radical cube zero selfinjective algebras of · radical cube zero selfinjective algebras of finite...

RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS OF

FINITE COMPLEXITY

KARIN ERDMANN AND ØYVIND SOLBERG

Abstract. One of our main results is a classification all the possible quivers of

selfinjective radical cube zero finite dimensional algebras over an algebraicallyclosed field having finite complexity. In the paper [5] we classified all weaklysymmetric algebras with support varieties via Hochschild cohomology satis-

fying Dade’s Lemma. For a finite dimensional algebra to have such a theoryof support varieties implies that the algebra has finite complexity. Hence thispaper is a partial extension of [5].

Introduction

This paper is a companion of [5], where all radical cube zero weakly symmetricalgebras with support varieties via the Hochschild cohomology satisfying Dade’sLemma were classified. In this paper we go half way with all selfinjective algebraswith radical cube zero, in that we classify which of these have finite complexity.This is half way for the following reasons. To get a theory of support using theHochschild cohomology ring satisfying Dade’s Lemma, for any known proof of thisthe Ext-algebra of all the simple modules must be a finitely generated module overthe Hochschild cohomology ring, which in turn needs to be Noetherian. Denotethis property by (Fg). By [4] a finite dimensional algebra satisfying (Fg) musthave finite complexity. In addition the trichotomy into finite, tame and wild repre-sentation type is characterized in two different ways as (i) λmax < 2, λmax = 2 andλmax > 2 and (ii) complexity of the algebra is 1, 2 or ∞, respectively, where λmax

is the eigenvalue of largest absolute value for the adjacency matrix of the algebra(λmax is a positive real number).

A selfinjective algebra Λ with radical cube zero over an algebraically closed fieldis either of finite or infinite representation type. If Λ has finite representation type,then it satisfies (Fg) by [3]. If Λ has infinite representation type, then it is a Koszulalgebra by [7, 8]. Using the results of [5] Λ has (Fg) if and only if the Koszul dualof Λ is a finitely generated module over the graded centre of the Koszul dual andthis is a Noetherian ring. This was the key argument in [5], where the results wereobtained through explicit calculations case by case. This approach is still availablefor selfinjective algebras with radical cube zero, however it seems to us that it isan almost new game to treat this class of algebras. And, as our results show, thisclass is seemingly much more complex than the weakly symmetric algebras with

Date: September 10, 2010.2010 Mathematics Subject Classification. 16P10, 16G20, 16L60, 16E05, 16P90; Secondary:

16S37.Key words and phrases. Selfinjective algebras, finite complexity, Koszul algebras.The authors acknowledge support from EPSRC grant EP/D077656/1 and NFR Storforsk grant

no. 167130.

1

2 ERDMANN AND SOLBERG

radical cube zero. Hence we seek a better method for characterizing which of theselfinjective algebras with radical cube zero and finite complexity satisfy (Fg).

By [11] a finite dimensional Koszul algebra Λ over a field k with degree zero partisomorphic to k, is selfinjective with finite complexity if and only if the Koszul dualis an Artin-Schelter regular Koszul algebra. An extension of this was proved in [8]for finite dimensional Koszul algebras over a field k with degree zero part isomorphicto a finite number of copies of k. It is natural to say that a (non-connected) Koszulk-algebra R = ⊕i≥0Ri is an Artin-Schelter regular algebra of dimension d, if

(i) dimk Ri < ∞ for all i ≥ 0,(ii) R0 ≃ kn for some positive integer n,(iii) gldimR = d,(iv) the Gelfand-Kirillov dimension of R is finite,(v) for all simple graded R-modules S we have

ExtiR(S,R) ≃(0), i 6= d,

S′, i = d and some simple graded R-module S′.

Classifying all selfinjective Koszul algebras of finite complexity d and Loewy lengthm+1 (up to isomorphism) is the same as classifying Artin-Schelter regular Koszulalgebras with Gelfand-Kirillov dimension d and global dimension m (up to isomor-phism) by [10]. Hence, by the results in this paper, we have classified the quiversof all indecomposable (non-connected) Artin-Schelter regular Koszul algebras ofdimension 2.

Now we describe the content of the paper section by section. The first section isdevoted to giving the combinatorial data of a finite dimensional selfinjective algebrawith radical cube zero in terms of the adjacency matrix, the Nakayama permutationand the contracted matrix of the adjacency matrix (See Section 1 for definition).Furthermore, the possible shapes of such combinatorial data are found. In Section2 the trichotomy into finite, tame and wild representation type is characterizedthrough the spectral radius of the adjacency matrix of the algebra. The nextsection is devoted to characterizing this trichotomy in terms of the complexityof the algebra. In Section 4 we carry out the main underlying classification, aswe find all possible contracted matrices of the adjacency matrix of a radical cubezero selfinjective algebra. The last two sections of the paper is devoted to givingthe complete classification of the adjacency matrices and corresponding Nakayamapermutation for all selfinjective algebras with radical cube zero. This extends theresults in [2] and some of the methods are generalizations of those by Benson.

1. The combinatorial data

For a radical cube zero selfinjective algebra over an algebraically closed field,there is a naturally associated adjacency matrix and Nakayama permutation ofthe algebra. This section is devoted to giving an initial description of the possibleadjacency matrices and an elementary property of them. Recall that such an algebrais Morita equivalent to a quotient of a path algebra of a quiver by an admissibleideal.

Let Λ = kQ/I be a radical cube zero selfinjective algebra, where Q is a connectedquiver with n vertices and I is an admissible ideal I in kQ for a field k. Thiswill be the standing assumption on our path algebra Λ = kQ/I throughout thepaper. Denote by S1, . . . , Sn all the non-isomorphic simple Λ-modules, and let

RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 3

E = (eij)n,ni,j=1 be the n × n-matrix given by eij = dimk Ext

1Λ(Sj , Si). We call the

matrix E the adjacency matrix of Λ. If Π: 1, 2, . . . , n → 1, 2, . . . , n denotes theNakayama permutation, then the radical layers of the indecomposable projectivecorresponding to vertex i is given by

Si

iiiiiiiiiisss

s NNNN

VVVVVVVVVVVVV

Se1i1

TTTTTTTTT Se2i2

JJJ· · · S

en−1,i

n−1

qqqSenin

hhhhhhhhhhh

SΠ(i)

Identify Si also with the i-th elementary column vector (0, . . . , 0, 1, 0, . . . , 0)T inRn, and identity Π with a permutation matrix Π such that ΠSi = SΠ(i) for all i.

From the above diagram it is immediate that ESi = (STΠ(i)E)T = ETSΠ(i). Hence

E = ETΠ. Using that Π−1 = ΠT , it follows that EΠ = ΠE.Suppose that the permutation Π is a product of t disjoint cycles. We label the

indices so that Π has block diagonal form, and the j-th diagonal block is the cycleΠj which corresponds to (tj , tj + 1, . . . , tj + dj − 1) for tj the smallest element inthe support of Πj . Then Πj has length dj . We also have a corresponding blockdecomposition of E as

E11 E12 ··· E1t

E21 E22 ··· E2t

......

...Et1 Et2 ··· Ett

.

Given that E = ETΠ we have that E must be of the following form, where thediagonal blocks of E are described in (b) and the off diagonal blocks of E aredescribed in (c) of the next result.

Lemma 1.1. Let E be an n × n-matrix over R, and let Π be a permutation of1, 2, . . . , n which is also viewed as an n × n-permutation matrix. Then the fol-lowing hold.

(a) E = ETΠ if and only if eij = eΠ(j)i for all i, j = 1, 2, . . . , n.

(b) If Π is the cyclic permutation (1, 2, . . . , n), then E = ETΠ if and only if Eis the circulant matrix with the first row of the form

(a1, a2, . . . , am−1, am, am, am−1, . . . , a2, a1),

when n = 2m, or

(a1, a2, . . . , am−1, am, am+1, am, am−1, . . . , a2, a1),

when n = 2m+ 1, where ai is in R for all i. In particular,

E = a1(I +Π) +

[n2 ]∑

i=2

ai(Πi +Πn−i+1)

for n even, and

E = a1(I +Π) +

[n2 ]∑

i=2

ai(Πi +Πn−i+1) + a[n2 ]+1Π

[n2 ]+1

for n odd, with ai in R.


(c) Viewing E according to the block decomposition of Π, then Eij = ETjiΠj for

all i, j = 1, 2, . . . , t. In particular, ΠiEij = EijΠj for all i, j = 1, 2, . . . , tand ers = eΠj(s)r = eΠi(r)Πj(s) for all ers in Eij.

Let di be the length of the cycle Πi for i = 1, 2, . . . , t, and let cl =∑ di

gcd(di,dj)−1

r=0 Πrdj

i (STl ) for some l = 1, 2, . . . , di. Then the matrix Eij is in

the linear span of the matrices Cl, where we view Cl as a sequence of djcolumn vectors

cl,Πi(cl), . . . ,Πdj−1i (cl).

There are gcd(di, dj) free parameters in a generic Eij for all (i, j), andthe minimal row sum and the minimal column sum in the generator Cl are

equal todj

gcd(di,dj)and di

gcd(di,dj), respectively.

Proof. Direct inspection and computations.

The matrix E and Π are combinatorial data of a selfinjective algebra with radicalcube zero. We want to classify the possible combinatorial data for this class ofalgebras. The matrix E is too large an object at the present, so we first want tomake a reduction. This reduction is obtained through the following result.

Lemma 1.2. Let E be a non-negative connected n× n-matrix over R, and Π be apermutation of 1, 2, . . . , n. Assume that E commutes with Π. Consider E as alinear transformation E : Cn → Cn. Then E has a Π-invariant eigenvector with allentries positive and eigenvalue λ = |λ| maximal.

Proof. Let λ be the eigenvalue for E with |λ| maximal. Then by Perron-FrobeniusTheorem the corresponding eigenspace is one-dimensional and can be generated bya vector v with all entries positive. Since E commutes with Π, we also have thatΠi(v) is an eigenvector with eigenvalue λ for all i. Therefore v′ =

∑n−1i=0 Πi(v) is a

(non-zero) eigenvector with eigenvalue λ for E. Hence v′ = αv for some α, and weinfer that v = Π(v).

Let E be the adjacency matrix of an indecomposable selfinjective algebra Λ =kQ/I with radical cube zero with the corresponding Nakayama permutation Π =Π1Π2 · · ·Πt. Let v = (v1, v2, . . . , vn)

T be the eigenvector of E with eigenvalue λ oflargest absolute value. We have seen that the vector v is Π-invariant, hence vΠl(i) isconstant for all l ≥ 0. In addition, by Lemma 1.1 (c) the row sum of any row in Eij

is the same. Denote the row sum in Eij by fij , and this give rise to a t× t-matrixF = (fij). It is then clear that F has v′ = (v′1, . . . , v

′t)

T as an eigenvector witheigenvalue λ, where v′i is the constant value of vj on the i-th block. An eigenvectorwith eigenvalue λ′ of F gives in a natural way rise to a Π-invariant eigenvector of Ewith eigenvalue λ′. Hence the eigenvalues with maximal absolute value for E andF coincide. We call the matrix F the contracted matrix of E, which we in Section 5and 6 use to split selfinjective algebras with radical cube zero into different classes.

2. Representation type and spectral radius

If Λ is a finite dimensional selfinjective algebra over an algebraically closed fieldof finite representation type, then it follows from [3] that Λ satisfies (Fg). Henceto further limit the classes of algebras we need to analyze in characterizing whena selfinjective algebra Λ = kQ/I with radical cube zero satisfies (Fg), we recall in


this section how the representation type of Λ is determined by the spectral radiusof the adjacency matrix of Λ.

