 or 1. In the present paper we study A-full matrix algebras having non (0, 1)-structure systems. In particular, we study the Frobenius Afull matrix algebras. Several infinite families of such algebras with nice properties are constructed in Section 4. 2007 Article On Frobenius full matrix algebras with structure systems / H. Fujita, Y. Sakai, D. Simson // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 1. — С. 24–39. — Бібліогр.: 13 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16G10, 16G30, 16G60. http://dspace.nbuv.gov.ua/handle/123456789/157356 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
Let n ≥ 2 be an integer. In [5] and [6], an n × n
A-full matrix algebra over a field K is defined to be the set Mn(K)
of all square n × n matrices with coefficients in K equipped with a
multiplication defined by a structure system A, that is, an n-tuple
of n × n matrices with certain properties. In [5] and [6], mainly
A-full matrix algebras having (0, 1)-structure systems are studied,
that is, the structure systems A such that all entries are 0 or 1.
In the present paper we study A-full matrix algebras having non
(0, 1)-structure systems. In particular, we study the Frobenius Afull matrix algebras. Several infinite families of such algebras with
nice properties are constructed in Section 4. |
| format |
Article |
| author |
Fujita, H. Sakai, Y. Simson, D. |
| spellingShingle |
Fujita, H. Sakai, Y. Simson, D. On Frobenius full matrix algebras with structure systems Algebra and Discrete Mathematics |
| author_facet |
Fujita, H. Sakai, Y. Simson, D. |
| author_sort |
Fujita, H. |
| title |
On Frobenius full matrix algebras with structure systems |
| title_short |
On Frobenius full matrix algebras with structure systems |
| title_full |
On Frobenius full matrix algebras with structure systems |
| title_fullStr |
On Frobenius full matrix algebras with structure systems |
| title_full_unstemmed |
On Frobenius full matrix algebras with structure systems |
| title_sort |
on frobenius full matrix algebras with structure systems |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157356 |
| citation_txt |
On Frobenius full matrix algebras with structure systems / H. Fujita, Y. Sakai, D. Simson // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 1. — С. 24–39. — Бібліогр.: 13 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT fujitah onfrobeniusfullmatrixalgebraswithstructuresystems AT sakaiy onfrobeniusfullmatrixalgebraswithstructuresystems AT simsond onfrobeniusfullmatrixalgebraswithstructuresystems |
| first_indexed |
2025-07-14T09:48:00Z |
| last_indexed |
2025-07-14T09:48:00Z |
| _version_ |
1837615263885819904 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2007). pp. 24 – 39
c© Journal “Algebra and Discrete Mathematics”
On Frobenius full matrix algebras with structure
systems
Hisaaki Fujita, Yosuke Sakai and Daniel Simson
Abstract. Let n ≥ 2 be an integer. In [5] and [6], an n × n
A-full matrix algebra over a field K is defined to be the set Mn(K)
of all square n× n matrices with coefficients in K equipped with a
multiplication defined by a structure system A, that is, an n-tuple
of n × n matrices with certain properties. In [5] and [6], mainly
A-full matrix algebras having (0, 1)-structure systems are studied,
that is, the structure systems A such that all entries are 0 or 1.
In the present paper we study A-full matrix algebras having non
(0, 1)-structure systems. In particular, we study the Frobenius A-
full matrix algebras. Several infinite families of such algebras with
nice properties are constructed in Section 4.
1. Introduction
Throughout this paper we freely use the rings, modules, and representa-
tion theory terminology introduced in [1], [2], [4], [9], [11], and [12]. In
particular, given a finite dimensional algebra R over a field K, we denote
by modR the category of all finite dimensional unitary right R-modules.
Given a module M in modR, we denote by soc M the socle of M .
Let K be a field and n ≥ 2 an integer. Let A = [A1, . . . , An] =
[a
(k)
ij ]i,j,k be an n-tuple of n× n matrices Ak = (a
(k)
ij ) ∈Mn(K) (1 ≤ k ≤
n) satisfying the following three conditions:
(A1) a
(k)
ij a
(j)
il = a
(k)
il a
(j)
kl , for all i, j, k, l ∈ {1, . . . , n},
(A2) a
(k)
kj = a
(k)
ik = 1, for all i, j, k ∈ {1, . . . , n}, and
2000 Mathematics Subject Classification: 16G10, 16G30, 16G60.
Key words and phrases: Frobenius algebra, quiver, module, socle, tame repre-
sentation type.
H. Fujita, Y. Sakai, D. Simson 25
(A3) a
(k)
ii = 0, for all i, k ∈ {1, . . . , n} such that i 6= k.
We denote by
(1.1) RA =
n
⊕
i,j=1
Kuij
a K-vector space, with basis {uij | 1 ≤ i, j ≤ n}, equipped with a multi-
plication (depending on A) defined by the formula
uikulj :=
{
a
(k)
ij uij , if k = l,
0, otherwise.
It is easy to check that RA is an associative, basic K-algebra
u11, . . . , unn are orthogonal primitive idempotents of RA and 1 = u11 +
· · · + unn is an identity element of RA, see [5, Proposition 1.1]. We call
RA an A-full matrix algebra and A a structure system of RA.
The reader is referred to the recent paper [7] for a degeneration-like
approach to the full matrix algebras RA with structure systems.
Since uiiRAujj 6= 0, for all 1 ≤ i, j ≤ n, then the K-algebra RA
is connected, that is, RA can not be decomposed into a product of two
subalgebras. Note also that the Jacobson radical J(RA) of RA has the
form
(1.2) J(RA) =
⊕
i6=j
uijK.
