UNIVERSIT Rachel Taillefer
email: rachel.taillefer@uca.fr

FIRST HOCHSCHILD COHOMOLOGY GROUP AND STABLE EQUIVALENCE CLASSIFICATION OF MORITA TYPE OF SOME TAME SYMMETRIC ALGEBRAS

Keywords: Mathematics Subject Classification. 17B60, 16E40, 16G99, 16D90. Key words and phrases. stable equivalence of Morita type, algebras of dihedral, semi-dihedral and quaternion type, first Hochschild cohomology group, Lie algebra. 1

We use the dimension and the Lie algebra structure of the first Hochschild cohomology group to distinguish some algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type. In particular, we complete the classification of algebras of dihedral type that was mostly determined by Zhou and Zimmermann.

INTRODUCTION

Erdmann has given a description, up to Morita equivalence, of some families of tame symmetric algebras, which include the blocks of finite group algebras of tame representation type, and that are defined essentially in terms of their Auslander-Reiten quivers. They are separated into three types, dihedral, quaternion and semi-dihedral (generalising tame blocks whose defect groups are dihedral, semidihedral or generalised quaternion). Holm then classified them up to derived equivalence in [11]. It is then natural to try to classify them up to stable equivalence, but there are many properties that are not preserved under stable equivalences. However, Rickard in [START_REF] Rickard | Derived categories and stable equivalence[END_REF] and Keller and Vossieck in [START_REF] Keller | Sous les catégories dérivées[END_REF] proved that a derived equivalence between selfinjective algebras induces a stable equivalence of a particular form, called stable equivalence of Morita type because it is induced by tensoring with some bimodules; since then, such stable equivalences (even for algebras that are not selfinjective) have been much studied. In particular, in [START_REF] Zhou | Classifying tame blocks and related algebras up to stable equivalences of Morita type[END_REF] and in [START_REF] Zimmermann | K ülshammer ideals of algebras of quaternion type[END_REF], Zhou and Zimmermann used various techniques (including K ülshammer invariants and stable Hochschild cohomology) in order to distinguish most of the algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type, but some questions remain. Our aim is to use the first Hochschild cohomology group and its Lie structure to answer some of these questions.

It was shown by Xi in [START_REF] Xi | Stable equivalences of adjoint type[END_REF] that if A and B are two selfinjective algebras and if there is a stable equivalence of Morita type between them, then for n 1, the Hochschild cohomology groups HH n (A) and HH n (B) are isomorphic. Moreover, as a consequence of a result of K önig, Le and Zhou in [START_REF] Koenig | Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture[END_REF], if A is a symmetric algebra, the Lie algebra structure of HH 1 (A) is also preserved under such an equivalence. We shall use these facts to distinguish some of the algebras above up to stable equivalence of Morita type. As a result, we are able to complete the classification for the algebras of dihedral type, and to improve it for the algebras of quaternion and semi-dihedral types.

The Lie algebra structure of the first Hochschild cohomology group has been described by Strametz in [START_REF] Strametz | The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra[END_REF], where she studied the Lie algebra HH 1 (A) for a monomial algebra A. Her results were then used by Sánchez-Flores in [START_REF] Ánchez-Flores | The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero[END_REF] to study the Gerstenhaber algebra structure of the Hochschild cohomology ring HH * (A) of a monomial algebra A. Strametz' description has also been used by Bessenrodt and Holm in [START_REF] Bessenrodt | q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras[END_REF]. The Lie algebra HH 1 (A) has also been studied for instance in [START_REF] Guil-Asensio | The group of outer automorphisms and the Picard group of an algebra[END_REF], and used for example in [START_REF] Linckelmann | Block algebras with HH 1 a simple Lie algebra[END_REF] to retrieve information on some blocks of a group algebra. We shall describe Strametz' construction in Section 2.1 and use it in this paper.

After summarising in Section 1 the results known on stable equivalence of Morita type of algebras of dihedral, semi-dihedral and quaternion type, as well as proving our main result for algebras of quaternion type with two simple modules, we give some general tools that we will use in Section 2: we first describe the Lie algebra structure on the first Hochschild cohomology group. Moreover, the usual algorithmic methods to compute a minimal projective resolution of an algebra given by quiver and relations relies on the fact that we have a minimal set of relations, which is not the case here. Therefore we describe our method to determine the beginning of a minimal projective resolution of a finite-dimensional associative algebra in order to compute the first Hochschild cohomology group. Finally, we shall use some constructions that are invariant under Lie algebra isomorphisms, which we recall in the last part of Section 2. We then study the cases of algebras of dihedral type in Section 3, of semi-dihedral type in Section 4, and of quaternion type in Section 5.

Throughout, K is an algebraically closed field. Set ⊗ = ⊗ K .

Acknowledgements. I wish to thank Alexander Zimmermann for asking me, many years ago, whether the dimension of HH 1 (A) could help with the classification problem for the local dihedral algebras, thus initiating this project.

THE QUESTIONS STUDIED IN THIS PAPER

In [START_REF] Zhou | Classifying tame blocks and related algebras up to stable equivalences of Morita type[END_REF], Zhou and Zimmermann proved that if A and B are algebras that are stably equivalent of Morita type, then A is of dihedral (respectively semi-dihedral, respectively quaternion) type if and only if B is also. Moreover, if A and B are of dihedral, semi-dihedral and quaternion type, then A and B have the same number of simple modules.

Since our methods did not enable us to improve on the existing results for algebras with three simple modules (the only question being for the algebras Q(3A) 2,2 1 (d) of quaternion type for which the Lie algebra structure of HH 1 (Q(3A) 2,2 1 (d)) does not depend on d), we shall restrict our study to the algebras with one or two simple modules.

1.1. The algebras involved. Let us first define the K-algebras that we are going to study, by quiver and relations. We shall need the following quivers:

• x % % y z z 1 β + + α $ $ 2 γ k k η z z 1A 2B
We shall only consider the local algebras when char(K) = 2, and they are defined as follows.

The quiver of all the local algebras is 1A. Moreover,

• the algebras D(1A) k 2 (d) of dihedral type, where k 2 is an integer and d ∈ {0, 1}, whose relations ideal is generated by

x 2 -(xy) k , y 2 -d(xy) k , (xy) k -(yx) k , (xy) k x and (yx) k y, • the algebras SD(1A) k 2 (c, d) of semi-dihedral type,
where k 2 is an integer and (c, d) ∈ K 2 , (c, d) = (0, 0), whose relations ideal is generated by

(xy) k -(yx) k , (xy) k x, y 2 -d(xy) k and x 2 -(yx) k-1 y + c(xy) k ,
• the algebras Q(1A) k 1 of quaternion type, where k 2 is an integer, whose relations ideal is generated by

(xy) k -(yx) k , (xy) k x, y 2 -(xy) k-1 x and x 2 -(yx) k-1 y and the algebras Q(1A) k 2 (c, d) of quaternion type,
where k 2 is an integer and (c, d) ∈ K 2 , (c, d) = (0, 0), whose relations ideal is generated by

x 2 -(yx) k-1 y -c(xy) k , y 2 -(xy) k-1 x -d(xy) k , (xy) k -(yx) k , (xy) k x and (yx) k y.
These algebras all have dimension 4k with basis the elements (xy) t (yx) t+1 y(xy) t x(yx) t for 0 t k -1, and the centre of all these algebras has dimension (k + 3). We no longer assume that char(K) = 2. The quiver of all the algebras with two simple modules is 2B and they are the following.