Let Λ = kQ/I be an indecomposable selfinjective algebra with radical cube zeroover a field k. Denote by S1, . . . , Sn all the non-isomorphic simple Λ-modules,and let E be the adjacency matrix of Λ. The representation type of Λ and Λ/SocΛis the same, since they only differ by |Q0| indecomposable modules, namely theindecomposable projective Λ-modules. The algebra Λ/SocΛ is a radical squarezero algebra, and therefore it is stably equivalent to a hereditary algebra. Thishereditary algebra is given as the path algebra of the separated quiver of Λ/SocΛ.

We construct the separated quiver Q of Λ/SocΛ as follows, where we suppose the

vertices in Q are labelled 1, 2, . . . , n. The vertices in Q are given by the disjointunion Q0∪Q′

0, where Q′0 = i′ | i ∈ Q0. For each arrow α : i → j in Q, there is an

arrow α : i → j′ in Q. Then the arrows in Q are given by the adjacency 2n × 2n-

matrix E = ( 0 0E 0 ). The representation type of Λ/SocΛ is determined by the type

of the quadratic form given by the matrix I − 12 (E + ET ), which corresponds to

the Tits form of the associated hereditary algebra. We recall next how the type isdetermined by the spectral radius of E (see [1, Chap. VIII, Theorem 6.9]).

Proposition 2.1. Let Λ = kQ/I be an indecomposable selfinjective algebra withradical cube zero over a field k, and let E be the adjacency matrix of the quiver Q.Let λ be the eigenvalue of largest absolute value for E. Then we have the following.

(a) λ > 2 if and only if Λ is of wild representation type.(b) λ = 2 if and only if Λ is of tame representation type.(c) λ < 2 if and only if Λ is of finite representation type.

Proof. Let Λ and E be as above. Then Λ is of finite type if and only if the separatedquiver of Q is a disjoint union of Dynkin quivers, and Λ is of tame type if and onlyif the separated quiver of Q is a disjoint union of Dynkin and extended Dynkindiagrams with at least one extended Dynkin diagram occurring.

The quadratic form given by the matrix A = I− 12 (E+ET ) is positive definite or

positive semidefinite if and only if Λ is of finite type or of tame type, respectively.Furthermore, A determines a positive definite or positive semidefinite quadraticform if and only if all eigenvalues of A are positive or all eigenvalues of A arenon-negative.

We have Av = αv for a scalar α if and only if (E + ET )v = 2(1 − α)v, hence

the largest eigenvalue µ = λmax(E + ET ) of E + ET corresponds to the smallesteigenvalue αmin of A. In particular αmin ≥ 0 if and only if µ ≤ 2.

Let λ = λmax(E), the largest eigenvalue. We are done if we show that λ = µ.(1) First let Ev = λv where v is an eigenvector with all entries > 0. Then by

Lemma 1.2 we know that Πv = v and it follows that(

0 ET

E 0

)( vv ) =

(EΠT vEv

)= (Ev

Ev ) = λ ( vv ) .

Hence λ is an eigenvalue of E + ET and therefore λ ≤ µ.(2) Next, we show µ ≤ λ.

Let ( vw ) be an eigenvector for E + ET with eigenvalue µ. Then we have that

(0 ET

E 0

)( vw ) =

(0 EΠT

E 0

)( vw ) =

(EΠT (w)E(v)

)= µ ( v

w ) .

It follows from this that E(w) = µΠ(v) and E(v) = µw. Furthermore, EE(w) =µEΠ(v) = µΠE(v) = µ2Π(w) and similarly E2(v) = µ2Π(v). Let E(µ) be the


eigenspace of E + ET for the eigenvalue µ. Let V be the vector space spanned by

Πi(v),Πi(w) | ( vw ) ∈ E(µ)t−1

i=0,

where t is the order of the Nakayama permutation Π. Then E2|V : V → V , andin addition (E2|V )t = µ2tIV , where IV is the identity on V . Hence the minimalpolynomial of E2|V divides p(x) = xt − µ2t. The roots of p(x) are uiµ2 for i =0, 1, . . . , t − 1, where u is a primitive t-th root of unity. Therefore E2|V has an

eigenvector with eigenvalue uixµ2 for some ix in [0, . . . , t−1]. It follows that√uixµ

or −√uixµ is an eigenvalue for E with absolute value |µ|. Hence |µ| ≤ λ. This

shows that λmax(E + ET ) = λmax(E). The claims follow from this.

Let Λ = kQ/I be an indecomposable selfinjective algebra with radical cube zeroover an algebraically closed field k. As we pointed out earlier, when Λ has finiterepresentation type, then Λ satisfies (Fg). Hence the only case left to analyze isthe case when the spectral radius of the adjacency matrix of Λ is greater or equalto 2. In this case Λ is also Koszul, as we have the following result.

Theorem 2.2 ([7, 8]). Let Λ = kQ/I be an indecomposable selfinjective algebrawith r

3 = (0) and r2 6= (0). Then Λ is Koszul if and only if Λ is of infinite

representation type.

Therefore, in the rest of the paper we can assume that Λ = kQ/I is an indecom-posable selfinjective Koszul algebra with radical cube zero.

3. Complexity

Let Λ be a finite dimensional algebra satisfying (Fg). By [4, Theorem 2.5]the complexity of any finitely generated module over Λ is bounded above by theKrull dimension of the Hochschild cohomology ring, hence finite. We define thecomplexity of a finite dimensional algebra Λ as the supremum of the complexitiesof the simple Λ-modules. Hence for a selfinjective algebra Λ with radical cube zero,a necessary condition to have (Fg) is that Λ has finite complexity. This section isdevoted to characterizing when an indecomposable selfinjective algebra Λ = kQ/Iwith radical cube zero has finite complexity.

Let Λ = kQ/I be an indecomposable selfinjective algebra with radical cube zero,and let E and Π be the adjacency matrix and the Nakayama permutation of Λ,respectively. In analyzing the complexity of Λ we need to compute resolutions ofmodules over Λ. Let M be an indecomposable non-projective Λ-module M withradical layers ((r1, . . . , rn), (s1, . . . , sn))

T , where this means that the top of M isisomorphic to ⊕n

i=1Srii and the radical of M is isomorphic to ⊕n

i=1Ssii . Then the

first syzygy of M is indecomposable and has radical layers (E(ri)T −(si)

T ,Π(ri)T ).

In other words, the radical layers of the first syzygy of M is given by(E −IΠ 0

)((r1, . . . , rn), (s1, . . . , sn))

T .

By Lemma 1.2 the eigenvector vmax of E with a eigenvalue of maximal absolutevalue is Π-invariant. We show next that

(E −IΠ 0

)has a (Π 0

0 Π )-invariant eigenvector,and such a eigenvector with eigenvalue of maximal absolute value is linked to vmax.

Proposition 3.1. Let Λ, E, and Π be as above. Consider the linear transforma-tions E : Cn → Cn and

(E −IΠ 0

): Cn ⊕ Cn → Cn ⊕ Cn. Then


(a) E has a Π-invariant eigenvector with eigenvalue λ, where λ = µ2+1µ

for µ 6=0 if and only if

(E −IΠ 0

)has a (Π 0

0 Π )-invariant eigenvector with eigenvalue

µ, where µ is a root in x2 − λx+ 1 = 0.(b) E has a Π-invariant eigenvector with all entries positive and eigenvalue

λ = |λ| maximal.(c)

(E −IΠ 0

)has a (Π 0

0 Π )-invariant eigenvector with eigenvector µ, where µ is a

root in x2 − λx+ 1 = 0 for some λ.(d) E has an eigenvector with eigenvalue λ, where |λ| is maximal if and only

if(E −IΠ 0

)has a (Π 0

0 Π )-invariant eigenvector with eigenvalue µ which is a

root in x2 − λx+ 1 = 0, where |µ2+1µ

| is maximal.

Proof. (a) Assume that v is a Π-invariant eigenvector with eigenvalue λ for E. Letµ be a root in x2 − λx+ 1 = 0. Then

(E −IΠ 0

)(µv, v)T = (µE(v)− v, µΠ(v))T = (λµv − v, µv)T

= (µ2v, µv)T = µ(µv, v)T .

Hence (µv, v)T is a (Π 00 Π )-invariant eigenvector of

(E −IΠ 0

)with eigenvalue µ, where

µ is a root in x2 − λx+ 1 = 0.Conversely, assume that (v, w)T is a (Π 0

0 Π )-invariant eigenvector with eigenvalue

µ for(E −IΠ 0

). This means

(E −IΠ 0

)(v, w)T = (E(v)− w,Π(v))T = µ(v, w)T ,

which implies that E(v) − w = µv and v = Π(v) = µw. Hence E(v) = µ2+1µ

v and

v is a Π-invariant eigenvector with eigenvalue λ = µ2+1µ

for E. The scalar µ is

non-zero, since(E −IΠ 0

)is invertible.

(b) This is Lemma 1.2.(c) This follows immediately from (a) and (b).(d) This also follows from the above.

Now we characterize when an indecomposable selfinjective algebra Λ with r3 =

(0) and r2 6= (0) has finite complexity in terms of the adjacency matrix E.

Proposition 3.2. Let Λ, E and Π be as above. Then we have

(a) Λ has finite representation type if and only if the complexity of Λ is 1.(b) Λ has tame representation type if and only if the complexity of Λ is 2.(c) Λ has wild representation type if and only if the complexity of Λ is ∞.

Proof. Let λ be the eigenvalue of E with largest absolute value, and let v be thecorresponding eigenvector (see Lemma 1.2).

Assume that Λ has wild representation type, or equivalently that λ > 2. ThenΛ is Koszul [7, 8] and no positive syzygy of a simple Λ-module is simple. Hence the

linear transformation(E −IΠ 0

)mcomputes the radical layers of the m-th syzygy of

any given simple Λ-module. If µ is the largest root of x2−λx+1, then µ > 1 and it isan eigenvalue for the eigenvector (µv, v)T of

(E −IΠ 0

). Suppose all simple Λ-modules

have finite complexity, that is, the total dimension (1, 1, . . . , 1)(E −IΠ 0

)m(Si, 0)

T of


ΩmΛ (Si) has polynomial growth with respect to m for i = 1, 2, . . . , n. We have that

(1, 1, . . . , 1)µm(µv, v)T = (1, 1, . . . , 1)(E −IΠ 0

)m(µv, v)T

= (1, 1, . . . , 1)((

E −IΠ 0

)m(µv, 0)T −

(E −IΠ 0

)m−1(v, 0)T

),

where the last terms have at most polynomial growth with respect to m. Sinceµ > 1, this is a contradiction and consequently Λ has infinite complexity.

Assume that Λ has tame representation type, or equivalently that λ = 2. View Eas a linear transformation E : Qn → Qn. Then the eigenvector v with eigenvalue 2of E can be chosen to be a rational vector, and then also an integral strictly positivevector. Let also µ be as above. In particular µ = 1. Let V = Q(v, v)T , (v, 0)T ⊆Q2n. Then

(E −IΠ 0

)|V = ( 1 1

0 1 ) ,

and, since Λ is of infinite type,(E −IΠ 0

)mcomputes the radical layers of the m-th

syzygy of any given simple Λ-module. The module with radical layers (v, 0)T is adirect sum of all simple modules with multiplicity at least 1. Then

(1, 1, . . . , 1)(E −IΠ 0

)m(v, 0)T = (2m+ 1)(1, 1, . . . , 1)v.

This shows that the complexity of Λ is 2.If Λ is of finite representation type, then all non-projective indecomposable mod-

ules are Ω-periodic, hence Λ has complexity 1.Using the trichotomy into finite, tame and wild representation type (see Propo-

sition 2.1), the claim in the proposition now follows easily.