If V is a simple right RA-module, then V uii 6= 0, for some 1 ≤ i ≤ n,
and V ∼= uiiRA/uiiJ(RA) . Therefore the RA-modules
u11RA/u11J(RA), . . . , unnRA/unnJ(RA)
are the representatives of all pairwise non-isomorphic simple right RA-
modules. Note that dimKV = 1, for any simple right RA-module V .
Let RA be an A-full matrix algebra (1.1) and let M be a right RA-
module in modRA. The dimension vector of M (or the dimension type
of M) is defined to be the n-tuple
(1.3) dimM = (d1, . . . , dn) ∈ Z
n = K0(RA)
of integers di = dimKMuii, with 1 ≤ i ≤ n, see [1] and [5].
26 Frobenius full matrix algebras with structure systems
2. When A-full matrix algebras are isomorphic?
In this section, we give a criterion for two A-full matrix algebras RA and
RB to be isomorphic. Moreover, we give a list of the representatives of
all non-isomorphic 3× 3 A-full matrix algebras.
The isomorphism problem of A-full matrix algebras is also studied in
[7] in terms of an action
∗ : Gn(K)× STn(K) → STn(K)
of an algebraic group Gn(K) (containing the symmetric group Sn) on the
algebraic K-variety STn(K) of the structure systems A.
Proposition 2.1. Let RA =
n
⊕
i,j=1
Kuij and RB =
n
⊕
i,j=1
Kvij be full matrix
algebras with structure systems
A = [A1, . . . , An] = [a
(k)
ij ]i,j,k and B = [B1, . . . , Bn] = [b
(k)
ij ]i,j,k,
respectively. There is a K-algebra isomorphism RA
∼= RB if and only
if there exist a matrix T = (tij) ∈ Mn(K) and a permutation σ :
{1, . . . , n} → {1, . . . , n} of the set {1, . . . , n} such that
tij 6= 0, tii = 1, a
(σ(k))
σ(i)σ(j)tij = b
(k)
ij tiktkj , for all 1 ≤ i, j, k ≤ n.
Proof. Suppose that there is a K-algebra isomorphism f : RA → RB.
Then f(u11), . . . , f(unn) are orthogonal primitive idempotents of RB such
that 1RB
= f(u11) + · · ·+ f(unn). It follows from [4, Theorem 3.4.1] that
there exist a permutation σ of the set {1, . . . , n} and an invertible element
b ∈ RB such that vii = bf(uσ(i)σ(i))b
−1, for all 1 ≤ i ≤ n. Hence there is
a K-algebra isomorphism g : RA → RB such that vii = g(uσ(i)σ(i)), for all
1 ≤ i ≤ n. Since g(uσ(i)σ(j)) = viig(uσ(i)σ(j))vjj , then g(uσ(i)σ(j)) = tijvij ,
for some 0 6= tij ∈ K (1 ≤ i, j ≤ n). Clearly, tii = 1, for all 1 ≤ i ≤ n.
Since
g(uσ(i)σ(k)uσ(k)σ(j)) = g(uσ(i)σ(k))g(uσ(k)σ(j)),
then we have a
(σ(k))
σ(i)σ(j)tij = b
(k)
ij tiktkj , for all 1 ≤ i, j, k ≤ n. It follows that
T := (tij) ∈Mn(K) is the desired matrix.
Conversely, suppose that there exist a matrix T = (tij) and a permu-
tation σ of the set {1, . . . , n} satisfying the above condition. Then the
K-linear map
f : RA → RB,
given by uij 7→ tσ−1(i)σ−1(j)vσ−1(i)σ−1(j), defines a K-algebra isomorphism.
H. Fujita, Y. Sakai, D. Simson 27
As an immedeate consequence of the proposition, we have the follow-
ing.
Corollary 2.2. Let RA =
n
⊕
i,j=1
Kuij and RB =
n
⊕
i,j=1
Kvij be full matrix
algebras with (0, 1)-structure systems
A = [A1, . . . , An] = [a
(k)
ij ]i,j,k and B = [B1, . . . , Bn] = [b
(k)
ij ]i,j,k,
respectively. Then RA is isomorphic to RB as K-algebras if and only if
there exists a permutation σ of the set {1, . . . , n} such that b
(k)
ij = a
(σ(k))
σ(i)σ(j)
for all 1 ≤ i, j, k ≤ n.
Lemma 2.3. Let n ≥ 3 be an integer, and let A = [A1, . . . , An] =
[a
(k)
ij ]i,j,k be a structure system. Then, for any distinct 1 ≤ i, j, k ≤ n, the
following equalities hold a
(k)
ij a
(j)
ik = 0 and a
(i)
kj a
(k)
ij = 0.
Proof. This follows from (A1) and (A3).
Example 2.4. By applying Lemma 2.3 and Corollary 2.2, one can verify
that, for n = 3, the following five (0, 1)-structure systems A
(1), A
(2), A
(3),
A
(4), A
(5):
1 1 1 0 1 0 0 0 1
1 0 1 1 1 1 0 0 1
1 0 0 0 1 0 1 1 1
,
1 1 1 0 1 0 0 0 1
1 0 1 1 1 1 0 0 1
1 1 0 0 1 0 1 1 1
,
1 1 1 0 1 1 0 0 1
1 0 0 1 1 1 0 0 1
1 1 0 0 1 0 1 1 1
,
1 1 1 0 1 0 0 1 1
1 0 1 1 1 1 0 0 1
1 0 0 1 1 0 1 1 1
,
1 1 1 0 1 0 0 0 1
1 0 0 1 1 1 0 0 1
1 0 0 0 1 0 1 1 1
.
provide a list of all (0, 1)-structure systems A such that every A-full ma-
trix algebra RA is isomorphic to any of the algebras R
A(1) , R
A(2) , R
A(3) ,
R
A(4) , R
A(5) .