• The algebras SD(2B) k,s 1 (c) of semi-dihedral type, where k 2 and s 1 are integers and c ∈ {0, 1}, whose relations ideal is generated by

γβ, ηγ, βη, α 2 -(βγα) k-1 βγ -c(αβγ) k , η s -(γαβ) k and (αβγ) k -(βγα) k . • The algebras SD(2B) k,s 2 (c) of semi-dihedral type,
where k 2 and s 1 are integers with k + s 4 and c ∈ {0, 1}, whose relations ideal is gener- ated by 2 and η 2 γ.

βη -(αβγ) k-1 αβ, ηγ -(γαβ) k-1 γα, γβ -η s-1 , α 2 -c(αβγ) k , βη
• The algebras Q(2B) k,s 1 (a, c) of quaternion type, where k 1 and s 3 are integers and (a, c) ∈ K 2 with a = 0 (and a = 1 if k + s = 4), whose relations ideal is generated by

γβ -η s-1 , βη -(αβγ) k-1 αβ, ηγ -(γαβ) k-1 γα, α 2 -a(βγα) k-1 βγ -c(βγα) k , α 2 β, γα 2 .
The algebras with two simple modules (of semi-dihedral and quaternion type) all have dimension 9k + s, and the following elements, where 0 t k -1 and 1 r s, form a basis of each algebra:

(αβγ) (c, d) or not. We shall give a partial answer to this question in Theorem 4.1.

In the case of algebras of semi-dihedral type with two simple modules, it was proved in [START_REF] Zhou | Classifying tame blocks and related algebras up to stable equivalences of Morita type[END_REF] that if two such algebras SD(2B) k,s i (c) and SD(2B) k ′ ,s ′ i ′ (c ′ ) are stably equivalent of Morita type, then the sets {k, s} and {k ′ , s ′ } are equal. Moreover, if char(K) = 2 and k = 2 and s 3 is odd then SD(2B) k,s 1 (0) and SD(2B) k ′ ,s ′ 1 (1) are not stably equivalent of Morita type, and if k and s are both odd, and if {k ′ , s ′ } = {k, s}, then SD(2B) k,s 2 (0) and SD(2B) k ′ ,s ′ 2 (1) are not stably equivalent of Morita type.

We go further in this classification in Theorem 4.10.

1.4. Algebras of quaternion type. For local algebras of quaternion type, the remaining questions are whether, for a fixed k, two algebras among Q(1A) k 1 and the Q(1A) k 2 (c, d) can be stably equivalent of Morita type or not. We shall study this situation in Section 5, whose main result is Corollary 5.6.

We now turn to the algebras of quaternion type with two simple modules. Zhou and Zimmermann showed in [START_REF] Zhou | Classifying tame blocks and related algebras up to stable equivalences of Morita type[END_REF]

that if Q(2B) k,s 1 (a, c) and Q(2B) k ′ ,s ′
1 (a ′ , c ′ ) are stably equivalent, then the sets {k, s} and {k ′ , s ′ } are equal. There remained some questions however.

First assume that char(K) = 2. If k + s > 4, the problem relating to the parameters a and c was solved recently by Zimmermann in [START_REF] Zimmermann | K ülshammer ideals of algebras of quaternion type[END_REF], where he proved that

Q(2B) k,s 1 (a, c) ∼ = Q(2B) k,s 1 (1, 0
) (recall that the field K is algebraically closed). Therefore, using [3, Lemma 5.7 (ii)], we need only consider the algebras Q(2B) 1,3 1 (a, 0) with a ∈ K * , a = 1. However, the methods in this paper do not provide any information to distinguish the stable equivalence classes of Morita type, therefore we shall assume that char(K) = 2. In this case, then by [START_REF] Erdmann | The stable Calabi-Yau dimension of tame symmetric algebras[END_REF]Lemma 5.7], if k + s > 4 we need only consider the algebras Q(2B) k,s 1 (1, c) for c ∈ K. Theorem 1.1 below can be obtained from the Lie algebra structure of HH 1 (Q(2B) k,s 1 (a, c)) and the techniques in this paper, using a minimal projective resolution from [START_REF] Erdmann | The stable Calabi-Yau dimension of tame symmetric algebras[END_REF] and computing the dimensions of the Hochschild cohomology groups as in Proposition 5.1 and the Lie algebra structure of the first Hochschild cohomology group as in the other cases. However we shall give a more elegant proof here using a result from [START_REF] Zimmermann | K ülshammer ideals of algebras of quaternion type[END_REF].

Theorem 1.1. Let K be an algebraically closed field of characteristic 2. Let k 1 and s 3 be integers, and let c be in

K * . If k + s > 4, then the algebras Q(2B) k,s 1 (1, c) and Q(2B) k,s 1 (1, 0)
are not stably equivalent of Morita type.

If k = 1 and s = 3 then, for any a, a ′ in K\ {0, 1}, the algebras Q(2B) k,s 1 (a, c) and Q(2B) k,s 1 (a ′ , 0) are not stably equivalent of Morita type. Before we prove this result, let us define the objects that we shall use. Let A be a symmetric algebra over a field of characteristic p, endowed with a non-degenerate symmetric associative bilinear form (, ). Let KA be the subspace of A generated by the commutators abba of elements a, b in A. Define T n (A) = {x ∈ A, x p ∈ KA} and let T n (A) ⊥ be the orthogonal space with respect to (, ), which is an ideal in the centre Z(A), called K ülshammer ideal.

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 the ideals 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)). Proof. In [START_REF] Zimmermann | K ülshammer ideals of algebras of quaternion type[END_REF]Theorem 7 (2)], Zimmermann describes the quotient

Z(Q(2B) k,s 1 (a, c))/T ⊥ 1 (Q(2B) k,s 1 (a, c
)) in all cases depending on the values and parity of k and s and on whether c = 0 or c = 0, and it follows that the algebras

Z(Q(2B) k,s 1 (a, c))/T ⊥ 1 (Q(2B) k,s 1 (a, c)) and Z(Q(2B) k,s 1 (a ′ , 0))/T ⊥ 1 (Q(2B) k,s 1 (a ′ , 0)
) are not isomorphic when c = 0 (see the proof of [START_REF] Zimmermann | K ülshammer ideals of algebras of quaternion type[END_REF]Corollary 10]). The result follows.

Remark. The same result when k 2 can be obtained as a consequence of the algebra structure of the whole Hochschild cohomology computed in [START_REF] Generalov | Hochschild cohomology of algebras of quaternion type. II: The family Q(2B) 1 in characteristic 2[END_REF]. We note that although the algebras HH * (Q(2B) k,s 1 (1, c)) and HH * (Q(2B) s,k 1 (1, c)) in [START_REF] Generalov | Hochschild cohomology of algebras of quaternion type. II: The family Q(2B) 1 in characteristic 2[END_REF] ap- pear to be different, there is an explicit isomorphism between them.

Remark. We should mention that the first Hochschild cohomology group does not separate algebras with different parameter a.

Thereofore, in Section 5 we shall only consider local algebras of quaternion type.

GENERAL FACTS ON THE FIRST HOCHSCHILD COHOMOLOGY GROUP AND ITS

COMPUTATION AND ON INVARIANTS OF LIE ALGEBRAS 2.1. Lie algebra structure on HH 1 (A). K önig, Le and Zhou proved in [13, The- orem 10.7] that the Batalin-Vilkoviskyi structure of the stable Hochschild cohomology HH * st (A) (that is, the Hochschild cohomology HH * (A) = n∈N HH n (A) modulo the projective centre of A) of a symmetric algebra A is invariant under stable equivalences of Morita type. In particular, the Lie algebra structure of HH 1 (A) is preserved under such an equivalence.

The Lie structure on HH 1 (A) is usually described on the Hochschild complex (obtained from the bar resolution). However, we will be working with minimal resolutions, so we will need a description of the Lie bracket when HH 1 (A) is computed from a minimal projective resolution. This is based on [START_REF] Strametz | The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra[END_REF].