4. The contracted matrix

Let Λ = kQ/I be an indecomposable selfinjective algebra with radical cube zeroover a field k with corresponding adjacency matrix E and Nakayama permutationΠ = Π1Π2 · · ·Πt written as a product of disjoint cycles. The decomposition ofΠ into a product of disjoint cycles gives a block decomposition of E, and recallfrom Section 1 the construction of the contracted matrix F of E. This section isdevoted to classifying the possible contracted matrices F of Λ when the spectralradius of E is at most 2, or equivalently when Λ has finite complexity and not ofwild representation type.

Let Λ, E, Π and F be as above. Let F be the largest symmetric matrix of the

same size as F such that F − F is non-negative. Since Λ is indecomposable, bothE and F are strongly connected matrices. Then we have the following.

Proposition 4.1. Let Λ = kQ/I be an indecomposable selfinjective algebra withradical cube zero over a field k with corresponding adjacency matrix E, contractedmatrix F and Nakayama permutation Π = Π1Π2 · · ·Πt written as a product ofdisjoint cycles. Assume that the spectral radius λmax of E is at most 2. Then wehave the following.

(a) If F is a symmetric matrix, then F is the adjacency matrix of a Euclidean

digram An (n ≥ 1), Dn (n ≥ 4), E6, E7, E8,

Zn : • • • •


with n+ 1 vertices for n ≥ 0 or

DZn : •SSSSSS

• • • ••

kkkkkk

with n + 1 vertices for n ≥ 2, if λmax = 2, or the adjacency matrix of aDynkin diagram An (n ≥ 1), Dn (n ≥ 4), E6, E7, E8 or the diagram

Zn : • • • •

with n+ 1 vertices for n ≥ 0, if λmax < 2.(b) If F is not symmetric (in particular t > 1), then F is one of the following

matrices if λmax = 2, and F is a smaller strongly connected submatrix of

the following matrices or the matrix

(0 1 0 01 0 a 00 b 0 10 0 1 0

)with ab = 2, if λmax < 2.

(i)

Type At

t F

2 ( 0 14 0 ), (

0 41 0 )

3(

0 a 0b 0 10 1 0

),(

0 1 01 0 a0 b 0

)with ab = 3, or

(0 c 0d 0 e0 f 0

)with cd = ef = 2

4

(0 a 0 0b 0 1 00 1 0 c0 0 d 0

)with ab = 2 = cd

5

(0 ab 0 11 0 1

1 0 cd 0

),

(0 11 0 a

b 0 11 0 1

1 0

),

(0 11 0 11 0 a

b 0 11 0

)with ab = 2 = cd

t ≥ 6

0 a 0 ··· ··· 0 0 0b 0 1 0 ··· ··· 0 0 00 1 0 1 0 ··· ··· 0 0 0.........

......

......

0 0 0 ··· ··· 0 1 0 1 00 0 0 ··· ··· 0 1 0 c0 0 0 ··· ··· 0 d 0

with ab = 2 = cd

for positive integers a, b, c, d, e and f .

(ii)

Type Zt−1

t F

t ≥ 2

1 1 0 ··· ··· 0 0 01 0 1 0 ··· ··· 0 0 00 1 0 1 0 ··· ··· 0 0 0.........

.. ... .

......

0 0 0 ··· ··· 0 1 0 1 00 0 0 ··· ··· 0 1 0 a0 0 0 ··· ··· 0 b 0

with ab = 2

for positive integers a and b.


(iii)

Type Dt

t F

3(

0 0 a0 0 cb d 0

),(

0 0 10 0 e1 f 0

),(

0 0 e0 0 1f 1 0

)with ab = 2 = cd and ef = 3

4

(0 0 a 00 0 1 0b 1 0 10 0 1 0

),

(0 0 1 00 0 a 01 b 0 10 0 1 0

),

(0 0 1 00 0 1 01 1 0 a0 0 b 0

)with ab = 2

t ≥ 5

0 0 1 ··· ··· 0 0 00 0 1 0 ··· ··· 0 0 01 1 0 1 0 ··· ··· 0 0 00 0 1 0.........

.. ... .

......

0 0 0 ··· ··· 0 1 0 1 00 0 0 ··· ··· 0 1 0 a0 0 0 ··· ··· 0 b 0

with ab = 2

for positive integers a, b, c, d, e and f .

Proof. (a) If F is symmetric, then this is Theorem 1.1 of [2].

(b) If F is not symmetric, then the spectral radius of F is strictly less than 2by the Perron-Frobenius Theorem (see [6, Theorem 8.8.1]). Hence by Theorem 1.1

(iii) of [2], the matrix F is the adjacency matrix of a Dynkin diagram An (n ≥ 1),Dn (n ≥ 4), E6, E7, E8 or Zn (n ≥ 0).

F of type At. Then F is of the form

0 a1 0 ··· ··· 0b1 0 a2 0 ··· ··· 00 b2 0 · ·· 0 · · · ·· · 0 ·· · at−2 00 ··· ··· 0 bt−2 0 at−1

0 ··· ··· 0 bt−1 0

where ai and bi are positive integers with minai, bi = 1 for all i = 1, 2, . . . , t− 1.

Let fij(x) be the characteristic polynomial of

0 ai 0 ··· ··· 0bi 0 ai+1 0 ··· ··· 00 bi+1 0 · ·

· 0 · · · ·· · 0 ·· · aj−2 00 ··· ··· 0 bj−2 0 aj−1

0 ··· ··· 0 bj−1 0

.

Then we have that f1j(x) = xf2j(x)− a1b1f3j(x), where fi+1,i(x) = 1 and fii(x) =x. Recall that the spectral radius of any proper submatrix of F is at most 2 bythe Perron-Frobenius Theorem. All the eigenvalues of proper square submatricesof F are strictly less than 2. In particular, all characteristic polynomials fij(x)evaluated in 2 are non-negative. We have that

f12(x) = x2 − a1b1,

f13(x) = x(x2 − (a1b1 + a2b2)),

f14(x) = x4 − (a1b1 + a2b2 + a3b3)x2 + a1b1a3b3,

f15(x) = x(x4 − (a1b1 + a2b2 + a3b3 + a4b4)x2 + a2b2a4b4 + a1b1a3b3 + a1b1a4b4).

It follows from this that

1 ≤ a1b1 ≤∗ 4,

2 ≤ a1b1 + a2b2 ≤∗ 4,

0 ≤∗ 16− 4(a1b1 + a2b2 + a3b3) + a1b1a3b3,


where equality in the inequality ≤∗ gives λmax = 2. The possible sequences(a1b1, a2b2, . . . , at−1bt−1) satisfying the above are

t (a1b1, a2b2, . . . , at−1bt−1) (a1b1, a2b2, . . . , at−1bt−1)2 (4) (i), with 1 ≤ i ≤ 33 (2, 2), (1, 3), (3, 1) (2, 1), (1, 2), (1, 1)4 (2, 1, 2) (2, 1, 1), (1, 2, 1), (1, 1, 2), (1, 1, 1)t λmax = 2 λmax < 2

For t = 5 and F with λmax ≤ 2, then, using the classification for t = 4 andthat the spectral radius is strictly smaller for a square submatrix, the sequence(a1b2, a2b2, a3b3, a4b4) must start like (2, 1, 1, ?), (1, 2, 1, ?), (1, 1, 2, ?) or (1, 1, 1, ?),and must end like (?, 1, 1, 2), (?, 1, 1, 1), (?, 2, 1, 1), or (?, 1, 2, 1). One easily checksthat (2, 1, 1, 2), (1, 1, 2, 1) and (1, 2, 1, 1) give λmax = 2, and all smaller sequencesgive λmax < 2.

Now we consider the case t ≥ 6. We show that the sequence(a1b1, a2b2, . . . , at−1bt−1) is in

(2, 1, . . . , 1, 2), (2, 1, . . . , 1), (1, . . . , 1, 2), (1, 1, . . . , 1).It suffices to show that the matrix for the sequence (2, 1, . . . , 1, 2) has λmax = 2,then by [6, Theorem 8.8.1] for the other three cases, the largest eigenvalue is < 2.

Let F be of type At corresponding to sequence (2, 1, . . . , 1, 2). The matrix F ′

obtained from F by deleting the first row and column from F , has maximum rowor column sum equal to 2. Hence the eigenvalue λ′ of with maximal absolute valueof F ′ satisfies λ′ = |λ′| ≤ 2. It follows from this that fit(x) > 0 for all i ≥ 2 andfor all x > 2. Using the recursion we have

f1t(x) = xf2t(x)− 2f3t(x)

= x(xf3t(x)− f4t(x))− 2f3t(x)

= (x2 − 4)f3t(x) + 2f3t(x)− xf4t(x)

By the following Lemma which is easy to prove (and which we will use again), thisis equal to

(∗) (x2 − 4)(f3t(x) + f5t(x) + · · · )where the last term is equal to ftt(x) or ft+1,t(x).

Lemma 4.2. Assume F of type At corresponds to a sequence (a1b1, 1, . . . , 1, 2)(with t ≥ 4). Then for 1 < j < t− 1 we have

2fjt(x)− xfj+1,t(x) = (x2 − 4)(fj+2,t(x) + fj+4,t(x) + . . .),

where the last term is either ftt(x) or ft+1,t(x).

It follows from (∗) that f1t(2) = 0 and f1t(x) > 0 for x > 2. Hence λmax ≤ 2 forF . This completes the proof for type At.

F of type Zt−1. The case t = 1 is covered by (a). For t ≥ 2, the matrix F is ofthe form

1 a1 0 ··· ··· 0b1 0 a2 0 ··· ··· 00 b2 0 · ·· 0 · · · ·· · 0 ·· · at−2 00 ··· ··· 0 bt−2 0 at−1

0 ··· ··· 0 bt−1 0

,


where ai and bi are positive integers with minai, bi = 1 for all i = 1, 2, . . . , t− 1.Let gt(x) be the characteristic polynomial of this matrix. Then direct calculationsshow that gt(x) = (x − 1)f2t(x) − a1b1f3t(x) for t ≥ 2, where fi,i−1(x) = 1 andfij(x) is as in (a) with fii(x) = x and fi,i+1(x) = x2 − aibi. We obtain that

g2(x) = x(x− 1)− a1b1,

g3(x) = x3 − x2 − (a1b1 + a2b2)x+ a2b2,

g4(x) = x4 − x3 − (a1b1 + a2b2 + a3b3)x2 + (a2b2 + a3b3)x+ a1b1a3b3.

Using similar arguments as above we deduce that

a1b1 ≤ 2,

2a1b1 + a2b2 ≤ 4,

4a1b1 + 2a2b2 + 2a3b3 ≤ 8 + a1b1a3b3,

where equality gives λmax = 2. For t ≥ 2 we have

t (a1b1, a2b2, . . . , at−1bt−1) (a1b1, a2b2, . . . , at−1bt−1)2 (2) (1)3 (1, 2) (1, 1)4 (1, 1, 2) (1, 1, 1)t λmax = 2 λmax < 2

For t = 5 and F with λmax ≤ 2, then, using the classification for t = 4, the sequence(a1b1, a2b2, a3b3, a4b4) must start like (1, 1, 1, ?), and it must end like (?, 2, 1, 1),(?, 1, 2, 1), (?, 1, 1, 2) or (?, 1, 1, 1) using the classification for the At-case. So thepossibilities are (1, 1, 1, 2) and (1, 1, 1, 1). We show that the first has λmax = 2 andthen by [6, Theorem 8.8.1] the second will have λmax < 2. Inductively, for t ≥ 6the possibilities are then just (1, 1, . . . , 2) and (1, 1, . . . , 1). So it suffices to showthat for any t ≥ 5 the matrix F associated to (1, 1, . . . , 2) has λmax = 2. Fix sucha matrix. For 1 ≤ j ≤ t, let gj(x) be the characteristic polynomial of the principalj × j submatrix. Then expanding now along the last row, we have

gt(x) = xgt−1(x)− 2gt−2(x)

gj(x) = xgj−1(x)− gj−2(x) (3 ≤ j < t)

With the method as for Lemma 4.2 we find that for t even,

gt(x) = (x2 − 4)(gt−2(x) + gt−4(x) + . . . g4(x)) + (x− 2)(x− 1),

and if t is odd then

gt(x) = (x2 − 4)(gt−2(x) + gt−4(x) + . . . g5(x)) + (x− 2)(x2 + x− 1).