Given an arbitrary structure system A = [A1, . . . , An] = [a
(k)
ij ]i,j,k, we
define a new one A = [A1, . . . , An] = [a
(k)
ij ]i,j,k, where
a
(k)
ij :=
{
1, if a
(k)
ij 6= 0,
0, otherwise.
It is easy to see that A is a structure system. Following [7, Definition
3.1], we call the A-full matrix algebra R
A
a (0, 1)-limit of RA.
28 Frobenius full matrix algebras with structure systems
Theorem 2.5. For n = 3, there are just five 3 × 3 A-full matrix alge-
bras RA, up to isomorphism, which are given by the five (0, 1)-structure
systems in Example 2.4.
Proof. Let A be a 3× 3 A-full matrix algebra, where A = [A1, A2, A3] =
[a
(k)
ij ], and let R
A
be the (0, 1)-limit of RA. Then we show that RA is
isomorphic to R
A
, using Proposition 2.1. We put σ = id and T = (tij) ∈
M3(K), where
tij :=
{
a
(k)
ij , if a
(k)
ij 6= 0, for k 6= i, j,
1, otherwise,
for distinct i, j ∈ {1, 2, 3}, and tii := 1 for i = 1, 2, 3. Then using Lemma
2.3, one can check that a
(k)
ij tij = a
(k)
ij tiktkj , for all 1 ≤ i, j, k ≤ n. This
completes the proof.
3. Frobenius A-full matrix algebras
In this section, we improve the characterization of Frobenius A-full matrix
algebras RA given by [5, Lemma 4.2], where structure systems are (0, 1)-
matrices.
Assume that RA is an A-full matrix algebra (1.1) and let M be a right
RA-module with dimM = (1, . . . , 1). Then M has a K-basis {v1, . . . , vn}
such that viuii = vi, for all 1 ≤ i ≤ n. Consider the matrix S = (sij) ∈
Mn(K) such that
(∗) viukj =
{
sijvj , if k = i,
0, otherwise,
for all 1 ≤ i, j, k ≤ n, and that
(∗∗) sii = 1 and sikskj = a
(k)
ij sij , for all 1 ≤ i, j, k ≤ n.
We call S a representation matrix of M with respect to a K-basis
{v1, . . . , vn}. Conversely, let M be a K-vector space with a K-basis
{v1, . . . , vn} and S = (sij) ∈ Mn(K) which satisfies the condition (∗∗).
Then, by (∗), M has a right RA-module structure with dimM = (1, . . . , 1),
see [5, Proposition 2.1].
Now we modify [5, Propositions 2.2, 2.3 and Lemma 4.2] to remove
the assumption of (0, 1)-structure systems. We begin with the following
lemma.
Lemma 3.1. Assume that RA is an A-full matrix algebra (1.1) and let
M , M ′ be right RA-modules, with dimM = dim M ′ = (1, . . . , 1) and with
H. Fujita, Y. Sakai, D. Simson 29
the representation matrices S = (sij) and S′ = (s′ij), respectively. There
exists an isomorphism M ∼= M ′ of right RA-modules if and only if there
exist t1, . . . , tn ∈ K such that
ti 6= 0 and sijtj = tis
′
ij , for all i, j ∈ {1, . . . , n}.
Proof. Let {vi | 1 ≤ i ≤ n}, {v′i | 1 ≤ i ≤ n} be associated K-bases of
M, M ′ with representation matrices S = (sij) and S′ = (s′ij), respectively.
First suppose that there is an isomorphism f : M → M ′. Since
v′j = v′jujj , for all j ∈ {1, . . . , n}, then there exists 0 6= ti ∈ K such that
f(vi) = f(vi)uii = tiv
′
i, for each i ∈ {1, . . . , n}. The equality f(viuij) =
f(vi)uij yields sijtj = tis
′
ij , for all i, j ∈ {1, . . . , n}.
Conversely, suppose that there exist t1, . . . , tn ∈ K satisfying the
above conditions. Since ti 6= 0, for all i ∈ {1, . . . , n}, we can define a K-
linear isomorphism f : M → M ′ by f(vi) := tiv
′
i, for all i ∈ {1, . . . , n}.
The latter condition implies that f is an RA-module homomorphism, so
that f : M →M ′ is an isomorphism.
Indecomposable projective RA-modules are characterized by their rep-
resentation matrices as follows, see [5, Proposition 2.2].
Lemma 3.2. Assume that RA is an A-full matrix algebra (1.1).
(i) For each indecomposable projective right RA-module uiiRA, we
have
• dimuiiRA = (1, . . . , 1) and
• the representation matrix of the module uiiRA, with respect to the
K-basis {uij | 1 ≤ j ≤ n}, is the n× n matrix (a
(k)
ij )k,j, where the (k, j)-
entry equals a
(k)
ij .
(ii) Let M be a right RA-module with dimM = (1, . . . , 1), and let
S = (sij) be a representation matrix of M with respect to a K-basis
{vi | 1 ≤ i ≤ n}. Then M is isomorphic to to the projective RA-module
ullRA if and only if slk 6= 0, for all k ∈ {1, . . . , n}.
Proof. (i) This follows from the definition of the multiplication of RA,
that is, uikukj = a
(k)
ij uij , for all i, j, k ∈ {1, . . . , n}. Note that (A2)
implies dimuiiRA = (1, . . . , 1).