Let A = KΓ/I be a finite dimensional algebra, where Γ is a quiver and I is an admissible ideal. Let Γ 0 be the set of vertices in Γ and Γ 1 be the set of arrows.

Using the bar resolution Bar(A), we get HH 1 (A) = Ker d 1 / Im d 0 where

0 → A d 0 -→ Hom K (A, A) d 1 -→ Hom K (A ⊗ K A, A) and d 0 (λ)(p) = λp -pλ and d 1 ( f )(a ⊗ b) = a f (b) -f (ab) + f (a)b.
The space HH 1 (A) is then endowed with a Lie bracket defined by

[ f , g] = f • g -g • f .
For all the algebras A we shall consider in this paper, there is a minimal projective resolution P of A that starts with

A ⊗ E KZ ⊗ E A ∂ 1 -→ A ⊗ E KΓ 1 ⊗ E A ∂ 0 -→ A ⊗ E A → A → 0 where E = KΓ 0 , Z is a set of relations in I and ∂ 0 (1 ⊗ E a ⊗ E 1) = a ⊗ E 1 -1 ⊗ E a for a ∈ Γ 1 ∂ 1 (1 ⊗ E n ∑ i=1 c i a 1,i • • • a s i ,i ⊗ E 1) = n ∑ i=1 s i ∑ j=1 c i a 1,i • • • a j-1,i ⊗ E a j,i ⊗ E a j+1,i • • • a i,s i
where the c i are in K, the a j,i are in Γ 1 and z = ∑ s i j=1 c i a j,i ∈ Z. As Bar(A) and P are projective resolutions of the A-bimodule A, there exist, by the Comparison Theorem, chain maps ξ : Bar(A) → P and ̺ : P → Bar(A). As in [START_REF] Strametz | The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra[END_REF], these maps induce inverse linear isomorphisms ξ * 1 and ̺ * 1 at the cohomology level between HH 1 (A, A) = Ker d 1 / Im d 0 and HH 1 (A, A) = Ker ∂ 1 / Im ∂ 0 given by the classes of

ξ 1 : Hom E-E (KΓ 1 , A) -→ Hom K (A, A) f -→ [a 1 • • • a n → ∑ n i=1 a 1 • • • a i-1 f (a i )a i+1 • • • a n ] ̺ 1 : Hom K (A, A) -→ Hom E-E (KΓ 1 , A) h → h |KΓ 1 .
This allows us to transfer the Lie algebra structure of Ker

d 1 / Im d 0 to Ker ∂ 1 / Im ∂ 0 ,
where the bracket is given by

[ f , g] : = ̺ 1 * ([ξ * 1 ( f ), ξ * 1 (g)]) = ξ * 1 ( f ) • g -ξ * 1 (g) • f for all f , g in Hom E-E (KΓ 1 , A).

Method used to determine the beginning of a minimal projective resolution of an algebra A as an A-A-bimodule.

Given a finite-dimensional K-algebra A = KΓ/I defined by quiver Γ and relations I, Happel's theorem [START_REF] Happel | Hochschild cohomology of finite-dimensional algebras[END_REF] gives the modules in a minimal projective resolution of an algebra A as an A-A-bimodule, but not the maps. The general methods to determine the beginning of a minimal projective resolution of an algebra A as an A-A-bimodule usually rely on the fact we have a minimal set of generators for the algebra I. However, most of the algebras of dihedral, semi-dihedral and quaternion type are not defined with a minimal set of relations, and it is not easy to extract such a minimal set. Therefore, we shall repeatedly use the following result of [8, Proposition 2.8] (see also [START_REF] Snashall | The Hochschild cohomology ring of a class of special biserial algebras[END_REF]Theorem 1.6] for a more detailed proof). Lemma 2.1. [START_REF] Green | Projective bimodule resolutions of an algebra and vanishing of the second Hochschild cohomology group[END_REF] Let A = KΓ/I be a finite-dimensional K-algebra defined by quiver Γ and relations I. For i in the set of vertices Γ 0 of Γ, denote by e i the corresponding idempotent and by S i the corresponding simple right A-module. Let

(P • , d • ) be a minimal projective right A-module resolution of A/ rad A. Let (Q • , ∂ • ) be a complex of A-A-bimodules with Q -1 = A and Q n = i∈Q 0 (Ae i ⊗ e j A) dim Ext n A (S i ,S j ) for n 0. Assume that ((A/ rad A) ⊗ A Q • , id ⊗ A ∂ • ) = (P • , d • ). Then (Q • , ∂ • ) is a

minimal projective resolution of A as an A-A-bimodule.

Remark. Note that once a space Q 2 satisfying the conditions in the Lemma is found, a minimal set of relations for the ideal I is then given by (A/ rad A) e ⊗ A e Q 2 .

2.3. Some Lie algebra invariants. Let g be a finite dimensional Lie algebra over K with bracket [, ]. We briefly recall here a few objects associated to g that we will use throughout the paper.

The lower central series of g, whose i-th term is denoted by L i (g), is defined inductively by

L 0 (g) = g, L 1 (g) = [g, g] and L i (g) = [g, L i-1 (g)] for i 2.
If L i (g) = 0 for i large enough, the Lie algebra g is nilpotent.

The derived series of g, whose i-th term is denoted by D i (g), is defined inductively by

D 0 (g) = g, D 1 (g) = [g, g] and D i (g) = [D i-1 (g), D i-1 (g)] for i 2.
The nilradical of g is the maximal nilpotent ideal in g.

The lower central series, derived series and nilradical are clearly preserved under isomorphisms of Lie algebras.

We now recall the Killing form of g. This is the bilinear form κ :

g × g → K defined by κ(x, y) = trace([x, [y, -]]).
If g and g ′ are isomorphic Lie algebras, then their Killing forms are equivalent. In particular, they have the same rank. Finally, we introduce generalised derivations of g, that were defined in [START_REF] Hrivn Ák | On (α, β, γ)-derivations of Lie algebras and corresponding invariant functions[END_REF] and that we will use in the proof of Proposition 4.28. Let λ, µ, ν be three elements in

K that are not all zero. A (λ, µ, ν)-derivation of g is a linear map D : g → g that satisfies λD([x, y]) = µ[D(x), y] + ν[x, D(y)] for all x, y, z in g.
Let der g (λ, µ, ν) denote the space of (λ, µ, ν)-derivations of g.

As was mentioned by Novotn ý and Hrivnák in [16, Proposition 3.1], if g and g ′ are isomorphic Lie algebras, then der g (λ, µ, ν) and der g ′ (λ, µ, ν) are isomorphic vector spaces.

ALGEBRAS OF DIHEDRAL TYPE

The only remaining question in the classification of the algebras of dihedral type up to stable equivalence of Morita type are whether the local algebras

D(1A) k 2 (d) with d ∈ {0, 1} in characteristic 2 are equivalent or not.
Fix an integer k 2. Consider the local tame symmetric algebras of dihedral type

Λ := D(1A) k 2 (d) = K x, y /I k d where I k d is the ideal generated by x 2 -(xy) k , y 2 -d(xy) k ; (xy) k -(yx) k ; (xy) k x; (yx) k y for d ∈ {0, 1} .
As we ex- plained in Subsection 1.2, we must determine whether these two algebras are equivalent or not. We shall see that the first cohomology group HH 1 (D(1A) k 2 (d)) enables us to do this.

