Nicole Snashall
email: n.snashall@mcs.le.ac.uk

Rachel Taillefer
email: rachel.taillefer@math.univ-bpclermont.fr

Clermont Universit

CLASSIFICATION OF SYMMETRIC SPECIAL BISERIAL ALGEBRAS WITH AT MOST ONE NON-UNISERIAL INDECOMPOSABLE PROJECTIVE

Keywords: 2010 Mathematics Subject Classification. Primary: 16D50, 16E35. Secondary: 16E40 Special biserial algebra, derived equivalence, stable equivalence of Morita type, generalised Brauer tree algebra

, where they study the weakly symmetric algebras of Euclidean type, as well as some algebras of dihedral type.

INTRODUCTION

Let K be an algebraically closed field. We consider in this paper a generalisation of symmetric Nakayama K-algebras. A symmetric Nakayama K-algebra is a symmetric K-algebra A such that all indecomposable projective modules are uniserial. These algebras are well-known and have been classified up to Morita equivalence: every symmetric Nakayama algebra is Morita equivalent to exactly one algebra N n m defined by the quiver

∆ n = • a 3 / / • a 4 & & M M M M • a 2 8 8 q q q q • . . . • a 1 E E • a n Y Y 3 3 3 3 a n-3
x x q q q q q • a n-1

f f M M M M • a n-2 o o
and the ideal of relations L m in K∆ n generated by all paths of length nm + 1. Note that in particular, the basic algebra associated to A is special biserial.

Our aim is to describe the basic indecomposable finite-dimensional K-algebras A which are symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. These algebras include the symmetric Nakayama algebras, certain algebras in [START_REF] Bocian | Derived equivalence classification of weakly symmetric algebras of Euclidean type[END_REF] that occur in the classification, up to derived equivalence, of all weakly symmetric algebras of Euclidean type, as well as some algebras of dihedral type, see [START_REF] Erdmann | Blocks of tame representation type and related algebras[END_REF]. In this paper we also distinguish, up to derived equivalence and up to stable equivalence of Morita type, the basic indecomposable finite-dimensional symmetric special biserial algebras which have at most one non-uniserial indecomposable projective module. It is well-known that all special biserial algebras are tame [START_REF] Wald | Tame biserial algebras[END_REF]. Morever, it was proved by Al-Nofayee in [START_REF] Al-Nofayee | Equivalences of derived categories for selfinjective algebras[END_REF] (and by Rickard [START_REF] Rickard | Derived equivalences as derived functors[END_REF] for the symmetric case) that if A and B are derived equivalent algebras, then A is selfinjective if and only if B is selfinjective. It was also proved by Pogorzały [21] that if A is a selfinjective special biserial algebra that is not a Nakayama algebra and if A and B are stably equivalent of Morita type, then B is also a selfinjective special biserial algebra. The algebras given in [START_REF] Bocian | Derived equivalence classification of weakly symmetric algebras of Euclidean type[END_REF] are Brauer graph algebras, and we recall that Brauer tree algebras play an important role in the Morita equivalence classification of blocks of group algebras of finite type (see [START_REF] Alperin | Local representation theory. Modular representations as an introduction to the local representation theory of finite groups[END_REF][START_REF] Benson | Representations and cohomology. I. Basic representation theory of finite groups and associative algebras[END_REF]). We use the theory of generalised Brauer tree algebras as part of the classification of our algebras up to derived equivalence. We refer the reader also to [START_REF] Skowro Ński | Selfinjective algebras: finite and tame type[END_REF], where Skowro ński discusses the extensive programme to determine the derived equivalence classes of all tame selfinjective algebras.

We begin this paper with some background and properties about basic symmetric algebras, so that, in Section 2, we can describe by quiver and relations all basic indecomposable finite-dimensional algebras which are symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. In order to distinguish our algebras up to derived equivalence and up to stable equivalence of Morita type, we use several invariants including Hochschild cohomology which we discuss in Section 3. Section 4 contains the full classification of our algebras up to derived equivalence, and in addition to the dimensions of the Hochschild cohomology groups, we use Cartan invariants (see [START_REF] Bocian | Weakly symmetric algebras of Euclidean type[END_REF]Proposition 1.5] for a proof of derived invariance) and K ülshammer invariants (or generalised Reynolds ideals, whose derived invariance was proved in [START_REF] Zimmermann | Invariance of generalized Reynolds ideals under derived equivalences[END_REF]). The final section gives the full classification of our algebras up to stable equivalence of Morita type, based on the classification up to derived equivalence of Section 4 and using similar invariants.

We assume throughout that A is a basic indecomposable finite-dimensional algebra over the algebraically closed field K so that A is isomorphic to KQ/I for some unique connected quiver Q and admissible ideal I of KQ. We let rad(A) denote the Jacobson radical of A.

For any two positive integers p and q with p q, we define the quiver Q (p,q) to be the quiver formed of two oriented cycles, of lengths p and q respectively, joined at one vertex labelled 1, as follows:

• α 4 y y s s s s

• α 3 o o • β 3 / / • β 4 & & M M M M . . . . • α 2 f f M M M M • β 2 8 8 q q q q • . . . Q (p,q) : 1 p q 1 • α 1 X X 0 0 0 0 β 1 F F α p-3 % % K K K K • α p F F • β q X X 0 0 0 0 0 β q-3 x x q q q q q • α p-2 / / • α p-1 8 8 q q q q • β q-1 f f M M M M • β q-2 o o
We denote the trivial path at the vertex i by e i . Paths are written from left to right. We write o(α) for the trivial path corresponding to the origin of the arrow α and write t(α) for the trivial path corresponding to the terminus of the arrow α. The vertices of the quiver Q (p,q) are labelled by 1, . . . , p + q -1, in such a way that o(α i ) = i for i = 1, . . . , p, and t(β j ) = p + j for j = 1, . . . , q -1. Thus t(α

i ) = i + 1 for i = 1, . . . , p -1, t(α p ) = 1, o(β 1 ) = 1 and o(β j ) = p + j -1 for j = 2, . . . , q. Set γ = α 1 α 2 • • • α p and δ = β 1 β 2 • • • β q .
We define the following two admissible ideals in KQ (p,q) : (a) for a positive integer r, let I r be the ideal generated by

α p α 1 , β q β 1 , (γδ) r -(δγ) r , α i • • • α p (δγ) r-1 δα 1 • • • α i for all 2 i p -1, β j • • • β q (γδ) r-1 γβ 1 • • • β j for all 2 j q -1;
(b) for a pair of positive integers (s, t), let J (s,t) be the ideal generated by

α p β 1 , β q α 1 , γ s -δ t , α i • • • α p γ s-1 α 1 • • • α i for all 2 i p -1, β j • • • β q δ t-1 β 1 • • • β j for all 2 j q -1,
where, if p = 1 then s 2, and, if q = 1 then p = 1, s 2 and t 2.

The algebras considered in [START_REF] Bocian | Derived equivalence classification of weakly symmetric algebras of Euclidean type[END_REF] are special cases of these algebras. Specifically A(p, q) = KQ (p,q) /I 1 so that r = 1, and Λ(n) = KQ (1,n) /J (2,2) so that p = 1, q = n and s = 2 = t. Moreover, some of these algebras are derived equivalent to algebras of dihedral type (see [START_REF] Erdmann | Blocks of tame representation type and related algebras[END_REF]) in the classification of Holm [START_REF] Holm | Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type[END_REF]: KQ (1,1) /I r = D(1A) r 1 , KQ (1,2) /I r that is derived equivalent to D(2B) 1,r (0), and KQ (2,2) /I r that is derived equivalent to D(3K) r,1,1 , all three of which come from tame blocks of finite groups when char(K) = 2 and r is a power of 2, as well as KQ (2,2) /J (s,t) that is derived equivalent to D(2R) 1,s,t,1 and which does not come from blocks, see [START_REF] Holm | Derived equivalent tame blocks[END_REF][START_REF] Holm | Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type[END_REF][START_REF] Linckelmann | A derived equivalence for blocks with dihedral defect groups[END_REF].

BACKGROUND

The following result and especially its consequences will be used repeatedly. They are given in [START_REF] Antipov | The Yoneda algebras of symmetric special biserial algebras are finitely generated[END_REF], but we include the proofs here for completeness. Proposition 1.1. Let Q be a quiver and let I be an admissible ideal in KQ such that A = KQ/I is a symmetric algebra. Let ρ be a path in Q with ρ = 0 in A. Then there exists a cycle ρρ 1 in Q with ρρ 1 and ρ 1 ρ non-zero in A.

Proof. Since A is a symmetric algebra there exists a symmetric form f : A → K on A whose kernel contains no non-zero left or right ideals of A. Then ρA is not contained in Ker f so there exists a path ρ 1 such that f (ρρ 1 ) = 0. In particular, ρρ 1 = 0 and t(ρ) = o(ρ 1 ). Moreover, since f is symmetric, f (ρ 1 ρ) = f (ρρ 1 ) = 0 so ρ 1 ρ = 0. Therefore t(ρ 1 ) = o(ρ). Hence ρρ 1 and ρ 1 ρ are cycles in Q which are non-zero in A.

Corollary 1.2.

Let A = KQ/I be an indecomposable finite-dimensional symmetric special biserial algebra that is not isomorphic to N

1 1 ∼ = K[X]/(X 2 ). Then:
(1) For any arrow α, there is a unique arrow α ′ such that α ′ α = 0 and a unique arrow α ′′ such that αα ′′ = 0. (2) For any vertex v in Q, the number of arrows that start at v is equal to the number of arrows that end at v, and this number is either 1 or 2.