(ii) First suppose that M is isomorphic to ullRA. Then it follows
from Lemma 3.1 that there exist t1, . . . , tn ∈ K such that ti 6= 0 and
sijtj = tia
(i)
lj , for all i, j ∈ {1, . . . , n}. Hence sljtj = tia
(l)
lj = ti 6= 0, so
that slj 6= 0, for all j ∈ {1, . . . , n}.
30 Frobenius full matrix algebras with structure systems
Conversely, suppose that slj 6= 0, for all j ∈ {1, . . . , n}. Since a
(i)
lj slj =
slisij , for all i, j ∈ {1, . . . , n}, then there is an RA-module isomorphism
f : ullRA →M, ulj 7→ sljvj (1 ≤ j ≤ n).
We denote the standard duality functor HomK(− , K) : mod RA →
modRop
A
by (− )∗. As a dual of Lemma 3.2, we obtain the following, see
[5, Proposition 2.3].
Lemma 3.3. Assume that RA is an A-full matrix algebra (1.1).
(i) For each indecomposable injective right RA-module (RAujj)
∗, we
have
• dim (RAujj)
∗ = (1, . . . , 1) and
• the representation matrix of the module (RAujj)
∗, with respect to
the dual K-basis {u∗
ij | 1 ≤ i ≤ n}, is the n × n matrix (a
(k)
ij )i,k, where
the (i, k)-entry equals a
(k)
ij .
(ii) Let M be a right RA-module with dim M = (1, . . . , 1), and let
S = (sij) be a representation matrix of M with respect to a K-basis
{vi | 1 ≤ i ≤ n}. Then M is isomorphic to the injective RA-module
(RAull)
∗ if and only if skl 6= 0, for all k ∈ {1, . . . , n}.
Proposition 3.4. Let RA be an A-full n × n matrix algebra, where
A = [A1, . . . , An] is the structure system and Ak = (a
(k)
ij ) (1 ≤ k ≤ n).
The following two conditions are equivalent.
(i) RA is a Frobenius algebra with Nakayama permutation σ.
(ii) There exists a permutation σ of the set {1, . . . , n} such that σ(i) 6=
i, for all i ∈ {1, . . . , n}, and that a
(k)
iσ(i) 6= 0, for all i, k ∈ {1, . . . , n}.
Proof. (i)⇒(ii) It follows from (i) that uiiRA
∼= (RAuσ(i)σ(i))
∗, for all
i ∈ {1, . . . , n}. Since uiiRA has a representation matrix (a
(k)
ij )k,j with
respect to a K-basis {ui1, . . . , uin} then Lemma 3.3 yields a
(k)
iσ(i) 6= 0, for
all i, k ∈ {1, . . . , n}. Since dimuiiRA = (1, . . . , 1) then soc(uiiRA) 6∼=
uiiRA/uiiJ(RA), so that σ(i) 6= i, for all i ∈ {1, . . . , n}.
(ii)⇒(i) Lemmas 3.2 and 3.3 yield the isomorphism uiiRA
∼=
(RAuσ(i)σ(i))
∗ of right RA-modules, for all 1 ≤ i ≤ n. Hence (i) follows.
4. Infinite families of A-full matrix algebras
In this section, for n = 4, 5 and n = 6, we construct several interesting
infinite families of A-full matrix algebras RA that are of infinite represen-
tation type. We also determine their representation type (tame or wild),
by applying the well-known representation theory diagrammatic criteria,
H. Fujita, Y. Sakai, D. Simson 31
see [1], [11] and [12]. We end the section by presenting an idea of a
construction of a large class of Frobenius A-full matrix algebras RA such
that dimK RA = n2, n ≥ 4, soc RA = J(RA)n−2 and J(RA)n−1 = 0. A
characterization of all Frobenius algebras RA with the above properties
remains an open problem.
Example 4.1. Assume that n = 4 and K is a field. Consider the one-
parameter family of Aµ-full matrix algebras Cµ = RAµ
, where µ ∈ K∗ =
K \ {0} and Aµ is the following structure system
Aµ =
1111 0100 0110 0101
10µ0 1111 0010 0011
1001 0101 1111 0001
1000 1100 1010 1111
.
A simple calculation shows that, given µ ∈ K∗, the matrix satisfies the
conditions (A1)–(A3). We show that the algebra Cµ is isomorphic to the
bound quiver K-algebra KQ/Ωµ (see [1]), where Q is the quiver
Q : ◦
1
◦
3
◦
2
◦
4
β13
β31
β32
β24
β42
β14
β21 β43
and Ωµ is the two-sided ideal of the path K-algebra KQ of Q generated
by the following relations:
• β21β13 − µ · β24β43,
• β13β32 − β14β42,
• β32β24 − β31β14,
• β43β31 − β42β21,
• β13β31, β31β13, β24β42, β42β24,
• β21β14, β43β32, β32β21, β14β43.
It is easy to check that the correspondences εj 7→ ujj and βij 7→ uij define
a K-algebra homomorphism h : KQ/Ωµ → Cµ, where εj is the primitive
idempotent of the path algebra KQ defined by the stationary path at the
vertex j, for every j ∈ Q0. Note that dimK KQ/Ωµ = 16 and the cosets
of the idempotents ε1, ε2, ε3, ε4, the eigth arrows βij ∈ Q1, together with
the four cosets β21β13, β13β32, β32β24, and β43β31 form a K-basis of the
quotient K-algebra KQ/Ωµ.
Since e23 = h(β21β13), e12 = h(β13β32), e34 = h(β32β24), and e41 =
h(β43β31) then the map h is surjective. Finally, since dimK KQ/Ωµ =
dimK Cµ = 16, the surjection is an isomorphism of K-algebras.