Lemma 3.1. Consider the sequence of Λ-Λ-bimodules

Q 2 = 2 i=0 (Λ ⊗ i Λ) ∂ 2 -→ Q 1 = (Λ ⊗ x Λ) ⊕ (Λ ⊗ y Λ) ∂ 1 -→ Q 0 = Λ ⊗ Λ ∂ 0 -→ Λ → 0 with the maps determined by ∂ 1 (1 ⊗ δ 1) = δ ⊗ 1 + 1 ⊗ δ for δ ∈ {x, y} ∂ 2 (1 ⊗ 0 1) = x ⊗ x 1 + 1 ⊗ x x + k-1 ∑ t=0 (xy) t ⊗ x y(xy) k-1-t + (xy) t x ⊗ y (xy) k-1-t ∂ 2 (1 ⊗ 1 1) = k-1 ∑ t=0 (xy) t ⊗ x y(xy) k-1-t + (xy) t x ⊗ y (xy) k-1-t +(yx) t ⊗ y x(yx) k-1-t + (yx) t y ⊗ x (yx) k-1-t ∂ 2 (1 ⊗ 2 1) = y ⊗ y 1 + 1 ⊗ y y + d k-1 ∑ t=0 (yx) t ⊗ y x(yx) k-1-t + (yx) t y ⊗ x (yx) k-1-t ,
where the subscripts under the tensor product symbols ⊗ denote the component of the free

Λ-Λ-bimodule Q n for n = 1, 2.
Then this is the beginning of a minimal projective Λ-Λ-bimodule resolution of Λ.

Proof. It is easy to check that it is a complex, and that applying (Λ/ rad Λ)⊗ Λ ? gives the beginning of a minimal projective right Λ-module resolution of K = Λ/ rad Λ. From this resolution, we may determine dim Ext n Λ (K, K) for n = 0, 1 and 2 and check that Q n is the module in Happel's theorem [START_REF] Happel | Hochschild cohomology of finite-dimensional algebras[END_REF]. We then apply Lemma 2.1.

We shall now determine HH 1 (Λ). Recall that the centre of Λ has dimension k + 3 and that it is isomorphic to HH 0 (Λ) = Ker(? • ∂ 1 ). Therefore the dimension of the image of the map (?

• ∂ 1 : Hom Λ-Λ (Q 0 , Λ) → Hom Λ-Λ (Q 1 , Λ)) is equal to dim Hom Λ-Λ (Λ ⊗ Λ, Λ) -dim HH 0 (Λ) = 4k -(k + 3) = 3k -3.
In order to determine the kernel of the map (?

⊗ ∂ 2 : Hom Λ-Λ (Q 1 , Λ) → Hom Λ-Λ (Q 2 , Λ)), we note that an element in Hom Λ-Λ (Q 1 , Λ) = i∈{x,y} Hom Λ-Λ (Λ ⊗ i Λ, Λ) is determined by f (1 ⊗ i 1) = k-1 ∑ t=0 λ (i) t (xy) t + µ (i) t (yx) t+1 + τ (i) t y(xy) t + σ (i) t x(yx) t
for i ∈ {x, y}, where λ

(i) t , µ (i) t , σ (i) t and τ (i) t are scalars. Note that f • ∂ 2 (1 ⊗ 1 1) = 0
for any f . We then determine the conditions on the coefficients for f • ∂ 2 to vanish, using standard linear algebra.

We obtain dim Ker(?

• ∂ 2 ) =          4k + 3 if k is even and d = 0 4k + 2 if k is odd and d = 0 or if k is even and d = 1 4k + 1 if k is odd and d = 1.
Hence we have the following result.

Proposition 3.2. The first cohomology group

HH 1 (D(1A) k 2 (d)) has dimension k + 6 -d if k is even k + 5 -d if k is odd.

Corollary 3.3. There is no stable equivalence of Morita type between the algebras D(1A) k

2 (0) and D(1A) k 2 (1). Remark. This completes the classification of the algebras of dihedral type up to stable equivalence of Morita type.

ALGEBRAS OF SEMI-DIHEDRAL TYPE

As we mentioned in Subsection 1.3, the classification is complete for algebras of semi-dihedral type with three simple modules. We shall start with the local algebras.

Local algebras of semi-dihedral type.

4.1.1. Dimension of the first Hochschild cohomology group. We assume here that the field K has characteristic 2. Fix an integer k 2.

For (c, d) ∈ K 2 , let I k (c, d) be the ideal in K x, y generated by (xy) k + (yx) k ; x 2 + (yx) k-1 y + c(yx) k ; y 2 + d(xy) k ; (xy) k x) .
For any local tame symmetric algebra of semi-dihedral type Λ, there is a stable equivalence of Morita type with one of the algebras SD(1A) k [22, Theorem 7.1]). However, for a fixed k, it is not known whether these algebras are stably equivalent of Morita type or not. Using isomorphisms of the form (x, y) → (λx, µy), we can assume that (c, d) ∈ {(1, 0); (c, 1); with c ∈ K}. Note that in all these algebras, we have the following identities:

1 = K x, y /I k (0, 0) and SD(1A) k 2 (c, d) = K x, y /I k (c, d) for (c, d) ∈ K 2 \ {(0, 0)} (see
xy 2 = 0 = y 2 x; y(xy) k = 0; x 2 y = 0 = yx 2 ; x 3 = (xy) k = (yx) k ; x 4 = 0; y 3 = 0.
The aim of this section is to prove the following theorem.

Q 2 = 1 i=0 (Λ ⊗ i Λ) ∂ 2 -→ Q 1 = (Λ ⊗ x Λ) ⊕ (Λ ⊗ y Λ) ∂ 1 -→ Q 0 = Λ ⊗ Λ ∂ 0 -→ Λ → 0
with the maps determined by

∂ 1 (1 ⊗ δ 1) = δ ⊗ 1 + 1 ⊗ δ for δ ∈ {x, y} ∂ 2 (1 ⊗ 0 1) = x ⊗ x 1 + 1 ⊗ x x + k-2 ∑ t=0 (yx) t y ⊗ x y(xy) k-2-t + k-1 ∑ t=0 (yx) t ⊗ y (xy) k-1-t + c(yx) t y ⊗ x (yx) k-1-t + c(yx) t ⊗ y x(yx) k-1-t ∂ 2 (1 ⊗ 1 1) = y ⊗ y 1 + 1 ⊗ y y + d k-1 ∑ t=0 (xy) t x ⊗ y (xy) k-1-t + (xy) t ⊗ x y(xy) k-1-t ,
where the subscripts on the tensor product symbols ⊗ denote the component of the free

Λ-Λ-bimodule Q n for n = 1, 2.
Then this is the beginning of a minimal projective Λ-Λ-bimodule resolution of Λ.

Proof. The proof is the same as that of Lemma 3.1.

Using this resolution, we may compute the Hochschild cohomology groups.

As in the case of the dihedral algebras D(1A) k 2 (d), we have dim Im(? • ∂ 1 ) = 3k -3 (we give a generating set explicitly in the proof of Lemma 4.5).

Moreover, it is easy to check that dim Ker(?

• ∂ 2 ) =      4k + 3 if k is even and d = 0 or if k is odd and c = 0 = d, 4k + 2 if k is even and d = 0 or if k is odd, c = 0 and d = 0, 4k + 1 if k is odd and d = 0.
Therefore we get the following dimensions for the first Hochschild cohomology group.

Proposition 4.3. Let Λ be one of the algebras SD

(1A) k 1 or SD(1A) k 2 (c, d). Then dim HH 1 (Λ) =      k + 6 if
(1A) or SD k 2 (1A)(c, d).
Let Γ be a quiver of type 1A, with arrows x and y. Then the local tame symmetric algebras of semi-dihedral type may be defined as KΓ/I k (c, d) for (c, d) ∈ K 2 or, as we mentioned above, (c, d) ∈ {(0, 0), (1, 0), (c, 1); c ∈ K} .