Proof. The second statement follows easily from the first and the definition of a special biserial algebra. Here we prove the first statement. First suppose that α is an arrow which is not a loop. Then α is a non-zero path, so, by Proposition 1.1, there exists a path ρ = α such that αρ and ρα are non-zero cycles. Therefore we can take α ′ to be the last arrow in ρ and α ′′ the first arrow in ρ. The uniqueness of these arrows follows from the definition of a special biserial algebra. Now suppose that α is a loop. Assume for contradiction that αβ = 0 for every arrow β in Q. Then α is in the socle of the indecomposable projective module o(α)A. If no other arrow starts at o(α) then, since A is indecomposable and α 2 = 0, we get A ∼ = K[X]/(X 2 ), a contradiction. Therefore there is another arrow ρ with o(α) = o(ρ). Choose a path σ which is maximal with the property ρσ = 0 so that ρσ is in the socle of o(α)A. Since A is a self-injective algebra, soc(o(α)A) is one-dimensional so that there exists a non-zero c ∈ K such that α = cρσ, which is a contradiction since the ideal I is admissible. Therefore there exists an arrow α ′ with α ′ α = 0. The proof of the existence of the other arrow α ′′ is similar.

CLASSIFICATION THEOREM

Our main theorem in this section is Theorem 2.2 where we classify, by quiver and relations, all basic indecomposable finite-dimensional symmetric special biserial algebras with at most one non-uniserial indecomposable projective module.

Proposition 2.1. The algebras KQ (p,q) /I r and KQ (p,q) /J (s,t) are symmetric special biserial algebras with at most one non-uniserial indecomposable projective module.

Proof. It is easy to see that these algebras are special biserial and that all but one of each of their indecomposable projective modules are uniserial. Moreover, the algebras are weakly symmetric, that is, the top and the socle of each indecomposable projective module are isomorphic. It remains to prove that the algebras are symmetric.

For A = KQ (p,q) /I r , the socle of A has a K-basis consisting of (δγ) r ,

α i • • • α p (δγ) r-1 α 1 • • • α i-1 and β j • • • β q (γδ) r-1 β 1 • • • β j-1 for 2
i p and 2 j q, that is, all the paths obtained from cyclic permutations of (γδ) r , where we recall that

γ = α 1 α 2 • • • α p and δ = β 1 β 2 • • • β q .
Complete this K-basis of soc(A) with paths in KQ (p,q) to obtain a basis of A, and define f : A → K on this basis by sending the elements in the socle to 1 and the others to 0. Then it follows from [13, Proposition 3.1], that Ker f contains no non-zero left or right ideals of A. Moreover, f is clearly symmetric, since the paths on which it is non-zero are all the cyclic permutations of a single path. Thus A is a symmetric algebra.

For A = KQ (p,q) /I (s,t) the argument is similar, but this time the socle of A is generated by all the cyclic permutations of the two paths γ s and δ t .

We now have the following result.

Theorem 2.2. Let A be a basic indecomposable finite-dimensional symmetric special biserial algebra with at most one non-uniserial indecomposable projective module. Then A is isomorphic to a Nakayama algebra N n

m for some positive integers m and n, or to KQ (p,q) /I r for some positive integers p, q and r, or to KQ (p,q) /J (s,t) for some positive integers p, q, s and t.

Proof. Set A = KQ/I for some quiver Q and some admissible ideal I. It is already known that a symmetric (special biserial) algebra with no non-uniserial indecomposable projective module is isomorphic to a Nakayama algebra N n m . We may therefore assume that all except one indecomposable projective A-modules are uniserial. Consequently, using Corollary 1.2(2), we must have one vertex that is the end point of exactly two arrows and the starting point of exactly two arrows, which we label 1, and the other vertices are the end point of exactly one arrow and the starting point of exactly one arrow. Therefore the quiver of A must be Q (p,q) for some positive integers p, q. Without loss of generality we may assume that p q. Now consider the composition α p α 1 . There are two cases: α p α 1 = 0 and α p α 1 = 0. First assume that α p α 1 = 0. Then it follows from Corollary 1.2(1) that α p β 1 = 0, β q α 1 = 0 and β q β 1 = 0. Now, for each vertex k with k = 1, e k A is uniserial and, since A is symmetric, top (e k A) ∼ = soc (e k A) ∼ = S k , the simple module at k. Therefore, for i = 1, we get a relation

α i • • • α p (δγ) u i δα 1 • • • α i = 0 with α i • • • α p (δγ) u i δα 1 • • • α i-1 = 0 for
some integer u i , and, for j = 1, we get a relation

β j • • • β p (γδ) v j γβ 1 • • • β j = 0 with β j • • • β p (γδ) v j γβ 1 • • • β j-1 = 0 for some integer v j . Now consider e 1 A. Since rad(e 1 A) = α 1 A + β 1 A and soc(e 1 A) ∼ = S 1 ,
there is an element in soc(e 1 A) of the form (γδ) r or (γδ) r γ for some integer r, and there is an element of soc(e 1 A) of the form (δγ) s or (δγ) s δ for some integer s. But, soc(e 1 A) is simple so we must have a relation of one of the following forms:

(i) (γδ) r γ = c(δγ) s = 0 for some non-zero c ∈ K. Note that since I is admissible we must have s 1. But then, if r s, we would have (γδ) r γ = γ(δγ) s (δγ) r-s = c -1 γ(γδ) r γ(δγ) r-s = 0, which is a contradiction, and if r < s we would have (δγ) s = δ(γδ) s-1-r (γδ) r γ = cδ(γδ) s-1-r (δγ) s = 0, which is also a contradiction. Therefore we cannot have this type of relation. (ii) (γδ) r = c(δγ) s δ for some non-zero c ∈ K. As in the previous case, this relation cannot occur. (iii) (γδ) r γ = c(δγ) s δ = 0 for some non-zero c ∈ K. Then, if r > s, we have (γδ

) r γ = γ(δγ) r = γ(δγ) s δ(γδ) r-1-s γ = c -1 γ(γδ) r γ(γδ) r-1-s γ = 0,
which is a contradiction, and if s > r we have a similar contradiction. Therefore s = r. Now we also have

α 2 • • • α p (δγ) u 2 δα 1 α 2 = 0
so, multiplying on the left by α 1 and on the right by

α 3 • • • α p , we get γ(δγ) u 2 +1 = 0 so that u 2 + 1 > r. But α 2 • • • α p (δγ) r δα 1 = c -1 α 2 • • • α p (γδ) r γα 1 = 0 so u 2 < r which is impossible.
Therefore a relation of this form cannot occur either. (iv) (γδ) r = c(δγ) s for some non-zero c ∈ K, and this is the only possible type of relation.

Here again, if r > s, then (γδ

) r = γ(δγ) s (δγ) r-s-1 δ = c -1 γ(γδ) r (δγ) r-s-1 δ = 0 which
is a contradiction and if s > r we get a similar contradiction. Therefore r = s so that the relation is (γδ) r = c(δγ) r for some r 1 and c ∈ K * . Given this relation, we are now able to determine the u i and the v j . Since

α i • • • α p (δγ) r δα 1 • • • α i-1 = c -1 α i • • • α p (γδ) r δα 1 • • • α i-1 = 0, we must have r > u i . Moreover, (γδ) u i +2 = α 1 • • • α i-1 (α i • • • α p (δγ) u i δα 1 • • • α i )α i+1 • • • α p δ = 0 so that u i + 2 > r.
Hence u i = r -1 for all i = 2, . . . , p. Similarly, v j = r -1 for all j = 2, . . . , q. Moreover, we note that the relations α p (δγ) r-1 δα 1 • • • α p = 0 (when i = p) and β p (γδ) r-1 γβ 1 • • • β p = 0 (when j = q) are superfluous, so are not required to give a minimal generating set of the ideal I r .

Finally, we show that c must be equal to 1. Since A is symmetric, there exists a symmetric linear map f : A → K whose kernel does not contain any non-zero left or right ideal. In particular, the socle of A is not contained in Ker f . But, from the relations obtained above, we see that the socle is generated as a K-vector space by all the paths obtained by cyclic permutations of (γδ) r . Since f is symmetric it follows that f ((γδ) r ) = 0. But we have

f ((γδ) r ) = f ((δγ) r ) = f (c(γδ) r ) = c f ((γδ) r ) so that c = 1.
Hence we have shown that A ∼ = KQ (p,q) /I r . Now assume that α p α 1 = 0. Then it follows from Corollary 1.2(1) that α p β 1 = 0, β q α 1 = 0 and β q β 1 = 0. The same methods as in the previous case show that we must have a relation of the form γ s = cδ t for some non-zero c ∈ K and some positive integers s and t (by considering the structure of e 1 A) and relations

α i • • • α p γ s-1 α 1 • • • α i for all 2 i p and β j • • • β q δ t-1 β 1 • • • β j for all 2 j
q (from the structure of the other indecomposable projectives and using the relation γ s = cδ t ). Moreover, since K is algebraically closed, we may replace α 1 by c ′ α 1 , where c ′ is a t-th root of c, and thus we may replace the relation γ s = cδ t by γ s = δ t . Again we may find a minimal set of relations, and so conclude that A ∼ = KQ (p,q) /J (s,t) .

HOCHSCHILD COHOMOLOGY GROUPS

Our aim is now to investigate the derived equivalences among these algebras. It is wellknown that Hochschild cohomology is an invariant under derived equivalence, and this section determines some of the Hochschild cohomology groups of the algebras KQ (p,q) /I r and KQ (p,q) /J (s,t) , so that a full classification up to derived equivalence can be given in Section 4.