It follows from the shape of Q and Ωµ that, for each µ ∈ K∗,
KQ/Ωµ
∼= Cµ is a special biserial algebra [13], and therefore it is
32 Frobenius full matrix algebras with structure systems
representation-tame, see [3, 5.2]. Since there is a cyclic walk
1
β13 // 3 4
β43oo
β42 // 2 3
β32oo
β31 // 1 2
β21oo
β24 // 4 1
β14oo
of the quiver Q then, according to the finite representation type criterion
in [13] (see see also [10, Proposition 3.7]), the algebra Cµ is of infinite
representation type. Note also that, for each µ ∈ K∗, Cµ is self-injective,
J(Cµ)3 = 0 and
J(Cµ)2 = soc(Cµ) = Kβ21β13 ⊕Kβ13β32 ⊕Kβ32β24 ⊕Kβ43β31,
see also [7, Section 5]. Consequently, the quotient algebras
Cµ = Cµ/soc Cµ and Cγ = Cγ/soc Cγ
are isomorphic, for each pair µ, γ ∈ K∗. In particular, it follows that the
numbers of the indecomposable Cµ-modules and Cγ-modules are equal
and the stable Auslander-Reiten quivers of Cµ and of Cγ are isomorphic.
Example 4.2. Assume that n = 6. Consider the one-parameter family
of Aµ-full matrix algebras Hµ = RAµ
, where µ ∈ K and
Aµ =
111111
100000
100111
101011
100000
100010
010000
111111
010111
011011
010000
010010
011000
001000
111111
001000
µ11000
111010
010100
000100
000100
111111
110100
110110
011110
001110
000110
001010
111111
000010
011101
001101
000101
001001
000001
111111
.
First we observe that:
(a) if K is infinite, then the family {Hµ}µ∈K\{0,1} is infinite, because
Hµ
∼= Hγ if and only µ = γ, for µ, γ ∈ K \ {0, 1} (apply Corollary 2.2),
(b) for each µ ∈ K \ {0, 1}, the algebra Hµ is not self-injective (the
right ideals u22Hµ and u55Hµ are not injective, by Lemma 3.3, and
(c) for each µ ∈ K \ {0, 1}, the Gabriel quiver Q(Hµ) of the algebra
Hµ is the following quiver Q (apply [5, Proposition 1.2]).
Q : ◦3 ◦
2
◦
1
◦
5
◦
6
◦ 4
β32 β21 β15 β56 β64
β31
β26
β25 β16 β54
β53 β41
β63 β42
H. Fujita, Y. Sakai, D. Simson 33
Now we show that, for each µ ∈ K∗, the algebra Hµ is of wild represen-
tation type, see [9, Section 14.2] and [12, Chapter XIX]. To see this, we
note that Hµ/J(Hµ)2 ∼= Hν/J(Hν)
2, for all µ, ν ∈ K∗, and the algebra
B := Hµ/J(Hµ)2 has J(B)2 = 0. It follows that Q(B) = Q(Hqµ
). Since
the separated quiver Qs(B) of B (see [2, Section X.2]) contains a wild
subquiver of the form
5
����
��
��
��
�� ��
??
??
??
??
6
������
��
��
��
6′ 3′ 4′
then, by [9, Theorems 14.14 and 14.15] and [12, Chapter XIX], the
algebra B is representation-wild and hence also Hµ is representation-
wild, for each µ ∈ K∗, because there is a fully faithful exact embedding
mod B →֒ mod Hµ.
Example 4.3. Assume that n = 4, K is a field and A is a structure
system such that RA is a Frobenius algebra and the Nakayama permu-
tation of RA is the cyclic permutation σ = (1, 2, 3, 4), see [5, Theorem
3.4] and [7, Theorem 5.5]. The structure system A and the associated
(0, 1)-structure system A have the following forms
A =
1 1 1 1 0 1 0 0 0 µ6 1 0 0 µ7 0 1
1 0 µ1 0 1 1 1 1 0 0 1 0 0 0 µ8 1
1 0 0 µ2 0 1 0 µ4 1 1 1 1 0 0 0 1
1 0 0 0 µ3 1 0 0 µ5 0 1 0 1 1 1 1
.
A =
1 1 1 1 0 1 0 0 0 1 1 0 0 1 0 1
1 0 1 0 1 1 1 1 0 0 1 0 0 0 1 1
1 0 0 1 0 1 0 1 1 1 1 1 0 0 0 1
1 0 0 0 1 1 0 0 1 0 1 0 1 1 1 1
,
where µ1, . . . , µ8 are arbitrary scalars in K \{0}. By Proposition 3.4 (see
also [5, Theorem 3.4] and [7, Theorem 5.5]), each of the algebras RA in
the defined eight parameter family is Frobenius and soc RA = J(RA)2.
One shows that there is a K-algebra isomorphism R
A
∼= KQ/Ωσ, where
Q is the quiver
Q : ◦1 ◦
2
◦
3
◦ 4
β21 β32 β43
β14
β13 β24
β31 β42
and Ωσ is the two-sided ideal of the path K-algebra KQ of Q generated
by the following elements:
34 Frobenius full matrix algebras with structure systems
1◦ β21β13−β24β43, β13β32−β14β42, β32β24−β31β14, β43β31−β42β21,
2◦ β13β31, β31β13, β24β42, β42β24, β21β14, β14β43, β32β21, and β43β32.
It follows that
• the zero relation α1α2α3 belongs to Ωσ, for each path • α1−→• α2−→•
α3−→• in Q,
• there is an algebra isomorphism RA
∼= KQ/Ωσ, and
• J(RA)3 = 0 and J(RA)2 = soc(RA).