Remark. It is possible (though laborious) in this case to compute dim HH n (Λ) for all n 0 (the case c = 0 = d may be found in [START_REF] Generalov | Hochschild cohomology of algebras of semidihedral type. I. Group algebras of semidihedral groups (Russian)[END_REF]). However, these dimensions do not give any more information than dim HH 1 (Λ).

We have Hom

Λ-Λ ((Λ ⊗ x Λ) ⊕ (Λ ⊗ y Λ), Λ) ∼ = Hom Λ-Λ (Λ ⊗ K KΓ 1 ⊗ K Λ, Λ) ∼ = Hom K (KΓ 1 , Λ) via the correspondence f ↔ g given by f (1 ⊗ x 1) = g(x) and f (1 ⊗ y 1) = g(y)
. We shall often identify g ∈ Hom K (KΓ 1 , Λ) with the pair (g(x), g(y)).

Define the following elements in Hom K (KΓ 1 , Λ) :

ϕ t = (x(yx) t , 0) for 0 t k -1, θ 0 = (1 + cx, cy + d(yx) k-1 )
,

θ 1 = (y(xy) k-1 , 0), θ -1 = (0, x(yx) k-1 ), θ 2 = ((xy) k , 0), θ -2 = (0, (xy) k ), ω = (y(xy) k-2 + c(yx) k-1 , 1), χ = (0, y).
Set B = {ϕ t , 1 t k -1; θ 1 ; θ -1 ; θ 2 ; θ -2 ; θ 0 } ⊂ Hom K (KΓ 1 , Λ) and

B ′ =                {ω; ϕ 0 } if k is odd and c = 0 = d, {ω; χ} if k is even and d = 0, {χ} if k is even and d = 0, {ω} if k is odd, c = 0 and d = 0, ∅ if k is odd and d = 0.
Lemma 4.5. With the notation above, B ∪ B ′ is a set of cocycle representatives of a basis of HH 1 (Λ).

Proof. The fact that the elements in B ∪ B ′ are cocycles can be checked easily (recall that char(K) = 2). Moreover, the classes of the cocycles

C := (x(yx) t , y(xy) t ); ((xy) t + (yx) t , 0); (0, (xy) t + (yx) t ); 1 t k -1
form a basis of Im(? • ∂ 1 ) (this basis will be useful when computing Lie brackets).

It is then straightforward to check that the cochains in B ∪ B ′ ∪ C are linearly independent, and the result follows, using the dimension of HH 1 (Λ) obtained previously.

As described in Subsection 2.1, we transport the usual Lie bracket on HH 1 (Λ) defined using the Bar resolution to a Lie bracket on HH 1 (Λ) defined using the minimal projective resolution. Note that we can identify

Q 2 with Λ ⊗ KΓ 0 KZ ⊗ KΓ 0 Λ with Z = x 2 -(yx) k-1 y + c(yx) k ; y 2 -d(xy) k .
Lemma 4.6. We use the same notation for a cocycle and for its cohomology class. The (potentially) non-zero brackets of basis elements are the following:

[ϕ t , ϕ t ′ ] = (t + t ′ )ϕ t+t ′ ; [θ -2 ; χ] = θ -2 ; [θ -1 , ϕ 0 ] = θ -1 ; [ϕ t , θ 0 ] =      θ 0 if t = 0 d(k -1)θ -2 if t = 1 0 if t > 1; [θ -2 , ϕ 0 ] = θ -2 ; [θ 1 ; ϕ 0 ] = θ 1 ; [ϕ t , ω] = (k -1)(θ 1 + cθ 2 ) if t = 1 0 if t = 1; [θ -2 , θ 0 ] = ϕ k-1 + cθ -2 ; [θ 0 , ω] = cω; [ϕ t , χ] = tϕ t ; [θ 2 , ω] = ϕ k-1 ; [ω, χ] = ω. [θ -2 , ω] = θ -1 if k > 2 θ -1 + θ 2 if k = 2; [θ 2 , θ 0 ] = θ 1 + cθ 2 ;
Proof. We refer to Lemma 4.15 for an example (in a non-local case) of the computation of a Lie bracket, the method here is similar.

We then compute the first two terms in the lower central series. Since they give no new information when d = 0, we only give the results for d = 0. Proposition 4.7. We keep the notation above and assume that d = 0. Then a basis of

L 1 (HH 1 (SD(1A) k 2 (c, d))
) is given by: (a) ϕ 2p+1 , 1 p k-3 2 ;

θ 1 + cθ 2 ; ϕ k-1 + cθ -2 if k is odd (there are no ϕ t if k = 3); (b) ϕ 2p+1 , 0 p k-2 2 ; θ 1 + cθ 2 ; θ -2 if k is even. Moreover, L 2 (HH 1 (SD(1A) k 2 (c, d))
) is generated by the following set: (a) ϕ 2p+1 , 2 p k-3 2 ; c(θ

1 + cθ 2 ); c(ϕ k-1 + cθ -2 ) if k is odd (there are no ϕ t if k = 3 or k = 5); (b) ϕ 2p+1 , 0 p k-2 2 ; c(θ 1 + cθ 2 ); θ -2 if k is even. where δ is the Kronecker symbol. In particular, dim L 2 (HH 1 (SD(1A) k 2 (c, d))) = k-1 2 -2δ c,0 + δ k,3 if k is odd, k 2 + 2 -δ c,0 if k is even.

Corollary 4.8.

There is no stable equivalence of Morita type between the algebras SD(1A) k 2 (0, 1) and SD(1A) k 2 (c, 1) for c = 0. In order to complete the proof of Theorem 4.1, we must prove the following result. Proposition 4.9. Assume that k is even. Then there is no stable equivalence of Morita type between the algebras SD(1A) k 1 and SD(1A) k 2 (1, 0). Proof. Let g be the Lie algebra HH 1 (SD(1A) k 1 ) and g ′ be the Lie algebra HH 1 (SD(1A) k 2 (1, 0)). These Lie algebras are not nilpotent (indeed, since [ϕ 1 , χ] = ϕ 1 , it follows that ϕ 1 is in all the terms of the lower central series for both Lie algebras).

Consider the subspace I of g generated by {ϕ t , 1 t k -1; θ 0 ; θ -1 ; θ 1 ; θ -2 ; θ 2 ; ω} and the subset I ′ of g ′ generated by {ϕ t , 1 t k -1; θ -1 ; θ 1 ; θ -2 ; θ 2 ; ω}. They are Lie ideals.

Moreover, the lower central series of I is given by L

1 (I) = span {ϕ t , 3 t k -1; θ 0 ; θ -1 ; θ 1 } and L i (I) = span ϕ t , 2 i+1 -1 t k -1 if i
2, so that it vanishes eventually and I is nilpotent. Similarly, the lower central series of I ′ is given by L

1 (I ′ ) = span {ϕ t , 3 t k -1; θ -1 ; θ 1 + θ 2 } and L i (I ′ ) = span ϕ t , 2 i+1 -1 t k -1 if i
2, so that it vanishes eventually and I ′ is nilpotent.

Since dim I = dim g -1 and g is not nilpotent, I is the nilradical of g. We now prove that I ′ is the nilradical of g ′ . Assume for a contradiction that it is not. Then it follows that there is a non-zero element in g ′ , that we can choose of the form u = λχ + µθ 0 , such that the subspace J generated by I ′ and u is a nilpotent ideal. Since [u, ω] = (λ + µ)ω and J is nilpotent, we must have λ + µ = 0 (otherwise ω would be in all the L i (J)). Therefore we may assume that u = χ + θ 0 .

We have [u,

θ 1 + θ 2 ] = θ 1 + θ 2 so that θ 1 + θ 2 ∈ L i (J) for all i, a contradiction. Therefore I ′ is the nilradical of g ′ .
It follows that the nilradicals of HH 1 (SD(1A) k 1 ) and HH 1 (SD(1A) k 2 (1, 0)) have different dimensions, and hence that HH 1 (SD(1A) k 1 ) and HH 1 (SD(1A) k 2 (1, 0)) are not isomorphic Lie algebras.