Let Γ(p, q; r) = KQ (p,q) /I r and Λ(p, q; s, t) = KQ (p,q) /J (s,t) . The special cases of the algebras Γ(p, q; 1) and Λ(1, n; 2, 2) were considered in [START_REF] Bocian | Derived equivalence classification of weakly symmetric algebras of Euclidean type[END_REF], where, in their notation, we have A(p, q) = KQ (p,q) /I 1 = Γ(p, q; 1) and

Λ(n) = KQ (1,n) /J (2,2) = Λ(1, n; 2, 2).
We begin by describing HH 0 (A) and HH 1 (A) for the algebras A = Λ(p, q; s, t) and A = Γ(p, q; r). Recall that HH 0 (A) = Z(A), the centre of the algebra A.

3.1. HH 0 (A) and HH 1 (A) for the algebra A = Λ(p, q; s, t). We begin with the algebra Λ(p, q; s, t) = KQ (p,q) /J (s,t) where 1

p q. Recall that γ = α 1 α 2 • • • α p and δ = β 1 β 2 • • • β q . Let γ i = α i • • • α p α 1 • • • α i-1 for i = 1, . . . , p, and let δ j = β j • • • β q β 1 • • • β j-1 for j = 1, . . . , q, so that γ = γ 1 and δ = δ 1 .
Proposition 3.1. Consider the algebra Λ(p, q; s, t) and let 1 p q. Let x = ∑ p i=1 γ i and y = ∑ q j=1 δ j . Then dim HH 0 (Λ(p, q; s, t)) = p + q + s + t -2 and the set 1, x, . . . , x s-1 , y, . . . , y t-1 , γ s i , δ t j for i = 1, . . . , p and j = 2, . . . , q is a K-basis of HH 0 (Λ(p, q; s, t)).

Proof. We note that γ i β j = 0 = β j γ i and δ j α i = 0 = α i δ j for all i = 1, . . . , p, j = 1, . . . , q so x, y ∈ Z(Λ(p, q; s, t)). Moreover Λ(p, q; s, t) is weakly symmetric so all socle elements are central, namely, γ s i and δ t j are central for i = 1, . . . , p, j = 2, . . . , q. The result now follows.

We remark that Proposition 3.1 may be simplified if p = q = 1, as then Λ(1, 1; s, t) is the commutative algebra K[α, β]/(αβ, α sβ t ) with s 2, t 2. Thus HH 0 (Λ(1, 1; s, t)) = Λ(1, 1; s, t) which has K-basis 1, α, . . . , α s , β, . . . , β t-1 and dimension s + t.

In order to compute the first Hochschild cohomology group, we now fix a minimal set of generators of the ideal J (s,t) , and denote this set by g 2 . Proposition 3.2. Consider the algebra Λ(p, q; s, t) with 1 p q.

If p 2 then the following elements form a minimal set of generators for the ideal J (s,t) :

g 2 1 = γ s -δ t g 2 i = α i • • • α p γ s-1 α 1 • • • α i for all 2 i p -1 g 2 p = α p β 1 g 2 p+1 = β q α 1 g 2 p+j = β j • • • β q δ t-1 β 1 • • • β j
for all 2 j q -1. If p = 1 then the following elements form a minimal set of generators for the ideal J (s,t) :

g 2 0 = α s -δ t g 2 1 = αβ 1 g 2 2 = β q α g 2 j+1 = β j • • • β q δ t-1 β 1 • • • β j for all 2 j q -1.
We now compute the first Hochschild cohomology group HH 1 (Λ(p, q; s, t)). We use the explicit description of the start of a minimal projective bimodule resolution (P * , d * ) for Λ(p, q; s, t) as given in [START_REF] Green | Projective bimodule resolutions of an algebra and vanishing of the second Hochschild cohomology group[END_REF]. All tensors are over K so we write ⊗ for ⊗ K . For ease of notation, write Λ for Λ(p, q; s, t).

Let P 2 = k Λo(g 2 k ) ⊗ t(g 2 k )Λ, P 1 = a arrow Λo(a) ⊗ t(a)Λ and P 0 = p+q-1 v=1
Λe v ⊗ e v Λ. Then the minimal projective bimodule resolution of Λ begins

• • • -→ P 2 d 2 -→ P 1 d 1 -→ P 0 d 0 -→ Λ -→ 0
with the following maps. The map d 0 is the usual multiplication map. The map d 1 : P 1 → P 0 is given by

d 1 : o(a) ⊗ t(a) → o(a) ⊗ a -a ⊗ t(a)
where the first term o(a) ⊗ a lies in the summand Λo(a) ⊗ o(a)Λ and the second term a ⊗ t(a) lies in the summand Λt(a) ⊗ t(a)Λ. Now, each element of g 2 is a linear combination of paths in KQ (p,q) so, for x ∈ g 2 , we may write where c j ∈ K and the a kj are arrows in Q (p,q) . With this notation for x ∈ g 2 , the map d 2 : P 2 → P 1 is given by

d 2 : o(x) ⊗ t(x) → r ∑ j=1 c j s j ∑ k=1 a 1j . . . a k-1j ⊗ a k+1j . . . a s j j
where the term a 1j . . . a k-1j ⊗ a k+1j . . . a s j j lies in the summand Λo(a kj ) ⊗ t(a kj )Λ of P 1 .

We now apply Hom Λ e (-, -) to this resolution, where Λ e = Λ ⊗ Λ op is the enveloping algebra of Λ. Let ∂ 1 : Hom Λ e (P 1 , Λ) → Hom Λ e (P 2 , Λ) be the map induced by d 2 and let ∂ 0 : Hom Λ e (P 0 , Λ) → Hom Λ e (P 1 , Λ) be the map induced by d 1 . Then HH 1

(Λ) = Ker ∂ 1 / Im ∂ 0 . Proposition 3.3. If q 2 then dim HH 1 (Λ(p, q; s, t)) = s + t if char K | gcd(s, t) s + t -1 otherwise. If q = 1 then p = 1 and dim HH 1 (Λ(1, 1; s, t)) = s + t + 1 if char K | gcd(s, t) s + t otherwise.
Proof. There are three cases to consider. Case I: p 2

We start by calculating Im ∂ 0 . Let ϕ ∈ Hom(P 0 , Λ(p, q; s, t)) so that ∂ 0 (ϕ) = ϕd 1 . Suppose that ϕ is given by ϕ :

e 1 ⊗ e 1 → c 1,0 e 1 + c 1,1 γ 1 + • • • + c 1,s γ s 1 + d 1,1 δ 1 + • • • + d 1,t-1 δ t-1 1 e i ⊗ e i → c i,0 e i + c i,1 γ i + • • • + c i,s γ s i for i = 2, . . . , p e p-1+i ⊗ e p-1+i → d i,0 e p-1+i + d i,1 δ i + • • • + d i,t δ t i for i = 2, .
. . , q where c i,j , d i,j ∈ K.

We have

ϕd 1 (o(α 1 ) ⊗ t(α 1 )) = ϕ(e 1 ⊗ α 1 -α 1 ⊗ e 2 ) = ϕ(e 1 ⊗ e 1 )α 1 -α 1 ϕ(e 2 ⊗ e 2 ) = (c 1,0 -c 2,0 )α 1 + (c 1,1 -c 2,1 )γ 1 α 1 + • • • + (c 1,s-1 -c 2,s-1 )γ s-1 1 α 1 .
In a similar way we get

ϕd 1 (o(α 2 ) ⊗ t(α 2 )) = ϕ(e 2 ⊗ α 2 -α 2 ⊗ e 3 ) = (c 2,0 -c 3,0 )α 2 + (c 2,1 -c 3,1 )γ 2 α 2 + • • • + (c 2,s-1 -c 3,s-1 )γ s-1 2 α 2 . . . ϕd 1 (o(α p ) ⊗ t(α p )) = ϕ(e p ⊗ α p -α p ⊗ e 1 ) = (c p,0 -c 1,0 )α p + (c p,1 -c 1,1 )γ p α p + • • • + (c p,s-1 -c 1,s-1 )γ s-1 p α p ϕd 1 (o(β 1 ) ⊗ t(β 1 )) = ϕ(e 1 ⊗ β 1 -β 1 ⊗ e p+1 ) = (c 1,0 -d 2,0 )β 1 + (d 1,1 -d 2,1 )δ 1 β 1 + • • • + (d 1,t-1 -d 2,t-1 )δ t-1 1 β 1 . . . ϕd 1 (o(β q ) ⊗ t(β q )) = ϕ(e p+q-1 ⊗ β q -β q ⊗ e 1 ) = (d q,0 -c 1,0 )β q + (d q,1 -d 1,1 )δ q β q + • • • + (d q,t-1 -d 1,t-1 )δ t-1 q β q . Thus dim Im ∂ 0 = (p -1)s + (q -1)t.
Now let ψ ∈ Ker ∂ 1 so that ψd 2 = 0, and suppose that ψ ∈ Hom(P 1 , Λ(p, q; s, t)) is given by

ψ : o(α i ) ⊗ t(α i ) → c i,0 α i + c i,1 γ i α i + • • • + c i,s-1 γ s-1 i α i o(β j ) ⊗ t(β j ) → d j,0 β j + d j,1 δ j β j + • • • + d j,t-1 δ t-1 j β j
for i = 1, . . . , p, j = 1, . . . , q and where c k,l , d k,l ∈ K.

From Proposition 3.2 with p 2, and recalling that γ = γ 1 and δ = δ 1 , it is easy to see that ψd 2

(o(g 2 k ) ⊗ t(g 2 k )
) is immediately zero for k = 2, . . . , p + q -1 and so we do not get any restrictions on the constants in the cases where g 2 k is a monomial. It remains to consider

g 2 1 = γ s -δ t . The condition ψd 2 (o(g 2 1 ) ⊗ t(g 2 1 )) = 0 gives that (s(c 1,0 + c 2,0 + • • • + c p,0 ) -t(d 1,0 + d 2,0 + • • • + d q,0 ))γ s = 0 so that s(c 1,0 + c 2,0 + • • • + c p,0 ) -t(d 1,0 + d 2,0 + • • • + d q,0 ) = 0. Hence dim Ker ∂ 1 = ps + qt if char K | gcd(s, t) ps + qt -1 otherwise.
Thus, for p 2, we have dim HH 1

(Λ(p, q; s, t)) = s + t if char K | gcd(s, t) s + t -1 otherwise.

