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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.
4 ERDMANN AND SOLBERG
(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
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 5
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
6 ERDMANN AND SOLBERG
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
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 7
(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
8 ERDMANN AND SOLBERG
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 : • • • •
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 9
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.
10 ERDMANN AND SOLBERG
(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,
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 11
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
,
12 ERDMANN AND SOLBERG
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
,
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 13
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.
14 ERDMANN AND SOLBERG
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
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 15
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
16 ERDMANN AND SOLBERG
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
llXXXXXXXXXXXXXXXXXXXXXX
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
,,XXXXXXXXXXXXXXXXXXXXXXX 24 · · · (d− 1)4
tthhhhhhhhhhh
;;wwww
,,XXXXXXXXXXXXXXXXXXXX(d)4
tthhhhhhhhhhhhhh
llXXXXXXXXXXXXXXXXXXXXXX
wwwww
13
33ggggggggggggggggggg
23
44iiiiiiiiiiiiiii
· · · (d− 1)3
22ffffffffffffffffffff
(d)3
ccGGGGG
15
jjVVVVVVVVVVVVVVVV
##GGGGG 25
jjVVVVVVVVVVVVVVVV · · · (d− 1)5
jjVVVVVVVVVVV
##GGGG
(d)5
jjVVVVVVVVVVVVVV
rrffffffffffffffffffffff
12
DD
22
· · · (d− 1)2
;;wwww
(d)2
kkWWWWWWWWWWWWWWWWWW
16
OO
##GGGGG 26
OO
· · · (d− 1)6
OO
##GGGG
(d)6
OO
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
17
;;wwwww27
· · · (d− 1)7
;;wwww(d)7
llXXXXXXXXXXXXXXXXXXXXXX
16
;;wwwww26
· · · (d− 1)6
;;wwww(d)6
llXXXXXXXXXXXXXXXXXXXXXX
15
;;wwwww25
· · · (d− 1)5
;;wwww(d)5
llXXXXXXXXXXXXXXXXXXXXXX
14
;;wwwww24
· · · (d− 1)4
;;wwww(d)4
llXXXXXXXXXXXXXXXXXXXXXX
13
;;wwwww
uukkkkkkkkkkkk
,,XXXXXXXXXXXXXXXXXXXXXXX 23 · · · (d− 1)3
tthhhhhhhhhhh
;;wwww
,,XXXXXXXXXXXXXXXXXXXX(d)3
tthhhhhhhhhhhhhh
llXXXXXXXXXXXXXXXXXXXXXX
wwwww
12
33ggggggggggggggggggg
22
44iiiiiiiiiiiiiii
· · · (d− 1)2
22ffffffffffffffffffff
(d)2
ccGGGGG
19
jjVVVVVVVVVVVVVVVV29
jjVVVVVVVVVVVVVVVV · · · (d− 1)9
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
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 17
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.
18 ERDMANN AND SOLBERG
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.
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 19
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
20 ERDMANN AND SOLBERG
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
ttiiiiiiiiiiiiiiiiiiiii
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 +
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 21
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
ttiiiiiiiiiiiiiiiiiiiii
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
22 ERDMANN AND SOLBERG
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.
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 23
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
24 ERDMANN AND SOLBERG
(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
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 25
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,
26 ERDMANN AND SOLBERG
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.
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 27
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
28 ERDMANN AND SOLBERG
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.
RADICAL CUBE ZERO SELFINJECTIVE ALGEBRAS 29
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.
30 ERDMANN AND SOLBERG
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.
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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-
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[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
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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]