Remark. It can be noted that when k is odd and cc ′ = 0, the Lie algebras HH 1 (SD(1A) k 2 (c, 1)) and HH 1 (SD(1A) k 2 (c ′ , 1)) are isomorphic. Indeed, if {ϕ t , 1 t k -1; θ 1 ; θ -1 ; θ 2 ; θ -2 ; θ 0 } is a basis of HH 1 (SD(1A) k 2 (c, 1)) and

ϕ ′ t , 1 t k -1; θ ′ 1 ; θ ′ -1 ; θ ′ 2 ; θ ′ -2 ; θ ′ 0 is a basis of HH 1 (SD(1A) k 2 (c ′ , 1
)), the isomorphism is defined by

ϕ t → ϕ ′ t , θ 1 → θ ′ 1 , θ -1 → θ ′ -1 , θ 0 → c c ′ θ ′ 0 , θ 2 → c ′ c θ ′ 2 , θ -2 → c ′ c θ ′ -2 .
In the remaining unresolved cases, we do not know whether the first Hochschild cohomology groups are isomorphic or not.

Algebras of semi-dihedral type with two simple modules.

We have defined the algebras SD(2B) k,s 1 (c) and SD(2B) k,s 2 (c) of semi-dihedral type with two simple modules in Section 1. Note that when k 2, the ideal of relations for SD(2B) k,2 2 (c) is not admissible; a definition with an admissible ideal can be obtained by removing the loop η and adapting the relations.

We shall use the Lie algebra structure of the first Hochschild cohomology group to improve on the results in [START_REF] Zhou | Classifying tame blocks and related algebras up to stable equivalences of Morita type[END_REF], and to give a partial answer to the question of whether the algebras SD(2B) k,s 1 (c) and SD(2B) k ′ ,s ′ 2 (c ′ ) are stably equivalent of Morita type or not.

The main result of this section is the following.

Theorem 4.10. Let k, k ′ , s, s ′ be integers with {k, s} = {k ′ , s ′ } and let c be an element in {0, 1}.

(1) Assume that char(K) = 2. Then, for a ∈ {1, 2}, the algebras SD(2B) k,s a (0) and SD(2B) k ′ ,s ′ a (1) are not stably equivalent of Morita type. (2) In each of the following cases, the algebras SD(2B) k,s 1 (c) and SD(2B) k ′ ,s ′ 2 (c ′ ) are not stably equivalent of Morita type:

(i) char(K) = 2 and ks is even;

(ii) char(K) = 2, ks is odd and (c, c ′ ) = (0, 0);

(iii) char(K) = 3; (iv) char(K) = 2, 3, ks = 0 in K; (v) char(K) = 2, 3, ks = 0 in K and k + s -2ks = 0 in K.
(vi) char(K) = 2, 3, λ := 3 -1 2ks = 0 in K, µ := 2ksks = 0 in K, and the following subsets of K are not equal:

sλ -1 , 2sλ -1 , kλ -1 , 2kλ -1 , (sλ -1 ) -1 , (2sλ -1 ) -1 , (kλ -1 ) -1 , (2kλ -1 ) -1
and sµ -1 , 2sµ -1 , kµ -1 , 2kµ -1 , (sµ -1 ) -1 , (2sµ -1 ) -1 , (kµ -1 ) -1 , (2kµ -1 ) -1 .

The remainder of Section 4.2 is devoted to the proof of this result.

The first Hochschild cohomology group of SD(2B) k,s 1 (c).

Let Λ be the algebra SD(2B) k,s 1 (c) and let Γ be the quiver of type 2B. Let e 1 and e 2 denote the idempotents in Λ corresponding to the vertices.

Lemma 4.11. Define a sequence of

Λ-Λ-bimodules Q 2 ∂ 2 -→ Q 1 ∂ 1 -→ Q 0 ∂ 0 -→ Λ → 0 as follows. The modules Q n are given by Q 2 = (Λe 1 ⊗ e 1 Λ) ⊕ (Λe 1 ⊗ e 2 Λ) ⊕ (Λe 2 ⊗ e 1 Λ) ⊕ (Λe 2 ⊗ 1 e 2 Λ) ⊕ (Λe 2 ⊗ 2 e 2 Λ), Q 1 = 2 i,j=1 Λe i ⊗ e j Λ Q 0 = 2 i=1 Λe i ⊗ e i Λ,
where the subscripts on the tensor product symbols ⊗ denote the component of the free Λ-Λ-bimodule Q 2 . The map ∂ 0 is multiplication and the other maps are determined by ∂ 1 (e i(δ) ⊗ e t(δ) ) = δ ⊗ e t(δ)e i(δ) ⊗ δ for δ ∈ Γ 1 with origin i(δ) and endpoint t(δ)

∂ 2 (e 1 ⊗ e 1 ) = e 1 ⊗ α + α ⊗ e 1 - k-1 ∑ t=0 (βγα) t (β ⊗ e 1 + e 1 ⊗ γ) (αβγ) k-1-t - k-2 ∑ t=0 (βγα) t βγ ⊗ βγ(αβγ) k-2-t + c k-1 ∑ t=0 (βγα) t (e 1 ⊗ γα + β ⊗ α + βγ ⊗ e 1 ) (βγα) k-1-t ∂ 2 (e 1 ⊗ e 2 ) = e 1 ⊗ η + β ⊗ e 2 ∂ 2 (e 2 ⊗ e 1 ) = e 2 ⊗ γ + η ⊗ e 1 ∂ 2 (e 2 ⊗ 1 e 2 ) = e 2 ⊗ β + γ ⊗ e 1 ∂ 2 (e 2 ⊗ 2 e 2 ) = s-1 ∑ r=0 η r ⊗ η s-1-r - k-1 ∑ t=0 (γαβ) t (e 2 ⊗ αβ + γ ⊗ β + γα ⊗ e 2 ) (γαβ) k-1-t .
Then this sequence is the beginning of a minimal projective Λ-Λ-bimodule resolution of Λ.

Proof. It is easy to check that it is a complex, and that applying S i ⊗ Λ ? gives the beginning of a minimal projective right Λ-module resolution of the simple module S i for i = 1, 2. From these resolutions, we may determine dim Ext n Λ (S i , S j ) for n = 0, 1 and 2 and i, j = 1, 2 and check that Q n is the module in Happel's theorem [10]. Noting that Λ/ rad Λ = S 1 ⊕ S 2 as a right Λ-module, we then apply Lemma 2.1.

Remark. We can identify

Q 1 with Λ ⊗ KΓ 0 KΓ 1 ⊗ KΓ 0 Λ via a ⊗ δ ⊗ a ′ → ae iδ ⊗ e t(δ) a ′ and similarly Q 2 with Λ ⊗ KΓ 0 KZ ⊗ KΓ 0 Λ where Z = α 2 -βγ(αβγ) k-1 -c(αβγ) k ; βη; ηγ; γβ; η s -(γαβ) k .
Using the resolution above, we may now compute the dimension of HH 1 (Λ). The proof is straightforward and is omitted. Proposition 4.12. Let Λ be the algebra SD(2B) k,s 1 (c). Then ) and SD(2B) k ′ ,s ′ 1 (1). In order to go further, we now consider the Lie algebra structure of HH 1 (Λ).

dim HH 1 (Λ) =                                        k + s + 3 if char(K) = 2
In the sequel, we identify a morphism f ∈ Hom Λ-Λ (Q 1 , Λ) with g ∈ Hom KΓ 0 -KΓ 0 (KΓ 1 , Λ) such that g(δ) = f (e i(δ) ⊗ e t(δ) ) for all δ ∈ Γ 1 , and with the quadruple (g(α), g(β), g(γ), g(η)).