Case II: p = 1 and q 2

To calculate Im ∂ 0 , let ϕ ∈ Hom(P 0 , Λ(1, q; s, t)) so that ∂ 0 (ϕ) = ϕd 1 . Suppose that ϕ ∈ Hom(P 0 , Λ(1, q; s, t)) is given by

ϕ : e 1 ⊗ e 1 → c 1,0 e 1 + c 1,1 α + • • • + c 1,s α s + d 1,1 δ 1 + • • • + d 1,t-1 δ t-1 1 e i ⊗ e i → d i,0 e i + d i,1 δ i + • • • + d i,t δ t i for i = 2, .
. . , q and where c 1,j , d 1,j , d i,j ∈ K.

We have

ϕd 1 (o(α) ⊗ t(α)) = ϕ(e 1 ⊗ α -α ⊗ e 1 ) = ϕ(e 1 ⊗ e 1 )α -αϕ(e 1 ⊗ e 1 ) = 0 since α ∈ Z(Λ(1, q; s, t)). Now, ϕd 1 (o(β 1 ) ⊗ t(β 1 )) = ϕ(e 1 ⊗ β 1 -β 1 ⊗ e 2 ) = (c 1,0 -d 2,0 )β 1 + (d 1,1 -d 2,1 )δ 1 β 1 + • • • + (d 1,t-1 -d 2,t-1 )δ t-1 1 β 1 . . . ϕd 1 (o(β q ) ⊗ t(β q )) = ϕ(e q ⊗ β q -β q ⊗ e 1 ) = (d q,0 -c 1,0 )β q + (d q,1 -d 1,1 )δ q β q + • • • + (d q,t-1 -d 1,t-1 )δ t-1 q β q . Thus dim Im ∂ 0 = (q -1)t.
Now let ψ ∈ Ker ∂ 1 so that ψd 2 = 0, and suppose that ψ ∈ Hom(P 1 , Λ(1, q; s, t)) is given by

ψ : o(α) ⊗ t(α) → c 1,0 e 1 + c 1,1 α + • • • + c 1,s α s + c1,1 δ 1 + • • • + c1,t-1 δ t-1 1 o(β i ) ⊗ t(β i ) → d i,0 β i + d i,1 δ i β i + • • • + d i,t-1 δ t-1 i β i
for i = 1, . . . , q and where c 1,j , c1,j , d i,j ∈ K.

From Proposition 3.2, the minimal generating set for J (s,t) is

{g 2 0 = α s -δ t , g 2 1 = αβ 1 , g 2 2 = β q α, g 2 j+1 = β j • • • β q δ t-1 β 1 • • • β j | 2 j q -1} where we recall that δ = δ 1 . Starting with g 2 1 , the equation ψd 2 (o(g 2 1 ) ⊗ t(g 2 1 )) = 0 gives that 0 = ψ(o(α) ⊗ t(α))β 1 + αψ(o(β 1 ) ⊗ t(β 1 )) = (c 1,0 e 1 + c1,1 δ 1 + • • • + c1,t-1 δ t-1 1 )β 1 . Hence c 1,0 = c1,1 = • • • = c1,t-1 = 0.
So we may immediately simplify our expression for ψ as

ψ : o(α) ⊗ t(α) → c 1,1 α + • • • + c 1,s α s o(β i ) ⊗ t(β i ) → d i,0 β i + d i,1 δ i β i + • • • + d i,t-1 δ t-1 i β i
for i = 1, . . . , q. It then follows that ψd 2 (o(g 2 j ) ⊗ t(g 2 j )) is zero for j = 2, . . . , q and so we do not get any restrictions on the constants here. Finally,

0 = ψd 2 (o(g 2 0 ) ⊗ t(g 2 0 )) = (sc 1,1 -t(d 1,0 + d 2,0 + • • • + d q,0 ))α s so that sc 1,1 -t(d 1,0 + d 2,0 + • • • + d q,0 ) = 0. Hence dim Ker ∂ 1 = s + qt if char K | gcd(s, t) s + qt -1 otherwise.
Thus, for q 2, we have dim HH 1

(Λ(1, q; s, t)) = s + t if char K | gcd(s, t) s + t -1 otherwise.

Case III: p = 1 = q

To calculate Im ∂ 0 , let ϕ ∈ Hom(P 0 , Λ(1, 1; s, t)) so that ∂ 0 (ϕ) = ϕd 1 . We have

ϕd 1 (o(α) ⊗ t(α)) = ϕ(e 1 ⊗ α -α ⊗ e 1 ) = ϕ(e 1 ⊗ e 1 )α -αϕ(e 1 ⊗ e 1 ) = 0 since Λ(1, 1; s, t) is commutative. Similarly ϕd 1 (o(β) ⊗ t(β)) = 0. Thus Im ∂ 0 = (0). Hence HH 1 (Λ(1, 1; s, t)) = Ker ∂ 1 .
Let ψ ∈ Ker ∂ 1 so that ψd 2 = 0, and suppose that ψ ∈ Hom(P 1 , Λ(1, 1; s, t)) is given by

ψ : o(α) ⊗ t(α) → c 1,0 e 1 + c 1,1 α + • • • + c 1,s α s + d 1,1 β + • • • + d 1,t-1 β t-1 o(β) ⊗ t(β) → c 2,0 e 1 + c 2,1 α + • • • + c 2,s α s + d 2,1 β + • • • + d 2,t-1 β t-1 where c i,j , d i,j ∈ K.
From Proposition 3.2, the minimal generating set for

J (s,t) is {g 2 0 = α s -β t , g 2 1 = αβ, g 2 2 = βα}. Starting with g 2 1 , the equation ψd 2 (o(g 2 1 ) ⊗ t(g 2 1 )) = 0 gives that c 1,0 β + d 1,1 β 2 + • • • + d 1,t-2 β t-1 + c 2,0 α + c 2,1 α 2 + • • • + c 2,s-2 α s-1 + (d 1,t-1 + c 2,s-1 )α s = 0. Hence c 1,0 = d 1,1 = • • • = d 1,t-2 = c 2,0 = c 2,1 = • • • = c 2,s-2 = 0 and d 1,t-1 + c 2,s-1 = 0.
So we may simplify our expression for ψ as

ψ : o(α) ⊗ t(α) → c 1,1 α + • • • + c 1,s α s + d 1,t-1 β t-1 ψ : o(β) ⊗ t(β) → -d 1,t-1 α s-1 + c 2,s α s + d 2,1 β + • • • + d 2,t-1 β t-1 .
We then have that

0 = ψd 2 (o(g 2 0 ) ⊗ t(g 2 0 )) = sc 1,1 α s -td 2,1 β t = (sc 1,1 -td 2,1
)α s and hence sc 1,1td 2,1 = 0. The final equation ψd 2 (o(g 2 0 ) ⊗ t(g 2 0 )) = 0 gives no new information. Hence ψ ∈ Ker ∂ 1 is given by

ψ : o(α) ⊗ t(α) → c 1,1 α + • • • + c 1,s α s + d 1,t-1 β t-1 ψ : o(β) ⊗ t(β) → -d 1,t-1 α s-1 + c 2,s α s + d 2,1 β + • • • + d 2,t-1 β t-1
with the additional linear dependency that sc 1,1td 2,1 = 0. Therefore dim HH 1

(Λ(1, 1; s, t)) = dim Ker ∂ 1 = s + t + 1 if char K | gcd(s, t) s + t otherwise.
This completes the proof.

3.2. HH 0 (A) and HH 1 (A) for the algebras A = Γ(p, q; r). We now turn to the algebras Γ(p, q; r) = KQ (p,q) /I r . Set

η i = α i • • • α p δα 1 • • • α i-1 for 1 i p (so that η 1 = γδ) and θ j = β j • • • β q γβ 1 • • • β j-1 for 1 j q (so that θ 1 = δγ).
Proposition 3.4. Let p q be positive integers and consider the algebra Γ(p, q; r). Set z = ∑

p i=1 η i + ∑ q j=1 θ j . Then dim HH 0 (Γ(p, q; r)) =      p + q + r -1 if p > 1 q + r + 1 if p = 1 and q > 1 r + 3 if p = 1 = q,
and a basis for HH 0 (Γ(p, q; r)) is given by 1; (γδ) r ; z k , 1 k r -1; η r i ; θ r j for 2 i p and 2 j q if p > 1 1; (γδ) r ; z k , 1 k r -1; (δγ) r-1 δ; θ r j for 2 j q if p = 1 and q > 1 1

; (γδ) r ; z k , 1 k r -1; (γδ) r-1 γ; (δγ) r-1 δ if p = 1 = q.
Proof. It is clear that (γδ) r , η r i and θ r j are in the centre of Γ(p, q; r) since they are socle elements, and it is easy to check that (γδ) r-1 γ and (δγ) r-1 δ are in the centre in the appropriate cases.

Conversely, a central element ζ must be in p+q-1 v=1 e v Γ(p, q; r)e v and therefore is a linear combination of (γδ) k for 0 k r, (δγ) k for 1 k r -1, (γδ) k γ and (δγ) k δ for 0 k r -1, and η k i and θ k j for 2 i p, 2 j q and 1 k r. Writing the equations α i ζ = ζα i and β j ζ = ζβ j gives the result (noting that

η k i = α i • • • α p (δγ) k-1 δα 1 • • • α i-1 and θ k j = β j • • • β q (γδ) k-1 γβ 1 • • • β j-1 ).
We now use the same method as for Λ(p, q; s, t) to compute HH 1 (Γ(p, q; r)), starting with a minimal set g 2 of generators of I r . Proposition 3.5. Consider the algebra Γ(p, q; r) with 1 p q.