For j < t the polynomial gj(x) is the characteristic polynomial for a matrix F ′ oftype Zj and by induction it has λ′

max < 2. It follows that gt(2) = 0 and gt(x) > 0for x > 2. Hence F has λmax = 2, and this gives rise to the matrices listed in (b)(i).

F of type Dt. Then F is of the form

0 0 a1 ··· ··· 00 0 a2 0 ··· ··· 0b1 b2 0 a3 ·· 0 b3 · · ·· · 0 ·· · at−2 00 ··· ··· 0 bt−2 0 at−1

0 ··· ··· 0 bt−1 0

,


where ai and bi are positive integers with minai, bi = 1 for all i = 1, 2, . . . , t− 1.Let ht(x) be the characteristic polynomial of this matrix. Direct computations showthat ht(x) = xf2t(x) − a1b1xf4t(x) for t ≥ 4 and h3(x) = x(x2 − (a1b1 + a2b2)).This implies that

t (a1b1, a2b2, . . . , at−1bt−1) (a1b1, a2b2, . . . , at−1bt−1)3 (2, 2), (1, 3), (3, 1) (2, 1), (1, 2), (1, 1)4 (2, 1, 1), (1, 2, 1), (1, 1, 2) (1, 1, 1)t λmax = 2 λmax < 2

For t ≥ 5 and λmax ≤ 2, then the sequence (a1b1, a2b2, . . . , at−1bt−1) must startlike (1, 1, 1, ?, . . . , ?) by the case t = 4 using the same arguments as in the previouscases. Deleting the first row and the first column of F gives a matrix F ′ withλ′max < 2, which is of type At−1. From this and the above we obtain for t ≥ 5 that

(a2b2, . . . , at−1bt−1) is (1, 1, . . . , 1, 2) or (1, 1, . . . , 1, 1). By the previous arguments,we only need to show that a matrix F corresponding to the sequence (1, 1, . . . , 2)has λmax = 2. Using the recursion, one checks that ht(x) = x[xf3t(x) − 2f4t(x)].We can now apply Lemma 4.2, and this shows that

ht(x) = x(x2 − 4)[f5t(x) + f7t(x) + . . .],

where the last term is either ftt(x) or ft+1,t(x). Now it follows by the arguments asin type At that λmax = 2. By the above observations, the claim in (b) (iii) follows.

F of type E6,7,8. Using the classification of matrices for the At-case, the matrix

F in this case is

0 a1 0 0 0 0b1 0 1 0 0 00 1 0 1 0 a30 0 1 0 a2 00 0 0 b2 0 00 0 b3 0 0 0

,

0 a1 0 0 0 0 0b1 0 1 0 0 0 00 1 0 1 0 0 a30 0 1 0 1 0 00 0 0 1 0 a2 00 0 0 0 b2 0 00 0 b3 0 0 0 0

or

0 a1 0 0 0 0 0 0b1 0 1 0 0 0 0 00 1 0 1 0 0 0 a30 0 1 0 1 0 0 00 0 0 1 0 1 0 00 0 0 0 1 0 a2 00 0 0 0 0 b2 0 00 0 b3 0 0 0 0 0

,

where (a1b1, a2b2) is in (2, 1), (1, 2), (1, 1). Direct computation of the character-istic polynomials of these matrices give

h6(x) = x6 − (a1b1 + a2b2 + a3b3 + 2)x4

+ (a1b1 + a1b1a2b2 + a1b1a3b3 + a2b2 + a2b2a3b3)x2

− a1b1a2b2a3b3,

h7(x) = x7 − (a1b1 + a2b2 + a3b3 + 3)x5

+ (2a1b1 + 2a2b2 + a1b1a2b2 + a2b2a3b3 + a3b3 + a1b1a3b3 + 1)x3

− (a1b1a2b2a3b3 + a1b1a2b2 + a1b1a3b3)x,

h8(x) = x8 − (a1b1 + b2a2 + a3b3 + 4)x6

+ (a2b2a3b3 + a1b1b2a2 + 3a2b2 + 3 + a1b1a3b3 + 2a3b3 + 3a1b1)x4

− (a1b1a2b2a3b3 + a2b2 + 2a1b1a2b2 + a2b2a3b3 + a1b1 + 2a1b1a3b3)x2

+ a1b1a2b2a3b3.

Directly checking the possible values of (a1b1, a2b2) shows that there are no casewith λmax ≤ 2 for E6,7,8, unless F is a symmetric matrix. Hence we are back in theclaim in (a). This completes the proof of the proposition.


5. Classification for symmetric contracted matrices

In this section we classify the quivers Q of all indecomposable selfinjective alge-bras Λ = kQ/I with radical cube zero and radical square non-zero such that the

underlying contracted matrix of the adjacency matrix is symmetric and of type An,

Dn, E6, E7, E8, Zn, or DZn. Hence we obtain in this case a classification of thequivers Q of all such algebras Λ = kQ/I with complexity 2.

Hypothesis 5.0. We assume throughout this section that Λ = kQ/I is an inde-composable selfinjective algebra with radical cube zero and radical square non-zero,and with Nakayama permutation Π = Π1Π2 . . .Πt with t disjoint cycles Πi and Πi

has length di. We denote by F its contracted matrix.

For each possible F , we determine all possible quivers. As for the converse, foreach quiver Q in the list, there is at least one algebra satisfying the hypothesis.Namely, take I the ideal generated by

(a) for each i, all paths of length 2 starting at vertex i and ending at a vertex6= π(i),

(b) for each i, the sum∑r

i=1 ηi where η1, . . . , ηr are all paths of length twostarting at i and ending at π(i).

There may be more such algebras, with scalars occuring in relations of type (b).When the Nakayama permutation Π = Π1Π2 · · ·Πt is written as a product of

disjoint cycles with di the length of Πi, then we call the sequence (d1, d2, . . . , dt)the Nakayama cycle type of Λ. When t ≥ 2 and the contracted matrix is symmetricand of the above type, then Λ is of Nakayama cycle type (d, d, . . . , d) for somepositive integer d. If d is 1, we are back in the weakly symmetric case, so weassume throughout that d > 1.

We start with the case that the contracted matrix is of type At.

Proposition 5.1. Assume that the contracted matrix F of Λ is of the type At.Then di = d for all i, and the quiver Q is isomorphic to the following quiver

11

''PPPPPPPPP 12oo · · · 1t−1

((QQQQQQQQQ 1too // ΠN1 (1)1

vvmmmmmmmm

21 22oo · · · 2t−1 2too // ΠN1 (2)1

(d− 1)1

''PPPPPPP(d− 1)2oo · · · (d− 1)t−1

((QQQQQQQ(d− 1)too // ΠN

1 (d− 1)1

vvmmmmmmm

(d)1

DD(d)2oo · · · dt−1

CC(d)too // ΠN

1 (d)1

[[88888888888888888

with Πi = (1i, 2i, . . . , (d − 1)i, (d)i) for i = 1, 2, . . . , t and for some non-negativeinteger N satisfying 0 ≤ N < d. Any such N occurs. The left and the rightcolumns are identified according to the permutation ΠN

1 .

Proof. The contracted matrix F is of the form

0 1 11 0

. . .0 11 0

. . .0 1

1 1 0

. By Lemma

1.1 we have that d1 = d2 = · · · = dt−1 = dt = d for some positive integer d and


Πi = (1i, 2i, . . . , (d−1)i, (d)i) for i = 1, 2, . . . , t. Furthermore the matrix E is given

by

0 Πi11 Π

it1

Π1−i11 0

. . .0 Π

ij1

Π1−ij1 0

. . .0 Π

it−11

Π1−it1 Π

1−it−11 0

, identifying Πi and Π1. Conjugating

this matrix with the diagonal block matrix

D = diag(Π−(i1+i2+···+it−1)1 , . . . ,Π

−(it−2+it−1)1 ,Π

−it−1

1 , I),

we obtain the matrix

0 Id ΠN1

Π1 0

. . .0 IdΠ1 0

. . .0 Id

Π1−N1 Π1 0

for some N with 0 ≤ N ≤ d−1.

This is the adjacency matrix of the quiver Q in the proposition.

In all the other cases for the type of the contracted matrix F the proofs arebasically the same, so we leave the details to the reader.

Proposition 5.2. Assume that the contracted matrix F of Λ is of the type Dt.Then di = d for all i, and the quiver Q is isomorphic to the following quiver

11

++VVVVVVVVVVVVVVVVVVVVVVVVV 21 · · · (d− 1)1

++XXXXXXXXXXXXXXXXXXXXXXXXX (d)1

||yyyyyyy12

xxqqqqqqqqqqqq 22 · · · (d− 1)2

yyrrrrrrrrr(d)2

qqddddddddddddddddddddddddddddddddddddddddddddddd

13

iiRRRRRRRRRRRRRRRR

""EEE

EEEE

E

44jjjjjjjjjjjjjjjjjjjj23

jjTTTTTTTTTTTTTTTTTTTT

44jjjjjjjjjjjjjjjjjjjj · · · (d− 1)3

jjTTTTTTTTTTTTTTT

""EEE

EEEE

55jjjjjjjjjjjjjj(d)3

jjTTTTTTTTTTTTTTTTTTT

ssgggggggggggggggggggggggggggg

55kkkkkkkkkkkkkkkk

14

OO

24

OO

· · · (d− 1)4

OO

(d)4

OO

1t−3

""EEE

EEEE

2t−3 · · · (d− 1)t−3

""EEE

EEEE

(d)t−3

ssgggggggggggggggggggggggggg

1t−2

OO

xxrrrrrrrrr

++XXXXXXXXXXXXXXXXXXXXXXXXXXXX 2t−2

OO

· · · (d− 1)t−2

xxqqqqqqqqq

OO

++WWWWWWWWWWWWWWWWWWWWWW(d)t−2

qqdddddddddddddddddddddddddddddddddddddddddddd

OO

||yyyy

yyyy

1t−1

55llllllllllllll2t−1

44jjjjjjjjjjjjjjjjjj · · · (d− 1)t−1

44jjjjjjjjjjjjj(d)t−1

44jjjjjjjjjjjjjjjjj1t

jjTTTTTTTTTTTTTTTTTTT2t

jjTTTTTTTTTTTTTTTTTTT · · · (d− 1)t

jjTTTTTTTTTTTTTT(d)t

iiSSSSSSSSSSSSSSS

with Πi = (1i, 2i, . . . , (d− 1)i, (d)i) for i = 1, 2, . . . , t.