First assume that char(K) = 2. We start with a basis for HH 1 (SD(2B) k,s 1 (c)). Lemma 4.14. We define cocycles in Hom KΓ 0 -KΓ 0 (KΓ 1 , Λ) as follows.

ϕ t = (α(βγα) t , 0, 0, 0) for 1 t k -1 θ r = (0, 0, 0, η r+1 ) for 1 r s -1 ψ = ((αβγ) k , 0, 0, 0) χ = (e 1 + cα, cβ, 0, 0) ω = ((βγα) k-1 βγ + c(αβγ) k , 0, 0, 0)
ϕ 0 = (0, β, 0, 0) and θ 0 = (0, 0, 0, η) if k and s are both even,

ζ 1 = (0, sβ, 0, kη) if k + s is odd ζ 0 = (α, 0, 0, η) if k
B ′ =          {ϕ 0 ; θ 0 } if k and s are both even {ζ 1 } if k + s is odd {ζ 0 }
if k and s are both odd and c = 0 ∅ otherwise.

form a basis of HH 1 (SD(2B) k,s 1 (c)).

Proof. This is proved by computing explicitly Ker(? • ∂ 2 ) and Im(? • ∂ 1 ). We omit the details, but we give the following basis of Im(? • ∂ 1 ), which is useful when computing brackets of elements in HH 1 (Λ);

(α(βγα) t , -β(γαβ) t , 0, 0), (α(βγα) t , 0, -γ(αβγ) t , 0), ((αβγ) t -(βγα) t , 0, 0, 0) (0, αβ(γαβ) t , -γα(βγα) t , 0), (0, αβ, -γα, 0) (0, β, -γ, 0)

with 1 t k -1.
We may now compute the brackets of these basis elements.

Lemma 4.15. We use the notation in the previous lemma. We describe the (potentially) non-zero brackets in HH 1 (SD(2B) k,s 1 (c)). For all k and s we have

[ϕ t , ϕ t ′ ] = (t + t ′ )ϕ t+t ′ if 0 t, t ′ k -1, t + t ′ k -1, [θ r , θ r ′ ] = (r + r ′ )θ r+r ′ if 0 r, r ′ s -1, r + r ′ s -1, [ψ, χ] = ω [χ, ω] = cω. If moreover k + s is odd, then [ϕ t , ζ 1 ] = tsϕ t and [θ r , ζ 1 ] = rkθ r .
If instead k and s are both odd and c = 0, then

[ϕ t , ζ 0 ] = tϕ t , [χ, ζ 0 ] = χ, [θ r , ζ 0 ] = rθ r , [ω, ζ 0 ] = ω.
Proof. In order to illustrate the method, let us determine the bracket [χ, ω]. We view χ and ω as maps in Hom KΓ 0 -KΓ 0 (KΓ 1 , Λ). First, for every δ ∈ Γ 1 , we replace each instance of δ in ω(α) by χ(δ), that is, we replace every α in turn with e 1 + cα and every β by cβ, and we add the results. Since char(K) = 2, we get

(k -1)c(βγα) k-1 βγ + kc 2 (αβγ) k + ce 1 βγ(αβγ) k + kc(βγα) k-1 βγ + kc 2 (αβγ) k = 0.
We apply the same procedure to ω(β) = 0, ω(γ) = 0 and ω(η) = 0, and we obtain 0 in all cases.

Next, we exchange the roles of χ and ω. We replace each instance of δ in χ(α) and χ(β) by ω(δ). We get

α →c(βγα) k-1 βγ + c 2 (αβγ) k β →0
and of course γ and η are sent to 0.

Finally, we subtract the two quantities, which gives the map

α →c(βγα) k-1 βγ + c 2 (αβγ) k β →0 γ →0 η →0, that is, cω. Therefore, [χ, ω] = cω.
The other brackets are computed in the same way. Note that we work modulo Im(? • ∂ 1 ).

It can be noted that the Lie algebras HH 1 (SD(2B) k,s 1 (c)) and HH 1 (SD(2B) s,k 1 (c)) for c ∈ {0, 1} (same c) are isomorphic. Indeed, if we consider the basis of HH 1 (SD(2B) k,s 1 (c)) given in Lemma 4.14 and the sim- ilar basis contained in ϕ

′ r , 0 r s -1, θ ′ t , 0 t k -1, ψ ′ , χ ′ , ω ′ , ζ ′ 0 , ζ ′ 1 of HH 1 (SD(2B) s,k 1 (c)), the isomorphism is given by ϕ t → θ ′ t , θ r → ϕ ′ r , ψ → ψ ′ , χ → χ ′ , ω → ω ′ , ζ 0 → ζ ′ 0 , ζ 1 → ζ ′ 1
on the elements that are actually present in each case. Therefore the Lie algebra structure of HH 1 (Λ) does not help to separate the pairs of parameters (k, s) and (s, k). We already know from Corollary 4.13 that if k and s are both odd and {k, s} = {k ′ , s ′ }, then SD(2B) k,s 1 (0) and SD(2B) k ′ ,s ′ 1 (1) are not stably equivalent of Morita type.

If k + s is odd, the second term in the lower central series has dimension k+s-5

2 + c + δ k,3 + 2δ k,1 (if k is odd and s is even, it is spanned by ϕ 2p+1 ; θ 2q+1 ; cω; 2 p k-3 2 , 0 q s 2 -1 )
, therefore it follows that HH 1 (SD(2B) k,s 1 (0)) and HH 1 (SD(2B) k ′ ,s ′ 1 (1)), with {k, s} = {k ′ , s ′ }, are not iso- morphic Lie algebras, and hence that SD(2B) k,s 1 (0) and SD(2B) k ′ ,s ′ 1 (1) are not stably equivalent of Morita type.

Similarly, if k and s are both even, the second term in the lower central series of HH 1 ) are not stably equivalent of Morita type.

(SD(2B) k,s 1 (c)) is spanned by ϕ 2p+1 ; θ 2q+1 ; cω; 0 p k 2 -1, 0 q s 2 -1 and has dimension k+s 2 + c, therefore SD(2B) k,s 1 (0) and SD(2B) k ′ ,s ′ 1 (1
We have therefore proved the following result.

Proposition 4.16. Assume that char(K) = 2. Then the algebras SD(2B) k,s 1 (0) and SD(2B) k ′ ,s ′ 1 (1), with {k, s} = {k ′ , s ′ }, are not stably equivalent of Morita type.

Next assume that char(K) = 3. The Lie algebra structure of HH 1 (SD(2B) k,s 1 (c)) is determined in the following lemma. Lemma 4.17. Define the following cocycles in Hom K (KΓ 1 , Λ).

ϕ t = (α(βγα) t , 0, 0, 0) if 1 t k -1 ϕ 0 = (0, β, 0, 0) θ r = (0, 0, 0, η r+1 ) if 0 r s -1 ψ = ((αβγ) k , 0, 0, 0) ω = (α + c(βγα) k-1 βγ + c(αβγ) k , -β, 0, 0)
Then a basis of HH 1 (SD(2B) k,s 1 (c)) is given by the cohomology classes of the elements in B ∪ B ′ where B = {ϕ t , θ r , ψ, ω;

1 t k -1, 1 r s -1} and B ′ =          {ϕ 0 , θ 0 } if k and s are both 0 in K, {ϕ 0 } if k is 0 and s is not 0 in K, {θ 0 } if k is not 0 and s is 0 in K, ∅ if ks is not 0 in K.