If p 2 then the following elements form a minimal set of generators for the ideal I r :

g 2 1 = (γδ) r -(δγ) r g 2 i = η r i α i for all 2 i p -1 g 2 p = α p α 1 g 2 p+1 = β q β 1 g 2 p+j = θ r j β j for all 2 j q -1.
If p = 1 then the following elements form a minimal set of generators for the ideal I r :

g 2 0 = (γδ) r -(δγ) r g 2 1 = α 2 g 2 2 = β q β 1 g 2
j+1 = θ r j β j for all 2 j q -1. The proof of the next result giving the first Hochschild cohomology group is a similar calculation to that of Proposition 3.3 and so is left to the reader. Proposition 3.6. If p 2 then dim HH 1 (Γ(p, q; r)) = r + 1.

If p = 1 and q 2 then dim HH 1

(Γ(1, q; r)) = r + 4 if char K = 2 r + 2 if char K = 2.
If q = 1 then p = 1 and

dim HH 1 (Γ(1, 1; r)) = 2r + 6 if char K = 2 2r + 2 if char K = 2.
3.3. Higher Hochschild cohomology groups for Γ(p, q; r). In order to distinguish the algebras of the form Γ(p, q; r) up to derived equivalence we need the dimensions of the Hochschild cohomology groups up to HH 2p-2 (Γ(p, q; r)). If p = 1, this is just HH 0 (Γ(1, q; r)) which we already know, so we shall assume that p > 1 in the remainder of this section. We begin by giving the start of a projective bimodule resolution of Γ(p, q; r) to enable us to find these groups.

For ease of notation, set Γ = Γ(p, q; r). The projective bimodules P n in a minimal projective bimodule resolution of Γ are known from [START_REF] Happel | Hochschild cohomology of finite-dimensional algebras[END_REF]; specifically, the multiplicity of Γe i ⊗ e j Γ as a direct summand of P n is equal to the dimension of Ext n Γ (S i , S j ), where S k is the simple module at the vertex k. We thus define projective Γ-Γ-bimodules (equivalently Γ e -modules) P 0 , P 1 , . . . , P 2p which will be the projectives in our minimal projective bimodule resolution for Γ. Definition 3.7. Let Γ = Γ(p, q; r) with p > 1. We define projective Γ-Γ-bimodules P 0 , P 1 , . . . , P 2p as follows.

P 0 = p+q-1 i=1 Γe i ⊗ e i Γ, P 1 = p-1 i=1 Γe i ⊗ e i+1 Γ ⊕ Γe p ⊗ e 1 Γ ⊕ q-1 j=2 Γe p+j-1 ⊗ e p+j Γ ⊕ Γe p+q-1 ⊗ e 1 Γ ⊕ Γe 1 ⊗ e p+1 Γ, P 2n = p-n i=2 Γe i ⊗ e i+n Γ ⊕ p-1 i=p-n+1 Γe i ⊗ e i+n+1-p Γ ⊕ Γe p ⊗ e n+1 Γ ⊕ q-n j=2 Γe p+j-1 ⊗ e p+j+n-1 Γ ⊕ q-1 j=q-n+1 Γe p+j-1 ⊗ e p+j+n-q Γ ⊕ Γe p+q-1 ⊗ e p+n Γ ⊕ Γe 1 ⊗ e 1 Γ, for 1 n < p, P 2n-1 = p-n i=2 Γe i ⊗ e i+n Γ ⊕ p-1 i=p-n+1 Γe i ⊗ e i+n-p Γ ⊕ Γe p ⊗ e n Γ ⊕ Γe 1 ⊗ e n+1 Γ ⊕ q-n j=2 Γe p+j-1 ⊗ e p+j+n-1 Γ ⊕ q-1 j=q-n+2 Γe p+j-1 ⊗ e p+j+n-q-1 Γ ⊕ Γe p+q-1 ⊗ e p+n-1 Γ ⊕ Γe 1 ⊗ e p+n Γ ⊕ Γe p+q-n ⊗ e 1 Γ, for 2 n < p P 2p-1 = p i=1 Γe i ⊗ e i Γ ⊕ q-p j=2 Γe p+j-1 ⊗ e 2p+j-1 Γ ⊕ q j=q-p+2 Γe p+j-1 ⊗ e 2p+j-q-1 Γ ⊕ Γe q ⊗ e 1 Γ ⊕ Γe 1 ⊗ e 2p Γ, P 2p = p-1 i=1 Γe i ⊗ e i+1 Γ ⊕ q-p j=2 Γe p+j-1 ⊗ e 2p+j-1 Γ ⊕ q j=q-p+1 Γe p+j-1 ⊗ e 2p+j-q Γ ⊕ Γe p ⊗ e 1 Γ ⊕ Γe 1 ⊗ e 1 Γ.
The first maps d i : P i → P i-1 , for i = 1, 2, 3 of a minimal projective bimodule resolution are given in [START_REF] Green | Projective bimodule resolutions of an algebra and vanishing of the second Hochschild cohomology group[END_REF]. We extend the resolution in [START_REF] Green | Projective bimodule resolutions of an algebra and vanishing of the second Hochschild cohomology group[END_REF] for our algebra Γ, in a similar way to [START_REF] Snashall | The Hochschild cohomology ring of a class of special biserial algebras[END_REF], to make the following definition of maps d i : P i → P i-1 for i = 1, . . . , 2p. Definition 3.8. Let Γ = Γ(p, q; r) with p > 1. We define Γ-Γ-bimodule homomorphisms d i : P i → P i-1 , for i = 1, . . . , 2p, as follows.

⊲ d 2n-1 : P 2n-1 → P 2n-2 , for 1 n < p,

• e i ⊗ e i+n → e i ⊗ α i+n-1 -α i ⊗ e i+n for 2 i p -n • e i ⊗ e i+n-p → (-1) i+p+1 e i ⊗ η r i+n-p + (-1) (p-i)(n-1) α i • • • α p ⊗ α 1 • • • α n-p+i-1 -η r i ⊗ e i+n-p -(-1) n (-1) (p-i)(n-1) p-i ∑ m=1 (-1) m(n-1) α i • • • α p-m ⊗ α n-m+1 • • • α p δ(γδ) r-1 α 1 • • • α n-p+i-1 +(-1) n (-1) (p-i)(n-1) n-1 ∑ m=p-i+2 (-1) m(n-1) α i • • • α p (δγ) r-1 δα 1 • • • α p-m ⊗ α n-m+1 • • • α n-p+i-1 for p -n + 2 i p • e p-n+1 ⊗ e 1 → e p-n+1 ⊗ α p + (-1) n n-1 ∑ m=1 (-1) m(n-1) α p-n+1 • • • α p-m ⊗ α n-m+1 • • • α p δ(γδ) r-1 -α p-n+1 • • • α p ⊗ e 1 • e 1 ⊗ e n+1 → (-1) n-1 e 1 ⊗ α 1 • • • α n + (-1) n n-1 ∑ m=1 (-1) m(n-1) δ(γδ) r-1 α 1 • • • α p-m ⊗ α n-m+1 • • • α n + (-1) n α 1 ⊗ e n+1 • e p+j-1 ⊗ e p+j+n-1 → -e p+j-1 ⊗ β j+n-1 + β j ⊗ e p+j+n-1 for 2 j q -n • e p+j-1 ⊗ e p+j+n-q-1 → (-1) j+q e p+j-1 ⊗ θ r j+n-q + (-1) (q-j)(n-1) β j • • • β q ⊗ β 1 • • • β n+j-q-1 -θ r j ⊗ e p+j+n-q-1 -(-1) n (-1) (q-j)(n-1) q-j ∑ m=1 (-1) m(n-1) β j • • • β q-m ⊗ β n-m+1 • • • β q (γδ) r-1 γβ 1 • • • β n-q+j-1 +(-1) n (-1) (q-j)(n-1) n-1 ∑ m=q-j+2 (-1) m(n-1) β j • • • β q γ(δγ) r-1 β 1 • • • β q-m ⊗ β n-m+1 • • • β n-q+j-1 for q -n + 2 j q • e p+q-n ⊗ e 1 → -e p+q-n ⊗ β q + (-1) n n-1 ∑ m=1 (-1) m(n-1) β q-n+1 • • • β q-m ⊗ β n-m+1 • • • β q (γδ) r-1 γ -β q-n+1 • • • β q ⊗ e 1 • e 1 ⊗ e p+n → (-1) n e 1 ⊗ β 1 • • • β n + (-1) n n-1 ∑ m=1 (-1) m(n-1) (γδ) r-1 γβ 1 • • • β q-m ⊗ β n-m+1 • • • β n + (-1) n β 1 ⊗ e p+n ⊲ d 2n : P 2n → P 2n-1 , for 1 n < p, • e i ⊗ e i+n → r ∑ k=0 η k i ⊗ η r-k i+n + r-1 ∑ k=0 p-i-n ∑ m=0 η k i α i • • • α i+m ⊗ α i+m+n+1 • • • α p δα 1 • • • α i+n-1 η r-k-1 i+n + i-2 ∑ m=0 η k i α i • • • α p δα 1 • • • α m ⊗ α m+n+1 • • • α i+n-1 η r-k-1 i+n +(-1) n q-n ∑ j=0 η k i α i • • • α p β 1 • • • β j ⊗ β j+n+1 • • • β q α 1 • • • α i+n-1 η r-k-1 i+n for 2 i p -n • e i ⊗ e i+n+1-p → e i ⊗ α i+n-p -(-1) n α i ⊗ e i+n+1-p for p + 1 -n i p • e 1 ⊗ e 1 → r-1 ∑ k=0 p-n ∑ i=0 δ(γδ) k α 1 • • • α i ⊗ α i+n+1 • • • α p (δγ) r-k-1 + (-1) n (γδ) k α 1 • • • α i ⊗ α i+n+1 • • • α p (δγ) r-k-1 δ + q-n ∑ j=0 γ(δγ) k β 1 • • • β j ⊗ β j+n+1 • • • β q (γδ) r-k-1 + (-1) n (δγ) k β 1 • • • β j ⊗ β j+n+1 • • • β q (γδ) r-k-1 γ
• e p+j-1 ⊗ e p+j+n-q → e p+j-1 ⊗ β j+n-q -(-1) n β j ⊗ e p+j+n-q for q + 1n j q