Now it is easy to see that RA is a special biserial algebra, and therefore
it is representation-tame, see [3, 5.2]. Since there is a cyclic walk
1
��
>>
>>
>>
> 2
����
��
��
�
��
>>
>>
>>
> 3
����
��
��
�
��
>>
>>
>>
> 4
����
��
��
�
��
>>
>>
>>
> 1
����
��
��
�
4 1 2 3
in Q then, according to [13], the Frobenius algebra RA is of infinite rep-
resentation type, see also [10, Proposition 3.7].
We end this section by presenting an idea of a construction, for n = 5,
of tame Frobenius A-full matrix algebras RA of infinite representation
type such that J(RA)4 = 0 and J(RA)3 = soc RA.
Example 4.4. Assume that n = 5 and K is a field. We construct a set of
structure systems q = [q(1), q(2), q(3), q(4), q(5)] such that Rq is a Frobenius
algebra, J(Rq)
4 = 0, J(Rq)
3 = soc (Rq), and σ =
(
1 2 3 4 5
4 5 1 2 3
)
is the
Nakayama permutation of Rq.
Suppose that q = [q(1), q(2), q(3), q(4), q(5)] = [q
(k)
ij ]i,j,k is such a struc-
ture system and let
Rq =
5
⊕
i,j=1
Keij
be the corresponding q-full matrix K-algebra with the basis
{eij | 1 ≤ i, j ≤ 5}. We recall that the elements e1 = e11, . . . , e5 = e55
form a complete set of pairwise orthogonal primitive idempotents of the
algebra Rq and 1 = e1 + e2 + e3 + e4 + e5 is the identity of Rq. We denote
by ·q the multiplication in Rq.
One shows that soc(ejRq) = Kej σ(j) (see [7, Theorem 5.3]) and there-
fore ej σ(j) ·q J(Rq) = 0 and J(Rq) ·q ej σ(j) = 0, for j = 1, . . . , 5. Hence
we get the equalities q
(σ(j))
jr = 0, for all r 6= σ(j), and q
(j)
s σ(j) = 0, for all
s 6= j, that is,
H. Fujita, Y. Sakai, D. Simson 35
• q
(1)
32 = q
(1)
34 = q
(1)
35 = 0 and q
(1)
34 = q
(1)
24 = q
(1)
54 = 0,
• q
(2)
41 = q
(2)
43 = q
(2)
45 = 0 and q
(2)
45 = q
(2)
35 = q
(2)
15 = 0,
• q
(3)
51 = q
(3)
52 = q
(3)
54 = 0 and q
(3)
51 = q
(3)
41 = q
(3)
21 = 0,
• q
(4)
12 = q
(4)
13 = q
(4)
15 = 0 and q
(4)
12 = q
(4)
52 = q
(4)
32 = 0,
• q
(5)
23 = q
(5)
24 = q
(5)
21 = 0 and q
(5)
23 = q
(5)
43 = q
(5)
13 = 0.
Consequently, the block matrix q has the form
q =
1 1 1 1 1 0 1 ∗ ∗ 0 0 ∗ 1 ∗ ∗ 0 0 0 1 0 0 ∗ 0 ∗ 1
1 0 ∗ 0 ∗ 1 1 1 1 1 0 0 1 ∗ ∗ ∗ 0 ∗ 1 ∗ 0 0 0 0 1
1 0 0 0 0 ∗ 1 0 ∗ 0 1 1 1 1 1 ∗ 0 0 1 ∗ ∗ ∗ 0 ∗ 1
1 ∗ ∗ 0 ∗ 0 1 0 0 0 0 ∗ 1 0 ∗ 1 1 1 1 1 ∗ ∗ 0 0 1
1 ∗ ∗ 0 0 ∗ 1 ∗ ∗ 0 0 0 1 0 0 ∗ 0 ∗ 1 0 1 1 1 1 1
.
Since we assume that soc(ejRq) = Kej σ(j) ⊆ ejJ(Rq)
3, for j =
1, . . . , 5, then Kej σ(j) = K(ej j1 ·q ej1 j2 ·q ej2 σ(j)), where j1 6= j2 and
j1, j2 /∈ {j, σ(j)}.
Assume, for simplicity, that there exist non-zero scalars
λ14, λ25, λ31, λ42, λ53 ∈ K such that
λ14e14 = e12 ·q e23 ·q e34,
λ25e25 = e23 ·q e34 ·q e45,
λ31e31 = e34 ·q e45 ·q e51,
λ42e42 = e45 ·q e51 ·q e12,
λ53e53 = e51 ·q e12 ·q e23.
Hence we conclude that
q
(3)
12 = q
(5)
12 = 0, q
(2)
34 = q
(5)
34 = 0, q
(1)
23 = q
(4)
23 = 0, q
(1)
45 = q
(3)
45 = 0,
q
(2)
51 = q
(4)
51 = 0.
Indeed, if we assume to the contrary that q
(3)
12 6= 0 then e13 ·q e32 =
q
(3)
12 e12 and then the non-zero element λ14q
(3)
12 e14 = e13 ·q e32 ·q e23 ·q
e34 belongs to J(Rq)
4 = 0, and we get a contradiction. The remaining
equalities follow in a similar way.
Moreover, since the elements λ14, λ25, λ31, λ42, λ53 ∈ K are non-zero
then, by the associativity of ·q, the equalities above yields
q
(2)
13 q
(3)
14 = q
(2)
14 q
(3)
24 6= 0, q
(3)
24 q
(4)
25 = q
(3)
25 q
(4)
35 6= 0, q
(4)
35 q
(5)
31 = q
(4)
31 q
(5)
41 6= 0,
q
(5)
41 q
(1)
42 = q
(5)
42 q
(1)
52 6= 0, q
(1)
52 q
(2)
53 = q
(1)
53 q
(2)
13 6= 0.