Proposition 5.3. Assume that the contracted matrix F of Λ is of the type Et fort = 6, 7, 8. Then di = d for all i, and the quiver Q is isomorphic to the following


quiver

15

25 · · · (d− 1)5

(d)5

14

;;wwwww

24 · · · (d− 1)4

;;wwww(d)4

llXXXXXXXXXXXXXXXXXXXXXX

13

;;wwwww

uukkkkkkkkkkkk

,,XXXXXXXXXXXXXXXXXXXXXXX 23 · · · (d− 1)3

tthhhhhhhhhhh

;;wwww

,,XXXXXXXXXXXXXXXXXXXX(d)3

tthhhhhhhhhhhhhh


wwwww

12

33ggggggggggggggggggg

22

44iiiiiiiiiiiiiii

· · · (d− 1)2

22ffffffffffffffffffff

(d)2

ccGGGGG

16

jjVVVVVVVVVVVVVVVV

##GGGGG 26

jjVVVVVVVVVVVVVVVV · · · (d− 1)6

jjVVVVVVVVVVV

##GGGG

(d)6

jjVVVVVVVVVVVVVV

rrffffffffffffffffffffff

11

DD21 · · · (d− 1)1

;;wwww(d)1

kkWWWWWWWWWWWWWWWWWW17

OO

27

OO

· · · (d− 1)7

OO

(d)7

OO

or

18

28

· · · (d− 1)8

(d)8

14

;;wwwww

uukkkkkkkkkkkk


tthhhhhhhhhhh

;;wwww


tthhhhhhhhhhhhhh


wwwww

13


23

44iiiiiiiiiiiiiii

· · · (d− 1)3


(d)3

ccGGGGG

15

jjVVVVVVVVVVVVVVVV

##GGGGG 25


jjVVVVVVVVVVV

##GGGG

(d)5

jjVVVVVVVVVVVVVV


12

DD

22

· · · (d− 1)2

;;wwww

(d)2

kkWWWWWWWWWWWWWWWWWW

16

OO

##GGGGG 26

OO

· · · (d− 1)6

OO

##GGGG

(d)6

OO


11

DD21 · · · (d− 1)1

;;wwww(d)1

kkWWWWWWWWWWWWWWWWWW17

OO

27

OO

· · · (d− 1)7

OO

(d)7

OO

or

18

28

· · · (d− 1)8

(d)8

17

;;wwwww27

· · · (d− 1)7

;;wwww(d)7


16

;;wwwww26

· · · (d− 1)6

;;wwww(d)6


15

;;wwwww25

· · · (d− 1)5

;;wwww(d)5


14

;;wwwww24

· · · (d− 1)4

;;wwww(d)4


13

;;wwwww

uukkkkkkkkkkkk


tthhhhhhhhhhh

;;wwww


tthhhhhhhhhhhhhh


wwwww

12


22

44iiiiiiiiiiiiiii

· · · (d− 1)2


(d)2

ccGGGGG

19

jjVVVVVVVVVVVVVVVV29


jjVVVVVVVVVVV(d)9

jjVVVVVVVVVVVVVV

11

DD21 · · · (d− 1)1

;;wwww(d)1

kkWWWWWWWWWWWWWWWWWW

with Πi = (1i, 2i, . . . , (d− 1)i, (d)i) for i = 1, 2, . . . , t.

Proposition 5.4. Assume that the contracted matrix F of Λ is of the type Zt−1

for t ≥ 1. Then di = d for all i, and the quiver Q is isomorphic to the following


quiver

11 //

++WWWWWWWWWWWWWWWWW Σ(1)1 // Σ2(1)1 // · · · // Σd−2(1)1 //

,,XXXXXXXXXXXXXXXXXXXX Σd−1(1)1 // 11

12

OO

++WWWWWWWWWWWWWWWWW Σ(1)2

OO

Σ2(1)2

OO

· · · Σd−2(1)2

OO

,,XXXXXXXXXXXXXXXXXXXX Σd−1(1)2

OO

12

OO

13

OO

Σ(1)3

OO

Σ2(1)3

OO

· · · Σd−2(1)3

OO

Σd−1(1)3

OO

13

OO

1t−2

++WWWWWWWWWWWWWWW Σ(1)t−2 Σ2(1)t−2 · · · Σd−2(1)t−2

,,XXXXXXXXXXXXXXXXXX Σd−1(1)t−2 1t−2

1t−1

OO

++WWWWWWWWWWWWWWWW Σ(1)t−1

OO

Σ2(1)t−1

OO

· · · Σd−2(1)t−1

OO

,,XXXXXXXXXXXXXXXXXXX Σd−1(1)t−1

OO

1t−1

OO

1t //

OO

Σ(1)t //

OO

Σ2(1)t //

OO

· · · // Σd−2(1)t //

OO

Σd−1(1)t //

OO

1t

OO

for some positive odd integer d and t ≥ 2, with Πi = (1i, 2i, . . . , (d − 1)i, (d)i) for

i = 1, 2, . . . , t and Σ = Π[ d2 ]+11 . The left and the right columns are identified.

For t = 1, the quiver Q is isomorphic to a quiver with d vertices labelled1, 2, . . . , d for a positive integer d, and arrows i → Πl(i) and i → Π1−l(i) fori = 1, 2, . . . , d and some l with 0 ≤ l ≤ d

2 . In particular, the last quiver is given bythe configuration

1

%%JJJJJJJJJJJJJJJJJJJJJJ

yyrrrrrrrrrrrrrrrrrrrrrrrr

d 2

d+ 2− l

33hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhl + 1

]];;;;;;;;;;

and this picture rotated by powers of Π1 = (1, 2, . . . , d− 1, d).

Proof. The case t = 1: Suppose Π = Π1 = (1, 2, . . . , d − 1, d) for some positiveinteger d. By Lemma 1.1 we have that the adjacency matrix E is given by Πl+Π1−l

for 0 ≤ l ≤ d2 . So there are arrows i → Πl(i) and i → Π1−l(i) for i = 1, 2, . . . , d.

The value of l need only vary in the interval [0, d2 ].

Now we give the final class of quivers when the contracted matrix is symmetric.


Proposition 5.5. Assume that the contracted matrix F of Λ is of the type DZt−1

for t ≥ 0. Then the quiver Q is isomorphic to the following quiver

11

!!CCC

CCCC

CCCC

CCCC

C Σ(1)1 Σ2(1)1 · · · Σd−2(1)1

%%KKKKKKKKKKKKKKKKKKKKΣd−1(1)1 11

12

Σ(1)2 Σ2(1)2 · · · Σd−2(1)2

Σd−1(1)2 12

13

OO

((QQQQQQQQQQQ

66mmmmmmmmmmmmmΣ(1)3

OO

55lllllllllllΣ2(1)3

55llllllllll

OO

· · · Σd−2(1)3

OO

**UUUUUUUUUUUUUUUU

44iiiiiiiiiiiiiΣd−1(1)3

OO

44jjjjjjjjjjj13

OO

66mmmmmmmmmmmmmm

14

OO

Σ(1)4

OO

Σ2(1)4

OO

· · · Σd−2(1)4

OO

Σd−1(1)4

OO

14

OO

1t−1

((QQQQQQQQQQΣ(1)t−1 Σ2(1)t−1 · · · Σd−2(1)t−1

**UUUUUUUUUUUUUUUΣd−1(1)t−1 1t−1

1t //

OO

Σ(1)t //

OO

Σ2(1)t

OO

· · · Σd−2(1)t //

OO

Σd−1(1)t //

OO

1t

OO

for some positive odd integer d and t ≥ 3, with Πi = (1i, 2i, . . . , (d − 1)i, (d)i) for

i = 1, 2, . . . , t, and Σ = Π[ d2 ]+11 . The left and the right columns are identified, and

the two extreme vertices in the second cycle both labelled 12 are identified.

6. Classification for non-symmetric contracted matrices

This section is devoted to classifying the quivers Q of all indecomposable self-injective algebras Λ = kQ/I with radical cube zero and radical square non-zero,where the contracted matrix associated to Λ is non-symmetric. Hence in this casewe obtain a classification of the quivers of all such algebras Λ = kQ/I with com-plexity 2. Let Π = Π1Π2 · · ·Πt be the Nakayama permutation written as a productof disjoint cycles Πi. As explained at the beginning of Section 5, for each suchquiver there is at least one such algebra.

The possible contracted matrices F were determined in Section 4. For the clas-sification of the quivers, we can reduce the list of matrices F which need to beconsidered.

Given matrices F1 and F2 in the list, whenever there is a permutation matrix Psuch that P−1F2P = F1, then we only need to consider F1, since by relabelling thecycles of Π, an algebra associated to F2 is the same as an algebra associated to F1.Therefore we do not need to consider matrices F for type Dt with t = 3 as each ofthese occurs after relabelling cycles in the list for type At with t = 3. On the listof type A3 we do not need to consider the second matrix F , and we do not need toconsider the last matrix for t = 5, again by the relabelling argument; and clearlywe only need to consider one of the matrices of type A2. Similarly, any two of thematrices F for type D4 are related by relabelling cycles. We will only consider thelast type, and we view this as part of the two infinite families. For type At, the lastshape for t = 3, the shape F for t = 4, the first shape for t = 5 together with thematrices for t ≥ 6 are families. We may assume a ≥ c and hence there are threefamilies.


In this section, we make again the same hypothesis as in Section 5, that is thealgebra Λ is as in Hypothesis 5.0. We first discuss the three exceptional cases leftfrom the above considerations, and then we treat the three families of contractedmatrices.

Proposition 6.1. Assume that the contracted matrix F of Λ is of the form ( 0 14 0 )

or ( 0 41 0 ). Then the quiver Q is isomorphic to the following quiver

1

AAA

AAAA

A 2

AAA

AAAA

A 3

AAA

AAAA

A d

:::

::::

d+ 1''PPP

d+ 2''PPP

d+ 3''PPP

2d%%LL

L

4d+ 1

>> 77nnn

''PPP

AAA

AAAA

4d+ 2

>> 77nnn

''PPP

AAA

AAAA

4d+ 3

>> 77nnn

''PPP

AAA

AAAA

4d+ 4 5d

EE==zz

!!DD

222

222 4d+ 1

2d+ 1

77nnn2d+ 2

77nnn2d+ 3

77nnn3d

99sss

3d+ 1

>>3d+ 2

>>3d+ 3

>>4d

BB

with Nakayama cycle type (4d, d), and where Π1 = (1, 2, . . . , 4d) and Π2 = (4d +1, 4d+ 2, . . . , 5d). Here the end vertices in the quiver are identified.

Proof. It is easy to see that the matrices ( 0 14 0 ) and ( 0 4

1 0 ) give rise to isomorphicquivers. Hence we only consider the first case. Let F = ( 0 1

4 0 ). Then we have thatd2 = d and d1 = 4d with d some positive integer, Π1 = (1, 2, . . . , 4d) and Π2 = (4d+

1, 4d+2, . . . , 5d). By Lemma 1.1 the matrix E is given by

0 0 0 0 Πi2

0 0 0 0 Πi2

0 0 0 0 Πi2

0 0 0 0 Πi2

Π1−i2 Π1−i

2 Π1−i2 Π1−i

2 0

for some integer i with 0 ≤ i ≤ d− 1. Conjugating with the matrix(

I 00 Πi

2

)shows

that E is permutation equivalent with

0 0 0 0 Id0 0 0 0 Id0 0 0 0 Id0 0 0 0 IdΠ2 Π2 Π2 Π2 0

. Then the quiver Q can

be represented as in the claim of the proposition.

Our proofs of the remaining exceptional cases use the same type of argumentsas we have seen above. Therefore we leave the details to the reader for the casesleft to discuss.