• e p+j-1 ⊗ e p+j+n-1 → r ∑ k=0 θ k j ⊗ θ r-k j+n + r-1 ∑ k=0 q-j-n ∑ m=0 θ k j β j • • • β j+m ⊗ β j+m+n+1 • • • β q γβ 1 • • • β j+n-1 θ r-k-1 j+n + j-2 ∑ m=0 θ k j β j • • • β q γβ 1 • • • β m ⊗ β m+n+1 • • • β j+n-1 θ r-k-1 j+n +(-1) n p-n ∑ i=0 θ k j β j • • • β q α 1 • • • α i ⊗ α i+n+1 • • • α p β 1 • • • β j+n-1 θ r-k-1 j+n for 2 j q -n ⊲ d 2p-1 : P 2p-1 → P 2p-2 • e i ⊗ e i → (-1) i e i ⊗ η r i + (-1) (p-i)(p-1) α i • • • α p ⊗ α 1 • • • α i-1 -η r i ⊗ e i -(-1) p (-1) (p-i)(p-1) p-i ∑ m=1 (-1) m(p-1) α i • • • α p-m ⊗ α p-m+1 • • • α p δ(γδ) r-1 α 1 • • • α i-1 +(-1) p (-1) (p-i)(p-1) p-1 ∑ m=p-i+2 (-1) m(p-1) α i • • • α p (δγ) r-1 δα 1 • • • α p-m ⊗ α p-m+1 • • • α i-1 for 2 i p • e 1 ⊗ e 1 → e 1 ⊗ γ - p-1 ∑ m=1 (-1) m(p-1) α 1 • • • α p-m ⊗ α p-m+1 • • • α p δ(γδ) r-1 + (-1) p p-1 ∑ m=1 (-1) m(p-1) δ(γδ) r-1 α 1 • • • α p-m ⊗ α p-m+1 • • • α p + (-1) p γ ⊗ e 1 • e p+j-1 ⊗ e 2p+j-1 → -e p+j-1 ⊗ β p+j-1 + β j ⊗ e 2p+j-1 for 2 j q -p • e p+j-1 ⊗ e 2p+j-q-1 → (-1) j+q e p+j-1 ⊗ θ r p+j-q + (-1) (q-j)(p-1) β j • • • β q ⊗ β 1 • • • β p+j-q-1 -θ r j ⊗ e 2p+j-q-1 -(-1) p (-1) (q-j)(p-1) q-j ∑ m=1 (-1) m(p-1) β j • • • β q-m ⊗ β p-m+1 • • • β q (γδ) r-1 γβ 1 • • • β p-q+j-1 +(-1) p (-1) (q-j)(p-1) p-1 ∑ m=q-j+2 (-1) m(p-1) β j • • • β q γ(δγ) r-1 β 1 • • • β q-m ⊗ β p-m+1 • • • β p-q+j-1 for q -p + 2 j q • e q ⊗ e 1 → -e q ⊗ β q + (-1) p p-1 ∑ m=1 (-1) m(p-1) β q-p+1 • • • β q-m ⊗ β p-m+1 • • • β q (γδ) r-1 γ -β q-p+1 • • • β q ⊗ e 1 • e 1 ⊗ e 2p → (-1) p e 1 ⊗ β 1 • • • β p + (-1) p p-1 ∑ m=1 (-1) m(p-1) (γδ) r-1 γβ 1 • • • β q-m ⊗ β p-m+1 • • • β p + (-1) p β 1 ⊗ e 2p ⊲ d 2p : P 2p → P 2p-1 • e i ⊗ e i+1 → e i ⊗ α i -(-1) p α i ⊗ e i+1 for 1 i p -1 • e p ⊗ e 1 → e p ⊗ α p -(-1) p α p ⊗ e 1 • e 1 ⊗ e 1 → r-1 ∑ k=0 (-1) p δ(γδ) k ⊗ (δγ) r-k-1 + (γδ) k ⊗ (δγ) r-k-1 δ - q-p ∑ j=0 γ(δγ) k β 1 • • • β j ⊗ β j+p+1 • • • β q (γδ) r-k-1 + (-1) p (δγ) k β 1 • • • β j ⊗ β j+p+1 • • • β q (γδ) r-k-1 γ • e p+j-1 ⊗ e p+j+p-q → e p+j-1 ⊗ β j+p-q -(-1) p β j ⊗ e p+j+p-q for q + 1 -p j q • e p+j-1 ⊗ e 2p+j-1 → r ∑ k=0 θ k j ⊗ θ r-k j+p + r-1 ∑ k=0 q-j-p ∑ m=0 θ k j β j • • • β j+m ⊗ β j+m+p+1 • • • β q γβ 1 • • • β j+p-1 θ r-k-1 j+p + j-2 ∑ m=0 θ k j β j • • • β q γβ 1 • • • β m ⊗ β m+p+1 • • • β j+p-1 θ r-k-1 j+p -θ k j β j • • • β q ⊗ β 1 • • • β j+p-1 θ r-k-1 j+p for 2 j q -p.
It remains to show that the projective bimodules and homomorphisms that we have defined do indeed give the start of a minimal projective bimodule resolution of Γ. Theorem 3.9. With the above notation,

• • • / / P 2p d 2p / / P 2p-1 d 2p-1 / / • • • / / P 1 d 1 / / P 0 / / Γ / / 0
is the beginning of a minimal projective resolution of Γ as a Γ-Γ-bimodule (when p > 1).

Proof. It may be verified directly from the definitions that d 2 = 0 and thus we have a complex. The strategy for proving exactness is identical to that of [START_REF] Snashall | The Hochschild cohomology ring of a class of special biserial algebras[END_REF]Theorem 1.6] (and see [9, Proposition 2.8]), whereby we apply (Γ/ rad(Γ) ⊗ -) to the complex and show that the resulting sequence is a minimal projective resolution of Γ/ rad(Γ) as a right Γ-module. Minimality is then immediate since we know that the projectives are those of a minimal projective resolution of Γ as a Γ-Γ-bimodule from [START_REF] Happel | Hochschild cohomology of finite-dimensional algebras[END_REF].

We are now in a position to give the dimensions of the Hochschild cohomology groups up to HH 2p-2 (Γ). We only give those in even degree since we shall not need the others. The details of the proof are left to the reader. Theorem 3.10. For 2 n < p q we have

dim HH 2n-2 (Γ) =          r if n is odd and char K ∤ 2r r + 1 if n is odd and char K | 2r r
if n is even and char K = 2 r + 1 if n is even and char K = 2, and for 2 p < q we have

dim HH 2p-2 (Γ) =          r -1 if p is odd and char K ∤ 2r r if p is odd, char K = 2 and char K | 2r r + 1 if p is even and char K = 2 r + 2 if char K = 2.

DERIVED EQUIVALENCE CLASSES

It was shown in [START_REF] Bocian | Derived equivalence classification of weakly symmetric algebras of Euclidean type[END_REF] that two algebras of the form A(p, q) = Γ(p, q; 1) or Λ(n) = Λ(1, n; 2, 2) are derived equivalent if and only if they are isomorphic. The main result in this section is to extend this to all algebras of the form Γ(p, q; r) and Λ(p, q; s, t), and hence to all basic indecomposable finite-dimensional K-algebras A which are symmetric special biserial algebras with at most one non-uniserial indecomposable projective module.

We start with some properties of these algebras, all of which are invariants under derived equivalence.

Proposition 4.1. Suppose 1 p q. The algebras Γ(p, q; r) and Λ(p, q; s, t) have the following properties.

(1) The number of simples of Γ(p, q; r) is p + q -1, Λ(p, q; s, t) is p + q -1.

(2) The Cartan invariants of (3) The Cartan determinant of Γ(p, q; r) is 4r, Λ(p, q; s, t) is s + t + (p + q -2)st.

                   Γ(p, q; r) are         
Proof.

(1) This is immediate from the number of vertices of the quiver Q (p,q) . (2) Let I m be the m × m identity matrix, let J n be the n × n matrix with all entries equal to 1, and set u = p + q -2. We start with the algebra Γ(p, q; r). The Cartan matrix of Γ(p, q; r) is the (p + q -1) × (p + q -1) matrix

C Γ =      4r 2r . . . 2r 2r . . . I u + rJ u 2r      . The Smith normal form for C Γ is                    I u 0 0 4r if ru is even    I u-1 0 0 0 2 0 0 0 2r    if ru is odd,
and thus the Cartan invariants of Γ(p, q; r) are Now consider the algebra Λ(p, q; s, t). The Cartan matrix of Λ(p, q; s, t) is the (p + q -1) × (p + q -1) matrix

         1, 1, . . . ,
C Λ =             s + t t . . . t s . . . s t . . . I p-1 + tJ p-1 0 t s . . . 0 I q-1 + sJ q-1 s             . The Smith normal form for C Λ is I u 0 0 s + t + ust ,
so the Cartan invariants of Λ(p, q; s, t) are 1, 1, . . . , 1, p+q-2 s + t + (p + q -2)st.

(3) This is immediate from [START_REF] Alperin | Local representation theory. Modular representations as an introduction to the local representation theory of finite groups[END_REF].