Equivalently, we get the equalities
q
(3)
24 =
q
(2)
13 q
(3)
14
q
(2)
14
,
q
(4)
35 =
q
(3)
24 q
(4)
25
q
(3)
25
=
q
(2)
13 q
(3)
14 q
(4)
25
q
(2)
14 q
(3)
25
,
q
(5)
41 =
q
(4)
35 q
(5)
31
q
(4)
31
=
q
(2)
13 q
(3)
14 q
(4)
25 q
(5)
31
q
(2)
14 q
(3)
25 q
(4)
31
,
q
(1)
52 =
q
(5)
41 q
(1)
42
q
(5)
42
=
q
(2)
13 q
(3)
14 q
(4)
25 q
(5)
31 q
(1)
42
q
(2)
14 q
(3)
25 q
(4)
31 q
(5)
42
,
36 Frobenius full matrix algebras with structure systems
q
(2)
13 =
q
(1)
52 q
(2)
53
q
(1)
53
=
q
(2)
13 q
(3)
14 q
(4)
25 q
(5)
31 q
(1)
42 q
(2)
53
q
(2)
14 q
(3)
25 q
(4)
31 q
(5)
42 q
(1)
53
.
It follows that if q
(2)
13 ∈ K∗ is arbitrary, then the remaining non-zero
scalars q
(s)
ij that appear in the equalities above satisfy the condition
(∗) q
(3)
14 q
(4)
25 q
(5)
31 q
(1)
42 q
(2)
53 = q
(2)
14 q
(3)
25 q
(4)
31 q
(5)
42 q
(1)
53
Now we show that q
(1)
43 = 0, q
(2)
54 = 0, q
(3)
15 = 0, q
(4)
21 = 0 and q
(5)
32 = 0.
To see this, assume to the contrary that q
(1)
43 6= 0. Then 0 6= q
(1)
43 e43 =
e41 ·q e13. It follows that the non-zero element e45 ·q e51 ·q e12 ·q e23 =
q
(5)
41 q
(2)
13 e41 ·q e13 = q
(5)
41 q
(2)
13 q
(1)
43 e43 belongs to J(Rq)
4 = 0, and we get a
contradiction. The equalities q
(2)
54 = 0, q
(3)
15 = 0, q
(4)
21 = 0, q
(5)
32 = 0 follow
in a similar way. Consequently, the block matrix q has the form
q =
1 1 1 1 1 0 1 q
(2)
13 q
(2)
14 0 0 0 1 q
(3)
14 0 0 0 0 1 0 0 0 0 ∗ 1
1 0 0 0 ∗ 1 1 1 1 1 0 0 1 q
(3)
24 q
(3)
25 0 0 0 1 q
(4)
25 0 0 0 0 1
1 0 0 0 0 1 0 0 0 1 1 1 1 1 q
(4)
31 0 0 1 q
(4)
35 q
(5)
31 0 0 0 1
1 q
(1)
42 0 0 0 0 1 0 0 0 0 ∗ 1 0 0 1 1 1 1 1 q
(5)
41 q
(5)
42 0 0 1
1 q
(1)
52 q
(1)
53 0 0 0 1 q
(2)
53 0 0 0 0 1 0 0 0 0 ∗ 1 0 1 1 1 1 1
.
Now we claim that each of the scalars q
(1)
25 , q
(2)
31 , q
(3)
42 , q
(4)
53 , q
(5)
14 is non-
zero. Assume, to the contrary, that some of them is zero, say q
(5)
14 = 0. It
follows from the shape of q that q
(5)
1r = 0, for all r 6= 5, and consequently
e15 ·q J(Rq) = 0. It follows that S′ = e15K ⊆ e1Rq is a simple submodule
of e1Rq; contrary to the assumption that soc(e1Rq) = e14K. This finishes
the proof of our claim. Consequently, the block matrix q has the form
q =
1 1 1 1 1 0 1 q
(2)
13 q
(2)
14 0 0 0 1 q
(3)
14 0 0 0 0 1 0 0 0 0 q
(5)
14 1
1 0 0 0 q
(1)
25 1 1 1 1 1 0 0 1 q
(3)
24 q
(3)
25 0 0 0 1 q
(4)
25 0 0 0 0 1
1 0 0 0 0 q
(2)
31 1 0 0 0 1 1 1 1 1 q
(4)
31 0 0 1 q
(4)
35 q
(5)
31 0 0 0 1
1 q
(1)
42 0 0 0 0 1 0 0 0 0 q
(3)
42 1 0 0 1 1 1 1 1 q
(5)
41 q
(5)
42 0 0 1
1 q
(1)
52 q
(1)
53 0 0 0 1 q
(2)
53 0 0 0 0 1 0 0 0 0 q
(4)
53 1 0 1 1 1 1 1
.
where q
(1)
25 , q
(2)
31 , q
(3)
42 , q
(4)
53 , q
(5)
14 and q
(2)
13 are arbitrary non-zero scalars in K,
the coefficients
q
(3)
14 , q
(4)
25 , q
(5)
31 , q
(1)
42 , q
(2)
53 , q
(2)
14 , q
(3)
25 , q
(4)
31 , q
(5)
42 , q
(1)
53
satisfy the equation (∗) and the coefficients q
(1)
52 , q
(2)
13 , q
(3)
24 , q
(4)
35 , q
(5)
41 depend
of the remaining ones by the formulas preceding the equation (∗).