Proposition 6.2. Assume that the contracted matrix F of Λ is of the form(

0 a 0b 0 10 1 0

),

(0 1 01 0 a0 b 0

), where ab = 3. Then the quiver Q is isomorphic to one of the following

quivers: if b = 3,

1

**TTTTTTTTTTTTTTTTTTTTTTT2 · · · d− 1

++VVVVVVVVVVVVVVVVVVVVVVVVVVVVVd

000

0000d+ 1

888

8888

8 d+ 2

. . . 2d− 1

222

2222

2 2d

uukkkkkkkkkkkkkkkkkkk 2d+ 1

vvllllllllllllllll2d+ 2 . . . 3d− 1

uukkkkkkkkkkkkkkkkkkk3d

rreeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

3d+ 1

hhPPPPPPPPPPPPPPPPP

888

8888

8

OO 44iiiiiiiiiiiiiiiiiiiiiii3d+ 2

iiSSSSSSSSSSSSSSSSSSSSS

OO 44iiiiiiiiiiiiiiiiiiiiiii· · · 4d− 1

jjTTTTTTTTTTTTTTTTTTTTTT

OO 44iiiiiiiiiiiiiiiiiiiiiii

222

2222

2 4d

jjUUUUUUUUUUUUUUUUUUUUUUUUUU

44iiiiiiiiiiiiiiiiiiiiiiiiii

OO

uukkkkkkkkkkkkkkkkkkk

4d+ 1

OO

4d+ 2

OO

· · · 5d− 1

OO

5d

OO


with Nakayama cycle type (3d, d, d) and Π1 = (1, 2, . . . , 3d), Π2 = (3d + 1, 3d +2, . . . , 4d) and Π3 = (4d+ 1, 4d+ 2, . . . , 5d), or, if a = 3,

1

uulllllllllllllllll

999

9999

++WWWWWWWWWWWWWWWWWWWWWWWWWWWW 2 · · · d− 1

uukkkkkkkkkkkkkkkkkk

222

222

++WWWWWWWWWWWWWWWWWWWWWWWWWW d

rrdddddddddddddddddddddddddddddddddddddddddd

uukkkkkkkkkkkkkkkkkk

222

222

d+ 1

999

9999

44iiiiiiiiiiiiiiiiiiiiiiid+ 2

333

3333

44iiiiiiiiiiiiiiiiiiiiiii· · · 2d− 1

222

222

44iiiiiiiiiiiiiiiiiiii2d

222

222

44iiiiiiiiiiiiiiiiiiiiiiii2d+ 1

999

9999

OO

2d+ 2

333

3333

OO

· · · 3d− 1

OO

222

222 3d

222

222

OO

3d+ 1

999

9999

jjUUUUUUUUUUUUUUUUUUUUUU3d+ 2

333

3333

jjUUUUUUUUUUUUUUUUUUUUUU· · · 4d− 1

222

222

jjUUUUUUUUUUUUUUUUUUUU4d

qqbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

jjUUUUUUUUUUUUUUUUUUUUUUUU

4d+ 1

OO

4d+ 2

OO

· · · 5d− 1

OO

5d

OO

5d+ 1

OO

5d+ 2

OO

· · · 6d− 1

OO

6d

OO

6d+ 1

OO

6d+ 2

OO

· · · 7d− 1

OO

7d

OO

with Nakayama cycle type (d, 3d, 3d) and Π1 = (1, 2, . . . , d− 1, d), Π2 = (d+ 1, d+2, . . . , 4d− 1, 4d) and Π3 = (4d+1, 4d+2, . . . , 7d− 1, 7d) for some positive integerd ≥ 1.

Proposition 6.3. Assume that the contracted matrix F of Λ is of the form(0 1 0 0 01 0 a 0 00 b 0 1 00 0 1 0 10 0 0 1 0

)or

(0 1 0 0 01 0 1 0 00 1 0 a 00 0 b 0 10 0 0 1 0

), where ab = 2. Then the quiver Q is isomorphic to

one of the following quivers: if b = 2,

1

===

====

= 2 · · · d− 1

777

7777

d

777

7777

d+ 1 · · · 2d− 1

666

6666

2d

qqdddddddddddddddddddddddddddddddddddddddddddddd

2d+ 1

OO

))TTTTTTTTTTTTTTTTT2d+ 2

OO

· · · 3d− 1

OO

((QQQQQQQQQQQQQQ 3d

yyrrrrrrrrrrr

OO

3d+ 1

OO

zzuuuuuuuuu· · · 4d− 1

OO

4d

OO

rrffffffffffffffffffffffffffffffff

4d+ 1

===

====

ffNNNNNNNNNNN

66llllllllllllll4d+ 2

ffNNNNNNNNNNN· · · 5d− 1

777

7777

ddIIIIIIIII

::uuuuuuuuu5d

ttiiiiiiiiiiiiiiiiiiiii

::uuuuuuuuuu

ddIIIIIIIIII

5d+ 1

OO

===

====

5d+ 2

OO

· · · 6d− 1

OO

777

7777

6d

OO


6d+ 1

OO

6d+ 2

OO

· · · 7d− 1

OO

7d

OO

with Nakayama cycle type (2d, 2d, d, d, d) and Π1 = (1, 2, . . . , 2d − 1, 2d), Π2 =(2d + 1, d + 2, . . . , 4d − 1, 4d), Π3 = (4d + 1, 4d + 2, . . . , 5d − 1, 5d), Π4 = (5d +


1, 5d+ 2, . . . , 6d− 1, 6d) and Π5 = (6d+ 1, 6d+ 2, . . . , 7d− 1, 7d), or, if a = 2,

1

;;;

;;;;

2 · · · d− 1

777

7777

d


d+ 1

OO

d+ 2

OO

· · · 2d− 1

OO

((PPPPPPPPPPPPPP 2d

OO

rrffffffffffffffffffffffffffffffffff

2d+ 1

===

====

88qqqqqqqqqqq2d+ 2

88qqqqqqqqqqq· · · 3d− 1

777

7777

::uuuuuuuuu3d

777

7777

::uuuuuuuuuu3d+ 1

hhQQQQQQQQQQQQQQ· · · 4d− 1

666

6666

ddIIIIIIIII

4d

rrddddddddddddddddddddddddddddddddddddddddddddd

ddIIIIIIIIII

4d+ 1

===

====

OO

4d+ 2

OO

· · · 5d− 1

OO

777

7777

5d

777

7777

OO

5d+ 1

OO

· · · 6d− 1

OO

666

6666

6d

rrddddddddddddddddddddddddddddddddddddddddddddd

OO

6d+ 1

OO

6d+ 2

OO

· · · 7d− 1

OO

7d

OO

7d+ 1

OO

· · · 8d− 1

OO

8d

OO

with Nakayama cycle type (d, d, 2d, 2d, 2d) and Π1 = (1, 2, . . . , d− 1, d), Π2 = (d+1, d + 2, . . . , 2d − 1, 2d), Π3 = (2d + 1, 2d + 2, . . . , 4d − 1, 4d), Π4 = (4d + 1, 4d +2, . . . , 6d− 1, 6d) and Π5 = (6d+1, 6d+2, . . . , 8d− 1, 8d) for some positive integerd.

This completes the results for the exceptional cases, and now we discuss thethree remaining families of algebras.

Proposition 6.4. Assume that the contracted matrix F of Λ is of type At and of

the form

0 a 0b 0 10 1 0 1

1

...0 1 01 0 c0 d 0

, where ab = 2 = cd and t ≥ 3. Then the quiver Q is

isomorphic to one of the following quivers: if a = 2 = c,

11

222

2222

21 · · · (2d− 1)1

,,YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY (2d)1

ttjjjjjjjjjjjjjjjjjjjjj (2d+ 1)1

ttiiiiiiiiiiiiiiiiiiiiiii(2d+ 2)1 · · · (4d− 1)1

(4d)1

qqcccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

12

222

2222

OO 33gggggggggggggggggggggggggggg 22

OO 33ffffffffffffffffffffffffffffffff · · · (2d− 1)2

""EEEEEEEE

kkXXXXXXXXXXXXXXXXXXXXXXXXXXXX

<<yyyyyyyy(2d)2

qqccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

llXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

AA

13

OO

23

OO

· · · (2d− 1)3

OO

(2d)3

OO

1t−2

222

22222t−2 · · · (d− 1)t−2

???

????

?(d)t−2

???

????

?(d+ 1)t−2 · · · (2d− 1)t−2

""EEEEEEEE(2d)t−2

qqccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

1t−1

''OOOOOOOOOOOOOO

OO

2t−1

OO

· · · (d− 1)t−1

OO

**UUUUUUUUUUUUUUUUUUUU(d)t−1

xxppppppppppppp

OO

(d+ 1)t−1

wwoooooooooooooo

OO

· · · (2d− 1)t−1

||yyyyyyyy

OO

(2d)t−1

qqddddddddddddddddddddddddddddddddddddddddddddddddd

OO

1t

aaBBBBBBBBB

55jjjjjjjjjjjjjjjjjjj2t

ccGGGGGGGGGG· · · (d− 1)t

ggOOOOOOOOOOOO

66lllllllllllll(d)t

hhPPPPPPPPPPPPPP

66llllllllllllllll


with Nakayama cycle type (4d, 2d, 2d, . . . , 2d, d) and Π1 = (11, 21, . . . , (4d−1)1, (4d)1),Π2 = (12, 22, . . . , (2d− 1)2, (2d)2),. . . , Πt−1 = (1t−1, 2t−1, . . . , (2d− 1)t−1, (2d)t−1)and Πt = (1t, 2t, . . . , (d− 1)t, (d)t), or, if b = 2 = c,

11

''NNNNNNNNNNNNNNN21 · · · (d− 1)1

**TTTTTTTTTTTTTTTTTTT(d)1

xxrrrrrrrrrrrr(d+ 1)1

xxqqqqqqqqqqqqq· · · (2d− 1)1

~~~~

~~~~

(2d)1

rreeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

12

__????????

55kkkkkkkkkkkkkkkkkk

;;;

;;;;

; 22

ddIIIIIIIIIII· · · (d− 1)2

ffMMMMMMMMMMM

77nnnnnnnnnnnn

AAA

AAAA

A(d)2

ggOOOOOOOOOOOOOO

77ooooooooooooo

ssgggggggggggggggggggggggggggg

13

OO

23

OO

· · · (d− 1)3

OO

(d)3

OO

1t−2

;;;

;;;;

2t−2 · · · (d− 1)t−2

AAA

AAAA

A(d)t−2

ssggggggggggggggggggggggggggg

1t−1

OO

2t−1

OO

· · · (d− 1)t−1

OO

**UUUUUUUUUUUUUUUUUUU(d)t−1

OO

rrffffffffffffffffffffffffffffffffffff

~~

1t

??2t

::uuuuuuuuuu · · · (d− 1)t

88qqqqqqqqqqq(d)t

77ooooooooooooo(d+ 1)t

iiSSSSSSSSSSSSSSSSS· · · (2d− 1)t

ggPPPPPPPPPPPP

(2d)t

ggOOOOOOOOOOOO

with Nakayama cycle type (2d, d, d, . . . , d, 2d) and Π1 = (11, 21, . . . , (2d−1)1, (2d)1),Π2 = (12, 22, . . . , (d− 1)2, (d)2),. . . , Πt−1 = (1t−1, 2t−1, . . . , (d− 1)t−1, (d)t−1) andΠt = (1t, 2t, . . . , (2d− 1)t, (2d)t), or, if a = 2 = d,

11

21 · · · (d− 1)1

**UUUUUUUUUUUUUUUUUUUUU (d)1

rreeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

12

<<xxxxxxxxxx

555

5555

22

::uuuuuuuuuuuu · · · (d− 1)2

77oooooooooooo

???

????

?d2

77oooooooooooooo

???

????

?(d+ 1)2

iiTTTTTTTTTTTTTTTTTTT· · · (2d− 1)2

ggOOOOOOOOOOOO

""DDDD

DDDD

(2d)2

hhQQQQQQQQQQQQQQQ

qqccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

13

OO

23

OO

· · · (d− 1)3

OO

d3

OO

(d+ 1)3

OO

· · · (2d− 1)3

OO

(2d)3

OO

1t−2

555

5555

2t−2 · · · (d− 1)t−2

???

????

?dt−2

???

????