We now consider isomorphism classes of algebras of the same form. It is clear that Λ(q, q; s, t) = Λ(q, q; t, s), and the next result shows that, with this one exception, two algebras both of the form Γ(p, q; r) or of the form Λ(p, q; s, t) are pairwise non-isomorphic. (1) The algebras of the form Γ(p, q; r) (with 1 p q) are pairwise nonisomorphic.

(2) The algebras of the form Λ(p, q; s, t) (with 1 p q) are pairwise non-isomorphic with the exception that Λ(q, q; s, t) = Λ(q, q; t, s).

Proof.

(1) First, suppose that the algebras Γ(p, q; r) and Γ(p ′ , q ′ ; r ′ ) are isomorphic with 1 p q, 1 p ′ q ′ . Since both algebras are basic, the quivers are uniquely determined and hence Q (p,q) = Q (p ′ ,q ′ ) . Thus p = p ′ and q = q ′ . From the Cartan determinant in Proposition 4.1(3), we have that r = r ′ .

(2) Suppose that Λ(p, q; s, t) ∼ = Λ(p ′ , q ′ ; s ′ , t ′ ) with 1 p q, 1 p ′ q ′ . Since both algebras are basic, we again have that p = p ′ and q = q ′ . Then, using the zero-th Hochschild cohomology group from Proposition 3.1, we have s + t = s ′ + t ′ . Equality of the Cartan invariants from Proposition 4.1 [START_REF] Alperin | Local representation theory. Modular representations as an introduction to the local representation theory of finite groups[END_REF] gives that st = s ′ t ′ . Hence s ′ , t ′ are the two roots of the equation x 2 -(s + t)x + st = 0. Thus we have either s = s ′ and t = t ′ (which gives us the algebra Λ(p, q; s, s)), or s = t ′ and t = s ′ . In the latter case we have the algebras Λ(p, q; s, t) and Λ(p, q; t, s), and it remains to show p = q when s = t. The K-dimension of Λ(p, q; s, t) is tp 2 + sq 2 + p + q -2 and that of Λ(p, q; t, s) is sp 2 + tq 2 + p + q -2. Thus tp 2 + sq 2 = sp 2 + tq 2 so that (ts)(p + q)(pq) = 0. Since p + q > 0 and s = t we have p = q, which is precisely the case Λ(q, q; s, t) = Λ(q, q; t, s).

Our next theorem classifies up to derived equivalence, all basic indecomposable finitedimensional K-algebras A which are symmetric special biserial algebras with at most one nonuniserial indecomposable projective module. (1) An algebra of the form Λ(p, q; s, t) (with 1 p q) is derived equivalent to exactly one algebra in the following list: (a) Λ(1, p + q -1; s, t) with 2 s t, (b) N p+q-1 M with p + q > 2 and min(s, t) = 1, max(s, t) = M.

(2) An algebra of the form Γ(p, q; r) (with 1 p q) is derived equivalent to an algebra of the form Λ(p, q; s, t) if and only if they are isomorphic. This is only the case for Γ(1, 1; 1) ∼ = Λ(1, 1; 2, 2) and char K = 2. (3) The algebras Γ(p, q; r) and Γ(p ′ , q ′ ; r ′ ) (with 1 p q and 1 p ′ q ′ ) are derived equivalent if and only if (p, q, r) = (p ′ , q ′ , r ′ ).

Proof.

(1) The algebra Λ(p, q; s, t) is the generalised Brauer tree algebra associated to the Brauer tree in Figure 1:

a b 1 2 3 p 1 1 1 1 1 1 1 1 p+1 p+q-2 1 1 1 1 1 1 1 1 p+q-1 c d 1 2 3 p+q-2 1 1 1 1 1 1 1 1 p+q-1 Figure 1
Figure 2 in which the vertices a and b have multiplicities respectively s and t (we refer to [START_REF] Benson | Representations and cohomology. I. Basic representation theory of finite groups and associative algebras[END_REF]Section 4.18] or [START_REF] Membrillo-Hernández | Brauer tree algebras and derived equivalence[END_REF] for the definition of a Brauer tree algebra and a generalised Brauer tree algebra).

It was proved in [START_REF] Membrillo-Hernández | Brauer tree algebras and derived equivalence[END_REF]Theorem 9.7] that generalised Brauer tree algebras up to derived equivalence depend only on the number of edges in the graph and the set of multiplicities. Therefore Λ(p, q; s, t) is derived equivalent to the generalised Brauer tree algebra associated to the Brauer tree in Figure 2, in which the vertices c and d have multiplicities respectively m = min(s, t) and M = max(s, t) and {a, b} = {c, d} . This algebra is equal to

• either K∆ p+q-1 /L M = N p+q-1 M if m = 1, that is, if (s, t) = (M, 1) or (s, t) = (1, M), with M 1, • or Λ(1, p + q -1; m, M) if m > 1, that is, if s 2 and t 2.
Moreover, none of these algebras are derived equivalent, again by [START_REF] Membrillo-Hernández | Brauer tree algebras and derived equivalence[END_REF]Theorem 9.7].

(2) First, we show that the algebras Γ(1, 1; 1) and Λ(1, 1; 2, 2) are isomorphic when char K = 2. Since K is algebraically closed, let ε be a square root of -1 in K. Then the map

ϕ : Γ(1, 1; 1) → Λ(1, 1; 2, 2) given by α → α + εβ β → α -εβ is an isomorphism of algebras.
Suppose that there is a derived equivalence between the algebras Γ(p, q; r) and Λ(p ′ , q ′ ; s, t). Then the algebras have the same number of simple modules so, from Proposition 4.1, we have p + q = p ′ + q ′ . From (1), the algebra Λ(p ′ , q ′ ; s, t) is derived equivalent to exactly one algebra in the list above. Moreover, s + t = m + M where m = min(s, t) and M = max(s, t).

Case I: m = min(s, t) = 1.
Let Γ = Γ(p, q; r) and Λ = Λ(p ′ , q ′ ; s, t). From Propositions 3.1 and 3.3, dim HH 0 (Λ) = p + q + M -1 and dim HH 1 (Λ) = M. We first assume that p = 1, so that q > 1. From Proposition 3.4, dim HH 0 (Γ) = q + r + 1 so that M = r + 1. However, Proposition 3.6 gives dim HH 1

(Γ) = r + 4 if char K = 2 r + 2 if char K = 2
which is a contradiction. So Γ(1, q; r) is not derived equivalent to Λ(p ′ , q ′ ; s, t).

On the other hand, if p > 1 then dim HH 0 (Γ) = p + q + r -1 so that M = r. But dim HH 1 (Γ) = r + 1 so that M = r + 1, a contradiction. Again Γ(p, q; r) is not derived equivalent to Λ(p ′ , q ′ ; s, t).

Case II: m = min(s, t) > 1. so that dim K r Λ ′ /r 2 Λ ′ = 2 if M 4 1 if M = 2 (and therefore m = 2).
Since we assumed that Λ and Γ are derived equivalent, the algebras Λ ′ and Γ ′ are isomorphic and hence we have dim

K r Γ ′ /r 2 Γ ′ = dim K r Λ ′ /r 2 Λ ′ , which implies that m = 2 = M, that is, s = 2 = t. Therefore r = s + t -1 = 3. We now use the Cartan determi- nants: 0 = det C Γ -det C Λ = 4r -(s + t + (p + q -2)st) = 12 -(4 + (p + q -2)4) = 4(4 -(p + q))
so that p + q = 4. Since 1 < p q, we must have p = 2 = q. Therefore Γ = Γ(2, 2; 3) and Λ is derived equivalent to Λ(2, 2; 2, 2). However, it was shown in [12, Section 3] that Γ(2, 2; 3) = D(3A) 3 1 and Λ(2, 2; 2, 2) = D(3A) 2,2 2 are not derived equivalent. Therefore Λ and Γ are not derived equivalent.

(3) If the algebras Γ(p, q; r) and Γ(p ′ , q ′ ; r ′ ) (with p q and p ′ q ′ ) are derived equivalent, then from the Cartan determinant and number of simples (Proposition 4.1) we know that r = r ′ and p + q = p ′ + q ′ . Assume for contradiction that (p, q) = (p ′ , q ′ ). We may suppose that p < p ′ . It follows that p < q (for otherwise p = q and hence q ′ = (p ′ + q ′ )p ′ = (p + q)p ′ = 2pp ′ < 2p ′p ′ = p ′ , a contradiction). Using Theorem 3.10 and Proposition 3.4, we then have HH 2p-2 (Γ(p, q; r)) = HH 2p-2 Γ(p ′ , q ′ ; r ′ ) which contradicts the fact that the algebras are derived equivalent. Thus (p, q) = (p ′ , q ′ ).

STABLE EQUIVALENCE OF MORITA TYPE CLASSES

Finally, we give a classification up to stable equivalence of Morita type of all algebras of the form Γ(p, q; r) and Λ(p, q; s, t), and hence of all basic indecomposable finite-dimensional K-algebras A which are symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. This is based on the classification up to derived equivalence of Theorem 4.3.

Theorem 5.1.

(1) An algebra of the form Λ(p, q; s, t) (with 1 p q) is stably equivalent of Morita type to exactly one algebra in the following list: (a) Λ(1, p + q -1; s, t) with 2 s t, (b) N p+q-1 M with p + q > 2 and min(s, t) = 1, max(s, t) = M.

(2) An algebra of the form Γ(p, q; r) (with 1 p q) is stably equivalent of Morita type to an algebra of the form Λ(p, q; s, t) if and only if they are isomorphic. This is only the case for Γ(1, 1; 1) ∼ = Λ(1, 1; 2, 2) and char K = 2.