Conversely, if q is a block matrix of the above form, where q
(1)
25 , q
(2)
31 ,
q
(3)
42 , q
(4)
53 , q
(5)
14 and q
(2)
13 are arbitrary non-zero scalars in K, and the re-
maining ones satisfy the above conditions then q is a structure system and
Rq is a Frobenius algebra such that J(Rq)
4 = 0 and soc(Rq) = J(Rq)
3.
The associated (0, 1)-matrix q structure system has the following form
q =
1 1 1 1 1 0 1 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1
1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1
1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 0 1
1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1 1 1 1 0 0 1
1 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 1
.
H. Fujita, Y. Sakai, D. Simson 37
It follows that the Gabriel quiver Q(Rq) of Rq has the form
β15
Q(Rq) : 1
β12−→←−
β21
2
β23−→←−
β32
3
β34−→←−
β43
4
β45−→←−
β54
5.
β51
To view the algebra Rq as a path algebra KQ/Ωq of a bound quiver, we
note that
q
(5)
14 e12 ·q e23 ·q e34 = q
(2)
13 q
(3)
14 e15 ·q e54,
q
(1)
25 e23 ·q e34 ·q e45 = q
(3)
24 q
(4)
25 e21 ·q e15,
q
(2)
31 e34 ·q e45 ·q e51 = q
(4)
35 q
(5)
31 e32 ·q e21,
q
(3)
42 e45 ·q e51 ·q e12 = q
(5)
41 q
(1)
42 e43 ·q e32,
q
(4)
53 e51 ·q e12 ·q e23 = q
(1)
52 q
(2)
53 e54 ·q e43.
To see the first equality, we note that e15 ·q e54 = q
(5)
14 e14 and e12 ·q e23 ·q
e34 = q
(2)
13 q
(3)
14 e14. Hence the first equality follows, and the remaining ones
follow in a similar way.
Now we prove that there is a K-algebra isomorphism Rq
∼= KQ/Ωq,
where Q = Q(Rq) and Ωq is the two-sided ideal of the path K-algebra
KQ of Q generated by the following relations:
• βj+1 jβj j+1 and βj j+1βj+1 j , for j = 1, . . . , 5, where j+1 is reduced
modulo 5.
• β1β2β3β4, if there is a path • β1−→ • β2−→ • β3−→ • β4−→• in Q.
• β21β15β54, β32β21β15, β43β32β21, β54β43β32, β15β54β43;
• q
(5)
14 β12β23β34 − q
(2)
13 q
(3)
14 β15β54,
• q
(1)
25 β23β34β45 − q
(3)
24 q
(4)
25 β21β15,
• q
(2)
31 β34β45β51 − q
(4)
35 q
(5)
31 β32β21,
• q
(3)
42 β45β51β12 − q
(5)
41 q
(1)
42 β43β32,
• q
(4)
53 β51β12β23 − q
(1)
52 q
(2)
53 β54β43.
It is easy to check that the correspondences εj 7→ ej and βij 7→ eij define
a K-algebra homomorphism h : KQ/Ωq → Rq, where εj is the primitive
idempotent of the path algebra KQ defined by the stationary path at
the vertex j, for every j ∈ Q0. Note that the map h is well defined and
surjective. Finally, since dimK KQ/Ωq = dimK Rq = 25, the surjection
h is an isomorphism of K-algebras.
38 Frobenius full matrix algebras with structure systems
Now it is easy to see that Rq/soc(Rq) ∼= Rq/soc(Rq) and the algebra
KQ/Ωq
∼= Rq is special biserial; hence Rq is representation-tame, see [3,
5.2]. Since there is a cyclic walk
1
β12 // 2
β23 // 3 4
β43oo
β45 // 5 1
β51oo
of the quiver Q and, according to the finite representation type criterion
in [13], the algebra Rq is of infinite representation type, see also [10,
Proposition 3.7].
Problem 4.5. Give a characterisation of the Frobenius A-full matrix
algebras RA such that dimK RA = n2, n ≥ 3, soc RA = J(RA)n−2 and
J(RA)n−1 = 0.
Remark 4.6. In connection with Problem 4.5, we recall that if RA is an
A-full matrix algebra and R
A
is the (0, 1)-limit of RA then
• J(RA)s = J(R
A
)s, for each s ≥ 1 (by [7, Proposition 3.2]),
• soc RA = soc R
A
(by [7, Proposition 5.1]), and
• RA is a Frobenius algebra if and only if the (0, 1)-limit R
A
of RA
is a Frobenius algebra (by [7, Theorem 5.3]).
It follows that a solution of the Problem 4.5 for (0, 1)-structure sys-
tems should help to find a solution for arbitrary structure systems A.
We recall from [5] that in case n = 5, a list of (0, 1)-structure systems
A such that RA is a Frobenius algebra is given in Examples 4.7(4) and
4.7(5) of [5]. It is shown there that, up to isomorphisms of the A-full ma-
trix algebras, there are precisely four Frobenius (0, 1)-structure systems
A. Note that one of them has the property soc RA = J(RA)3, compare
with the (0, 1)-limit algebra Rq in Example 4.4.
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reine angew. Math., 345(1985), 480–500.
Contact information
H. Fujita Institute of Mathematics, University of
Tsukuba, Ibaraki 305-8571
E-Mail: fujita@math.tsukuba.ac.jap
Y. Sakai Institute of Mathematics, University of
Tsukuba, Ibaraki 305-8571
E-Mail: ysksakai@math.tsukuba.ac.jap
D. Simson Faculty of Mathematics and Computer Sci-
ences, Nicolaus Copernicus University, 87-
100 Toruń, Poland
E-Mail: simson@mat.uni.torun.pl
Received by the editors: 29.10.2006
and in final form 28.05.2007.
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