?(d+ 1)t−2 · · · (2d− 1)t−2

""DDDD

DDDD

(2d)t−2

qqccccccccccccccccccccccccccccccccccccccccccccccccccccccc

1t−1

OO

((PPPPPPPPPPPPPPPP2t−1

OO

· · · (d− 1)t−1

OO

**TTTTTTTTTTTTTTTTTTTdt−1

OO

wwpppppppppppppp(d+ 1)t−1

OO

· · · (2d− 1)t−1

OO

~~~~

~~~~

(2d)t−1

OO

rrdddddddddddddddddddddddddddddddddddddddddddddd

1t

bbFFFFFFFFF

55jjjjjjjjjjjjjjjjjjj2t

ddJJJJJJJJJJJ· · · (d− 1)t

ggOOOOOOOOOOOO

77oooooooooooo(d)t

ggOOOOOOOOOOOOO

66mmmmmmmmmmmmmm

with Nakayama cycle type (d, 2d, 2d, . . . , 2d, d) and Π1 = (11, 21, . . . , (d− 1)1, (d)1),Π2 = (12, 22, . . . , (2d− 1)2, (2d)2),. . . , Πt−1 = (1t−1, 2t−1, . . . , (2d− 1)t−1, (2d)t−1)and Πt = (1t, 2t, . . . , (d− 1)t, (d)t) for some positive integer d.


Proof. Up to permutation equivalence and isomorphism of the quiver Q there arethree cases to consider, (a, b, c, d) in (1, 2, 1, 2), (1, 2, 2, 1), (2, 1, 1, 2).

The case (a, b, c, d) = (1, 2, 1, 2): By Lemma 1.1 we have that d1 = 4d, d2 = d3 =· · · = dt−1 = 2d and dt = d, and Π1 = (1, 2, . . . , 4d − 1, 4d), Π2 = (4d + 1, 4d +2, . . . , 6d−1, 6d),. . . , Πt−1 = (2(t−1)d+1, 2(t−1)d+2, . . . , 2td−1, 2td), and Πt =(2td+1, 2td+2, . . . , (2t+1)d−1, (2t+1)d). Using similar arguments as before, we in-

fer that the matrix E is permutation equivalent to

0I2dI2d

Π2 Π2 I2dΠ2

. . .I2d

Π2 0IdId

Πt Πt 0

.

This is the adjacency matrix of the first quiver Q in the proposition.The case (a, b, c, d) = (1, 2, 2, 1): By Lemma 1.1 we have that d1 = 2d, d2 = d3 =

· · · = dt−1 = d and dt = 2d, and Π1 = (1, 2, . . . , 2d − 1, 2d), Π2 = (2d + 1, 2d +2, . . . , 3d − 1, 3d),. . . , Πt−1 = ((t − 1)d + 1, (t − 1)d + 2, . . . , td − 1, td), and Πt =(td+1, td+2, . . . , (t+2)d−1, (t+2)d). Using similar arguments as before, we infer

that the matrix E is permutation equivalent to

0IdId

Π2 Π2 IdΠ2

. . .Id

Π2 0 Id IdΠ2

Π20

.

This is the adjacency matrix of the second quiver Q in the proposition.The case (a, b, c, d) = (2, 1, 1, 2): By Lemma 1.1 we have that d1 = d, d2 =

d3 = · · · = dt−1 = 2d and dt = d, and Π1 = (1, 2, . . . , d − 1, d), Π2 = (d + 1, d +2, . . . , 3d− 1, 3d),. . . , Πt−1 = (2td− 5d+1, (t− 1)d+2, . . . , 2td− 3d− 1, 2td− 3d),and Πt = (2td− 3d+ 1, 2td− 3d+ 2, . . . , 2td− 2d− 1, 2td− 2d). Furthermore the

matrix E is given by

0 Πi11 Π

i11

Π1−i11

Π1−i11

Πi22

Π1−i22

. . .Π

it−22

Π1−iit−22 0

Πit−1t

Πit−1t

Π1−it−t Π

1−it−1t

0

. Using

similar arguments as before, we infer that the matrix E is permutation equivalent

to

0 Id IdΠ1

Π1I2d

Π2

. . .I2d

Π2 0 ΠN2

(

IdId

)

( Πt Πt )Π−N2 0

with N = i2+i3+· · ·+it−2. We leave

it to the reader to prove that ΠN2

(IdId

)=(

ΠNt

ΠNt

)and (Πt Πt )Π−N

2 = (Π1−Nt Π1−N

t ).

Using this we get that the last 2 × 2-block of the matrix E can by changed to


(0

IdId

Πt Πt 0

). This is the adjacency matrix of the third quiver Q in the proposition.

The proof for the contracted matrix F of type Dt offers nothing new comparedto the previous case, so we leave the details to the reader.

Proposition 6.5. Assume that the contracted matrix F of Λ is of type Dt and

of the form

0 0 10 0 11 1 0 1

1

.. .1 0

1 0 a0 b 0

, where ab = 2 and t ≥ 4. Then the quiver Q is

isomorphic to one of the following quivers: if b = 2,

11

AAA

AAAA

A 21 · · · (2d− 1)1

,,ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ (2d)1

uullllllllllllllllll12

ssggggggggggggggggggggggggggggg 22 · · · (2d− 1)2

(2d)2

qqcccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

13

WW.......

222

2222

33ffffffffffffffffffffffffffffffffff 23

XX2222222

22ffffffffffffffffffffffffffffffffffff· · · (d− 1)3

>>>

>>>>

(d)3

>>>

>>>>

(d+ 1)3 · · · (2d− 1)3

==zzzzzzzz

llXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

!!DDD

DDDD

D(2d)3

AA

llXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

qqccccccccccccccccccccccccccccccccccccccccccccccccccccccc

14

OO

24

OO

· · · (d− 1)4

OO

(d)4

OO

(d+ 1)3

OO

· · · (2d− 1)4

OO

(2d)4

OO

1t−2

222

22222t−2 · · · (d− 1)t−2

>>>

>>>>

(d)t−2

>>>

>>>>(d+ 1)t−2 · · · (2d− 1)t−2

!!DDD

DDDD

D(2d)t−2

qqcccccccccccccccccccccccccccccccccccccccccccccccccccccc

1t−1

OO

))RRRRRRRRRRRRRRRRRR2t−1

OO

· · · (d− 1)t−1

OO

))TTTTTTTTTTTTTTTTT(d)t−1

OO

wwnnnnnnnnnnnnnn(d+ 1)t−1

OO

· · · (2d− 1)t−1

OO

(2d)t−1

OO

rrdddddddddddddddddddddddddddddddddddddddddddd

1t

ddHHHHHHHHHH

44iiiiiiiiiiiiiiiiiiii2t

ggOOOOOOOOOOOOOO· · · (d− 1)t

ggNNNNNNNNNNN

88qqqqqqqqqqq(d)t

ffMMMMMMMMMMMM

77nnnnnnnnnnnnn


with Nakayama cycle type (2d, 2d, 2d, . . . , 2d, d) and Πi = (1i, 2i, . . . , (2d−1)i, (2d)i)for i = 1, 2, . . . , t− 1 and Πt = (1t, 2t, . . . , (d− 1)t, (d)t), or, if a = 2,

11

%%LLLLLLLLLLLLL21 · · · (d− 1)1

++VVVVVVVVVVVVVVVVVVVVVVVVV(d)1

yysssssssssss12

yytttttttttttt22 · · · (d− 1)2

(d)2

rrddddddddddddddddddddddddddddddddddddddddddddddddd

13

]];;;;;;;;

66mmmmmmmmmmmmmmmmmm

999

9999

23

ccFFFFFFFFFF

55kkkkkkkkkkkkkkkkkkkk · · · · · · (d− 1)3

iiSSSSSSSSSSSSSSS

77ooooooooooo

???

????

?(d)3

88qqqqqqqqqqqq

iiTTTTTTTTTTTTTTTTTTTT

ssffffffffffffffffffffffffffffffffff

14

OO

24

OO

· · · · · · (d− 1)4

OO

(d)4

OO

1t−2

999

9999

2t−2 · · · · · · (d− 1)t−2

???

????

?(d)t−2

ssffffffffffffffffffffffffffffffff

1t−1

OO

**VVVVVVVVVVVVVVVVVVVVVVV 2t−1

OO

· · · · · · (d− 1)t−1

OO

))TTTTTTTTTTTTTTTTT

xxpppppppppppp(d)t−1

OO

rreeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

wwnnnnnnnnnnnn

1t

AA2t

;;xxxxxxxxxx· · · (d− 1)t

55kkkkkkkkkkkkkkk(d)t

55jjjjjjjjjjjjjjjjjjjj(d+ 1)t

hhQQQQQQQQQQQQQQQ

(d+ 2)t

iiSSSSSSSSSSSSSSSSS· · · (2d− 1)t

ggOOOOOOOOOOO

(2d)t

ffMMMMMMMMMMM

with Nakayama cycle type (d, d, d, . . . , d, 2d) and Πi = (1i, 2i, . . . , (d− 1)i, (d)i) fori = 1, 2, . . . , t− 1 and Πt = (1t, 2t, . . . , (2d− 1)t, (2d)t) for some positive integer d.

The last case to investigate is when the contracted matrix F is of type Zt−1.Here one form of the contracted matrix is not possible due to fact that the size ofthe first cycle must be odd. We include some of the details of the proof for thiscase.

Proposition 6.6. Assume that the contracted matrix F of Λ is of type Zt−1 and

of the form

1 1 01 0 10 1

. ..1 0

1 0 a0 b 0

, where ab = 2 and t ≥ 2. Then we must have a = 2,


and the quiver Q is isomorphic to the following quiver

11 //

))SSSSSSSSSSSSS Σ1(1)1 // Σ2(1)1 Σd−3(1)1 //

++WWWWWWWWWWWWWWWWWWW Σd−2(1)1 //

qqddddddddddddddddddddddddddddddddddddd Σd−1(1)1

qqcccccccccccccccccccccccccccccccccccccccccc

12

OO

))SSSSSSSSSSSSS Σ1(1)2

OO

Σ2(1)2

OO

Σd−3(1)2

OO

++WWWWWWWWWWWWWWWWWWW Σd−2(1)2

qqddddddddddddddddddddddddddddddddddddd

OO

Σd−1(1)2

OO

qqcccccccccccccccccccccccccccccccccccccccccc

13

OO

Σ1(1)3

OO

Σ2(1)3

OO

Σd−3(1)3

OO

Σd−2(1)3

OO

Σd−1(1)3

OO

1t−2

))SSSSSSSSSSSΣ1(1)t−2 Σ2(1)t−2 Σd−3(1)t−2

++WWWWWWWWWWWWWWWWW Σd−2(1)t−2

qqddddddddddddddddddddddddddddddddddd Σd−1(1)t−2

qqcccccccccccccccccccccccccccccccccccccccc

1t−1

OO

))SSSSSSSSSSSSS

((QQQQQQQQQQQQQQQQQQQQQQQQQΣ1(1)t−1

OO

Σ2(1)t−1

OO

Σd−3(1)t−1

OO

++WWWWWWWWWWWWWWWWWW

**UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUΣd−2(1)t−1

qqdddddddddddddddddddddddddddddddddddd

OO

uukkkkkkkkkkkkkkkkkkkkkkkkkkkkΣd−1(1)t−1

OO

ttjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

qqccccccccccccccccccccccccccccccccccccccccc

1t

OO

Σ1(1)t

OO

Σ2(1)t

OO

Σd−3(1)t

OO

Σd−2(1)t

OO

Σd−1(1)t

OO

(d+ 1)t

XX111111111111

(d+Σ(1))t

\\8888888888888

(d+Σ2(1))t

^^<<<<<<<<<<<<<

(d+Σd−3(1))t

__@@@@@@@@@@@@@@

(d+Σd−2(1))t

aaBBBBBBBBBBBBBBB

(d+Σd−1(1))t

aaBBBBBBBBBBBBBBB

with Nakayama cycle type (d, d, . . . , d, 2d) and Πi = (1i, 2i, . . . , (d − 1)i, (d)i) fori = 1, 2, . . . , t − 1 and Πt = (1t, 2t, . . . , (2d − 1)t, (2d)t), where d is some positive

odd integer and Σ = Π[ d2 ]+11 .