(3) The algebras Γ(p, q; r) and Γ(p ′ , q ′ ; r ′ ) (with 1 p q and 1 p ′ q ′ ) are stably equivalent of Morita type if and only if (p, q, r) = (p ′ , q ′ , r ′ ).

Proof. It was proved by Rickard [START_REF] Rickard | Derived categories and stable equivalence[END_REF] and Keller-Vossieck [START_REF] Keller | Sous les catégories dérivées[END_REF] that if two selfinjective K-algebras are derived equivalent, then they are stably equivalent of Morita type. Therefore to prove the result, we need only prove that the algebras listed in Theorem 4.3 are not stably equivalent of Morita type since they are all selfinjective. We shall use the following invariants of stable equivalences of Morita type repeatedly:

• the dimension of HH n (Λ) for n 1 for artin K-algebras (Xi,[START_REF] Xi | Stable equivalences of adjoint type[END_REF]Theorem 4.2]);

• the number of simple Λ-modules, where Λ is an indecomposable selfinjective special biserial algebra (Pogorzały,[START_REF] Pogorzały | Algebras stably equivalent to self-injective special biserial algebras[END_REF]); • the dimension of the centre Z(Λ) ∼ = HH 0 (Λ) where Λ is an indecomposable symmetric special biserial algebra (using a result of Liu, Zhou and Zimmermann, [19, Corollary 1.2]); • the absolute value of the Cartan determinant of Λ (Xi, [28, Proposition 5.1]).

All our algebras are indecomposable symmetric special biserial algebras. We now prove the theorem.

(1) Assume first that Λ(1, a; s, t) and Λ(1, b; s ′ , t ′ ), with a 1, b 1, 2 s t and 2 s ′ t ′ , are stably equivalent of Morita type. Then a = b since the numbers of simples are equal and, using the dimensions of the centres, we have s + t = s ′ + t ′ . Moreover, their Cartan determinants are equal (they are positive) so that st = s ′ t ′ . Finally, we have (s, t) = (s ′ , t ′ ) so that Λ(1, a; s, t) = Λ(1, b; s ′ , t ′ ). M is a symmetric Nakayama algebra, Λ(1, a; s, t) must be a Brauer tree algebra by [START_REF] Gabriel | Group representations without groups[END_REF], and hence derived equivalent to a symmetric Nakayama algebra by [START_REF] Membrillo-Hernández | Brauer tree algebras and derived equivalence[END_REF], which we know is not the case. Therefore Λ(1, a; s, t) and N b M are not stably equivalent of Morita type. This concludes the proof of (1). (2) The proof is essentially the same as in Theorem 4.3 since almost all the derived invariants used there are invariants of stable equivalence of Morita type between indecomposable symmetric special biserial algebras, the exception being the K ülshammer ideal T ⊥ 1 (A) of an algebra A. However, for symmetric algebras, the algebra Z(A)/T ⊥ 1 (A) is a stable invariant of Morita type. Indeed, let Z st (A) = End A e (A) be the stable centre of A (the endomorphisms of A in the stable A e -module category) and let Z pr (A) = Ker(End A e (A) → End A e (A)) be the projective centre of A. Then Z st (A) and T ⊥ 1 (A)/Z pr (A) are invariants of stable equivalences of Morita type for symmetric algebras (see [START_REF] Liu | Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture[END_REF][START_REF] Koenig | Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture[END_REF]), and moreover Z(A)/T ⊥ 1 (A) ∼ = Z st (A)/(T ⊥ 1 (A)/Z pr (A)). Therefore the proof of Theorem 4.3 (2) still holds for stable equivalences of Morita type.

(3) The proof is the same as in Theorem 4.3 since the dimensions of the Hochschild cohomology groups in positive degrees are invariants of stable equivalences of Morita type.

Remark.

Recall from [START_REF] Gabriel | Group representations without groups[END_REF] that the Nakayama algebras are the distinct representatives of the stable equivalence classes of Brauer tree algebras. It follows from [START_REF] Pogorzały | Algebras stably equivalent to self-injective special biserial algebras[END_REF] and [START_REF] Liu | Summands of stable equivalences of Morita type[END_REF] that any algebra which is stably equivalent of Morita type to one of the Γ(p, q; r) or Λ(p, q; s, t) is a symmetric Brauer graph algebra. Moreover, the list of algebras given in Theorem 5.1 are "normal forms" for derived equivalences of Brauer graph algebras (a specific type of generalised star as in [START_REF] Kauer | Derived equivalence of graph algebras[END_REF] and [24, Theorem 5.7]), and hence, since Brauer graph algebras are selfinjective, for stable equivalences of Morita type. Indeed, the algebra Γ(p, q; r) is the Brauer graph algebra associated to the graph

  1j . . . a kj . . . a s j j

  Now assume that N a M and N b M ′ , with a > 1, b > 1, M 1 and M ′ 1, are stably equivalent of Morita type. Then a = b since the numbers of simples are equal and, using the Cartan determinant, we have 1+ M + (a -1)M = 1 + M ′ + (a -1)M ′ so that M = M ′ . Therefore N a M = N b M ′ . Finally, if Λ(1, a; s, t) and N b M , with a 1, b > 1, 2 s t and M 1, are stably equivalent of Morita type then, since N b

  1,

		4r	if ru is even
	p+q-2
	1, 1, . . . , 1,	2, 2r if ru is odd.
	p+q-3

& & & & &

ACKNOWLEDGEMENTS

The first author would like to thank the Université Jean Monnet (Saint-Etienne) and the Université Blaise Pascal (Clermont-Ferrand 2) for two months of invited professorships during which this collaboration took place.

We use Propositions 3.1, 3.3, 3.4 and 3.6 without further comment. We begin with the case p = 1. If q = 1 then dim HH 0 (Λ) = s + t and dim HH 0 (Γ) = r + 3. From Proposition 4.1 [START_REF] Antipov | The Yoneda algebras of symmetric special biserial algebras are finitely generated[END_REF], the Cartan determinant of Γ is 4r and of Λ is s + t. Hence r + 3 = s + t = 4r so that r = 1 and s = 2 = t. If char K = 2 then we have an isomorphism Γ(1, 1; 1) ∼ = Λ(1, 1; 2, 2) from above. If char K = 2, then dim HH 1 (Λ) = 5 and dim HH 1 (Γ) = 8, which is a contradiction and Γ(1, 1; 1) is not derived equivalent to Λ(1, 1; 2, 2). In the case where q > 1 (keeping p = 1), a similar consideration of the zero-th and first Hochschild cohomology groups and the Cartan determinant shows that Γ(1, q; r) is not derived equivalent to Λ(p ′ , q ′ ; s, t). Now suppose that p > 1. We have dim HH 0 (Λ) = p + q + s + t -2 and dim HH 0 (Γ) = p + q + r -1 so that r = s + t -1. So dim HH 1 (Γ) = r + 1 = s + t, and we must have char K = ℓ| gcd(s, t). From Proposition 4.1(3), the Cartan determinant of Γ is 4r and the Cartan determinant of Λ is s + t + (p + q -2)st so that 3(s + t) = (p + q -2)st -4. Thus ℓ|4 so that ℓ = 2, and s, t are both even. Thus we are in the situation where Γ(p, q; r) is derived equivalent to Λ(p ′ , q ′ ; s, t), char K = 2, both s, t are even, r = s + t -1, and p + q = p ′ + q ′ . We shall use K ülshammer invariants and the same arguments as in [13, Subsection 4.5.2] for this case. Recall from Proposition 2.1 and its proof that the algebras Λ(p ′ , q ′ ; s, t) and Γ(p, q; r) are symmetric; moreover, once we have fixed a K-basis of paths for the socle of a symmetric algebra A and completed it to a K-basis B A of paths for A, the linear map f : A → K that is defined on B A by sending socle elements to 1 and the rest to 0 defines a symmetric non-degenerate associative bilinear form on A. Orthogonals will be taken with respect to this bilinear form. Let κ(A) be the commutator subspace of A and, for any nonnegative integer n, define T n (A) = x ∈ A; x 2 n ∈ κ(A) . It was proved in [START_REF] Zimmermann | Invariance of generalized Reynolds ideals under derived equivalences[END_REF] that the generalised Reynolds ideals (or K ülshammer invariants) T n (A) ⊥ are derived invariant. Note that soc(A) ⊆ T n (A) ⊥ Z(A) for every n. It is well-known that the centre Z(A) is a derived invariant. Given a vector space V, let B V denote a basis of V.

• We start with the algebra Γ = Γ(p, q; r). A basis B Γ of paths of Γ is given by the union over all i, j with 1 i, j p + q -1 of all paths from e i to e j of length at most (p + q)r except (δγ) r . Recall from Proposition 2.1 the basis B soc(Γ) = (γδ) r ; η r i , 2 i p; θ r i , 2 j q ⊂ B Γ of soc(Γ) and from Proposition 3.4 the basis B Z(Γ) = 1; (γδ) r ; z k , 1 k r -1; η r i ; θ r j for 2 i p and 2 j q of Z(Γ) where z = ∑

2

. We now consider the Jacobson radical r Γ ′ of the algebra Γ ′ , which is spanned by z k ; 1 k r-1 2 , and its square r 2 Γ ′ , which is spanned by z k ; 2 k r-1 2 so that dim K r Γ ′ /r 2 Γ ′ = 1. • We now turn to the algebra Λ = Λ(p ′ , q ′ ; s, t). Since they are derived equivalent, we may assume that Λ = Λ(1, n; m, M) with n = p + q -1 (to simplify notation). Note that m and M are even. Set α = α 1 . We follow the same method as for Γ, using Propositions 2.1 and 3.1. with p -1 edges inside the loop and q -1 edges outside the loop and with multiplicity r at the central vertex. However, it is still an open question in general whether two such normal forms are derived equivalent or not.