Proof. By Lemma 1.1 we have that d1 = d2 = · · · = dt−1 is a positive odd integer,so that the only possible configuration for (a, b) is (2, 1) (otherwise dt−1 = 2dt). Inthis case dt = 2d and d1 = d. Hence Πi = (1i, 2i, . . . , (d−1)i, di) for i = 1, 2, . . . , t−1and Πt = (1t, 2t, . . . , (2d− 1)t, (2d)t).

Using similar arguments as before, we infer that the matrix E is permutation

equivalent to

Π[ d2]+1

1 Id 0Π1 0 Id0 Π1

. . .Id 0

Π1 0 Id Id

0Π1

Π10

, for some positive odd integer d. In

particular for t = 2 we get the matrix

(Π

[ d2]+1

1 I I

Π1 0 0Π1 0 0

). In either situation we obtain

the the quiver Q given in the proposition.

7. Types and algebras for smallest parameter

In this final section we review the classification of the quivers of the radical cubezero selfinjective algebras with symmetric and non-symmetric contracted matricesgiven in Section 5 and 6. Recall that we defined the Nakayama cycle type of such analgebra, and here we abbreviate this by just saying cycle type. For each symmetriccontracted matrix F the cycle type is (d, d, . . . , d), so we only list the type of thealgebra. For each non-symmetric contracted matrix F we list the cycle type andthe type, and we draw the quiver for the smallest possible cycle type. To this endwe make the following definition of the type of the algebra.


Definition 7.1. For each of the algebra in question, we have an associated cycletype, which is the cycle type of the Nakayama permutation Π. Furthermore, onecan easily see that for all components of the separated quiver, the underlying graphsare the same. We call this graph the type of the algebra.

First we list the type of algebras with symmetric contracted matrix consideredin Section 5. We leave the details to the reader.

Proposition Type

5.1 Ast−1 when s | d and any s occurs.

A2a+1st−1 when d = 2ad1 with d1 odd, s | d1, where any s occurs.

5.2 Dt−1

5.3 E6,7,8

5.4 A2t−1 when t > 1.

A2d−1 when t = 1.

5.5 D2t−1

For the non-symmetric cases we also draw the quiver in the smallest possible cycletypes. We make the convention that the vertices drawn in a given column belongto the same cycle, and the cycles come in order from left to right. The review isdone according to the propositions in Section 6.

Proposition 6.1F = ( 0 1

4 0 )

Cycle type(d, 4d), type D4.

1

""EEE

EEEE

EEEE

2

))RRRRRRRRR

5

bbEEEEEEEEEEEiiRRRRRRRRR

uulllllllll

||yyyy

yyyy

yyy

3

55lllllllll

4

<<yyyyyyyyyyy

Proposition 6.2

F =(

0 1 03 0 10 1 0

)F =

(0 3 01 0 10 1 0

)

Cycle type (3d, d, d), type D4. Cycle type (d, 3d, 3d), type E6.

1

((QQQQQQQQQ

2 // 4

hhQQQQQQQQQoo

vvmmmmmmmmm// 5oo

3

66mmmmmmmmm

2

vvmmmmmmmmm

((QQQQQQQQQ 5oo

1

66mmmmmmmmm //

((QQQQQQQQQ 3oo

((QQQQQQQQQ 6oo

4

hhQQQQQQQQQ

==7oo


Proposition 6.3

F =

(0 11 0 12 0 11 0 11 0

)F =

(0 11 0 21 0 11 0 11 0

)

Cycle type (2d, 2d, d, d, d), type E6. Cycle type (d, d, 2d, 2d, 2d), type E7.

1

777

7777

7 3oo

$$JJJJJ

5

ddJJJJJ

zzttttt// 6oo // 7oo

2

CC4oo

::ttttt

3

zzttttt

777

7777

7 5oo // 7

1 // 2oo

::ttttt

$$JJJJJ

4

ddJJJJJ

CC6oo // 8

[[77777777

Proposition 6.4, type At

First family, first two terms.

F =(

0 2 01 0 20 1 0

)F =

(0 2 0 01 0 1 00 1 0 20 0 1 0

)

Cycle type (d, 2d, 4d), type D6. Cycle type (d, 2d, 2d, 4d), type D8.

4

vvmmmmmmmmm

2

777

7777

7777

777

vvmmmmmmmmm// 5

1

66mmmmmmmmm

((QQQQQQQQQ

3

hhQQQQQQQQQ

CC// 6

aaCCCCCCCCCCC

7

hhQQQQQQQQQ

6

vvmmmmmmmmm

2

vvmmmmmmmmm

!!CCC

CCCC

CCCC

4

777

7777

7777

777

oo // 7

1

66mmmmmmmmm

((QQQQQQQQQ

3

hhQQQQQQQQQ

==5oo

CC// 8

aaCCCCCCCCCCC

9

hhQQQQQQQQQ

In general, for cycle type (d, 2d, . . . , 2d, 4d) with t disjoint cycles the algebra has

type D2t.

Second family, first two terms.

F =(

0 1 02 0 20 1 0

)F =

(0 12 0 11 0 21 0

)

Cycle type (2d, d, 2d), type D4. Cycle type (2d, d, d, 2d), type D5.

1

!!CCC

C 4

3

aaCCCC

==

!!CCC

C

2

==5

aaCCCC

1

!!CCC

C 5

3

aaCCCC

// 4

==

!!CCC

Coo

2

==6

aaCCCC

The Nakayama permutation permutes the simple modules at both ends of thequiver.

In general, for cycle type (d, 2d, . . . , 2d, d) with t disjoint cycles the algebra has type

Dt+1.

Third family, first two terms.

F =(

0 2 01 0 10 2 0

)F =

(0 2 0 01 0 1 00 1 0 10 0 2 0

)

Cycle type (d, 2d, d), type A3. Cycle type (d, 2d, 2d, d), type A5.


2

!!C

CCC

1

==

!!CCC

C 4

aaCCCC

3

aaCCCC==

2

vvmmmmmmmmm

!!CCC

CCCC

CCCC

4oo

((QQQQQQQQQ

1

66mmmmmmmmm

((QQQQQQQQQ 6

hhQQQQQQQQQ

vvmmmmmmmmm

3

hhQQQQQQQQQ

==5oo

66mmmmmmmmm

In general, for cycle type (d, 2d, . . . , 2d, d) with t disjoint cycles the algebra has type

A2t−3.

Proposition 6.5, type Dt.First family.

F =

(0 0 1 00 0 1 01 1 0 10 0 2 0

)F =

(0 0 10 0 11 1 0 1 0

1 0 10 2 0

)

Cycle type (2d, 2d, 2d, d), type D6. Cycle type (2d, 2d, 2d, 2d, d), type D8.

3

///

////

////

1

$$JJJJJJJJJJJJJ 5oo

ddJJJJJ

$$JJJJJ

7

ddJJJJJ

zzttttt

2

::ttttttttttttt6oo

zzttttt

::ttttt

4

GG

3

///

////

////

1

$$JJJJJJJJJJJJJ 5oo

ddJJJJJ

777

7777

7 7oo

$$JJJJJ

9

ddJJJJJ

zzttttt

2

::ttttttttttttt6oo

zzttttt

CC8oo

::ttttt

4

GG

In general, for cycle type (2d, 2d, . . . , 2d, d) with t disjoint cycles the algebra has

type D2t−2.

Second family.

F =

(0 0 1 00 0 1 01 1 0 20 0 1 0

)F =

(0 0 10 0 11 1 0 1 0

1 0 20 1 0

)

Cycle type (d, d, d, 2d), type D4 Cycle type (d, d, d, d, 2d), type D5

1

))RRRRRRRRR 4

||xxxx

3

iiRRRRRRRRR

||xxxx

<<xxxx

""FFF

F

2

<<xxxx5

bbFFFF

1

++VVVVVVVVVVVVVVV 5

xxqqqqqqq

3

kkVVVVVVVVVVVVVVV

xxqqqqqqq// 4oo

88qqqqqqq

&&MMMMMMM

2

88qqqqqqq6

ffMMMMMMM

The Nakayama permutation permutes the simple modules at one end of the quiverbut not at the other end.In general, for cycle type (d, d, . . . , d, 2d) with t disjoint cycles the algebra has type

Dt.


Proposition 6.6, type Zt

First two terms.

F = ( 1 21 0 ) F =

(1 1 01 0 20 1 0

)

Cycle type (d, 2d), type D5. Cycle type (d, d, 2d), type D7.

2

1::

==

!!CCC

C

3

aaCCCC

3

1::// 2oo

==

!!CCC

C

4

aaCCCC

In general, for cycle type (d, d, . . . , d, 2d) with t disjoint cycles the algebra has type

D2t+1.

References

[1] Auslander, M., Reiten, I. and Smalø, S. O., Representation theory of artin algebras, Cam-bridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995.

[2] Benson, D., Resolutions over symmetric algebras with radical cube zero, J. Algebra 320(2008), no. 1, 48–56.

[3] Dugas, A., Periodic resolutions and self-injective algebras of finite type, J. Pure Appl. Alg.,vol. 214, no. 6 (2010), 990–1000.

[4] Erdmann, K., Holloway, M., Snashall, N., Solberg, Ø., Taillefer, R., Support varieties for

selfinjective algebras, K-Theory, vol. 33, no. 1 (2004), 67–87.[5] Erdmann, K., Solberg, Ø., Radical cube zero weakly symmetric algebras and support varieties,

preprint 2009, J. Pure Appl. Alg., to appear.[6] Godsil, C., Royle, G., Algebraic graph theory, Graduate Texts in Mathematics, 207. Springer-

Verlag, New York, 2001. xx+439 pp.[7] Guo, J. Y., Translation algebras and their applications, J. Algebra 255 (2002), no. 1, 1–21.[8] Martınez-Villa, R., Applications of Koszul algebras: the preprojective algebra, Representa-

tion theory of algebras (Cocoyoc, 1994), 487–504, CMS Conf. Proc., 18, Amer. Math. Soc.,Providence, RI, 1996.

[9] Martinez-Villa, R., Graded, selfinjective, and Koszul algebras, J. Algebra 215 (1999), no. 1,34–72.

[10] Mori, I., An introduction to noncommutative algebraic geometry, Proceedings of the 40thSymposium on Ring Theory and Representation Theory, 53–59, Symp. Ring Theory Repre-sent. Theory Organ. Comm., Yamaguchi, 2008.

[11] Smith, S. P., Some finite-dimensional algebras related to elliptic curves, Representation the-

ory of algebras and related topics (Mexico City, 1994), 315–348, CMS Conf. Proc., 19, Amer.Math. Soc., Providence, RI, 1996.

[12] Snashall, N., Solberg, Ø., Support varieties and Hochschild cohomology rings, Proc. London

Math. Soc. (3) 88 (2004), no. 3, 705–732.[13] Snashall, N., Taillefer, R., The Hochschild cohomology ring of a class of special biserial

algebras, J. Alg. Its Appl., vol. 9, no. 1, (2010), 73–122.[14] Zhang, F., Matrix theory. Basic results and techniques. Universitext. Springer-Verlag, New

York, 1999. xiv+277 pp. ISBN: 0-387-98696-0

Karin Erdmann, Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, England

E-mail address: [email protected]

Øyvind Solberg, Institutt for matematiske fag, NTNU, N–7491 Trondheim, Norway

E-mail address: [email protected]

radical cube zero selfinjective algebras of · radical cube zero selfinjective algebras of finite...

Documents