The (potentially) non-zero brackets are given by

[ϕ t , ϕ t ′ ] = (t ′ -t)ϕ t+t ′ if 0 t, t ′ k -1, t + t ′ k -1, [θ r , θ r ′ ] = (r ′ -r)θ r+r ′ if 0 r, r ′ s -1, r + r ′ s -1, [ψ, ω] = ψ.
It is easy to check that the Lie algebras HH 1 (SD(2B) k,s 1 (c)) and HH 1 (SD(2B) k ′ ,s ′ 1 (c ′ )) are isomorphic if {k, s} = {k ′ , s ′ } and c, c ′ ∈ {0, 1}. The Lie algebra structure does not provide any new information at this point, however is will be useful in order to distinguish the algebras SD(2B) k,s 1 (c) and SD(2B) k,s 2 (c) later.

The situation when char(K) = 2, 3 is similar. Nevertheless, we give the Lie algebra structure, since it will be used later.

Lemma 4.18. Define the following cocycles in Hom K (KΓ 1 , Λ).

ϕ t = (α(βγα) t , 0, 0, 0) if 1 t k -1 ϕ 0 = (0, β, 0, 0) θ r = (0, 0, 0, η r+1 ) if 0 r s -1 ψ = ((αβγ) k , 0, 0, 0) ω = (ksα + ksc(βγα) k-1 βγ, (3 -k)sβ, 0, 3kη).
Then a basis of HH 1 (SD(2B) k,s 1 (c)) is given by the cohomology classes of the elements in B ∪ B ′ where B = {ϕ t , θ r , ψ; 1 t k -1, 1 r s -1} and B ′ = {ϕ 0 , θ 0 } if k and s are both 0 in K, {ω} if k and s are not both 0 in K. The (potentially) non-zero brackets are given by

[ϕ t , ϕ t ′ ] = (t ′ -t)ϕ t+t ′ if 0 t, t ′ k -1, t + t ′ k -1, [θ r , θ r ′ ] = (r ′ -r)θ r+r ′ if 0 r, r ′ s -1, r + r ′ s -1, [ω, ϕ t ] = 3tsϕ t for 1 t k -1, if k = 0 or s = 0 in K [ω, θ r ] = 3rkθ r for 1 r s -1, if k = 0 or s = 0 in K [ω, ψ] = 2ksψ (only if k = 0 or s = 0 in K).

The first Hochschild cohomology group of SD(2B) k,s

2 (c). Let Λ be the algebra SD(2B) k,s 2 (c) and let Γ be the quiver of type 2B.

Lemma 4.19. Define a sequence of Λ-Λ-bimodules

Q 2 = 2 z∈Z Λe i(z) ⊗ e t(z) Λ ∂ 2 -→Q 1 = 2 δ∈Γ 1 Λe i(δ) ⊗ e t(δ) Λ ∂ 1 -→ ∂ 1 -→ Q 0 = 2 i=1 Λe i ⊗ e i Λ ∂ 0 -→ Λ → 0 as follows. The set Z is α 2 -c(βγα) k , βη -αβ(γαβ) k-1 , ηγ -γα(βγα) k-1 , γβ -η s-1 if s > 2 and α 2 -c(αβγ) k , βγβ -(αβγ) k-1 αβ, γβγ -(γαβ) k-1 γα if s = 2. The map ∂ 0 is multiplication, ∂ 1 is determined by ∂ 1 (e i(δ) ⊗ e t(δ) ) = δ ⊗ e t(δ) -e i(δ) ⊗ δ for δ ∈ Γ 1 and ∂ 2 is determined by e 1 ⊗ e 1 →e 1 ⊗ α + α ⊗ e 1 -c k-1 ∑ t=0 (αβγ) t (α ⊗ γ + αβ ⊗ e 1 + e 1 ⊗ βγ) (αβγ) k-1-t e 1 ⊗ e 2 →e 1 ⊗ η + β ⊗ e 2 - k-1 ∑ t=0 (αβγ) t (e 1 ⊗ β + α ⊗ e 2 ) (γαβ) k-1-t - k-2 ∑ t=0 (αβγ) t αβ ⊗ αβ(γαβ) k-2-t e 2 ⊗ e 1 →e 2 ⊗ γ + η ⊗ e 1 - k-1 ∑ t=0 (γαβ) t (e 2 ⊗ α + γ ⊗ e 1 ) (βγα) k-1-t - k-2 ∑ t=0 (γαβ) t γα ⊗ γα(βγα) k-2-t e 2 ⊗ e 2 →e 2 ⊗ β + γ ⊗ e 2 - s-2 ∑ r=0 η r ⊗ η s-2-r if s > 2 and by e 1 ⊗ e 1 → e 1 ⊗ α + α ⊗ e 1 -c k-1 ∑ t=0 (βγα) t (e 1 ⊗ γα + β ⊗ α + βγ ⊗ e 1 ) (βγα) k-1-t e 1 ⊗ e 2 → e 1 ⊗ γβ + β ⊗ β + βγ ⊗ e 2 - k-1 ∑ t=0 (αβγ) t (e 1 ⊗ β + α ⊗ e 2 ) (γαβ) k-1-t - k-2 ∑ t=0 (αβγ) t αβ ⊗ αβ(γαβ) k-2-t e 2 ⊗ e 1 → e 2 ⊗ βγ + γ ⊗ γ + γβ ⊗ e 1 - k-1 ∑ t=0 (γαβ) t (e 2 ⊗ α + γ ⊗ e 1 ) (βγα) k-1-t - k-2 ∑ t=0 (γαβ) t γα ⊗ γα(βγα) k-2-t if s = 2.
Then this sequence is the beginning of a minimal projective Λ-Λ-bimodule resolution of Λ.

Proof. The proof is the same as that of Lemma 4.11.

Using the resolution above, we may now compute the dimension of HH 1 (Λ).

The proof is straightforward and is omitted.

Proposition 4.20. Let Λ be the algebra SD(2B) k,s 2 (c). Then dim HH 1 (Λ) =                k + s + 3 -c if char(K) = 2

and k and s are both even

k + s + 2 -c if char(K) = 2 and k + s is odd k + s + 2 -2c if char(K) = 2
[ψ 1 ,

ϕ 0 ] = ψ 1 , [ψ 1 , θ 0 ] = ψ 1 , [ψ 0 , ϕ 0 ] = (1 -c)ψ 0 , [ψ 0 , θ 0 ] = (1 -c)ψ 0 , [ψ 1 , ψ 0 ] = (1 -c)ω,
and, if ks = 0 in K and c = 0,

[ϕ t , χ] = tϕ t (t 1), [θ r , χ] = rθ r (r 1), [ω, χ] = ω, [ψ 0 , χ] = ψ 0 .
Remark. It is easy to check that HH 1 (SD(2B) k,s 2 (c)) and HH 1 (SD(2B) s,k 2 (c)) are isomorphic Lie algebras.

Corollary 4.25. Assume that char(K) = 2. Let k, k ′ , s, s ′ be integers such that {k, s} = {k ′ , s ′ } and let c, c ′ be in {0, 1}. Suppose that one of the following holds: (i) k and s are both even and cc ′ = 0; (ii) k + s is odd; (iii) ks is odd and (c, c ′ ) = (0, 0).

Then there is no stable equivalence of Morita type between the algebras SD(2B) k,s 2 (c) and SD(2B) k ′ ,s ′ 1 (c ′ ).

Proof. Set g = HH 1 (SD(2B) k,s 2 (c)) and g ′ = HH 1 (SD(2B) k ′ ,s ′ 1 (c ′ )). Let L i (g) and L i (g ′ ) be the i th term in the lower central series of g and g ′ respectively. Write the basis elements in g ′ with dashes.

In case (i) L 2 (g) is the span of the set

ϕ 2p+1 ; θ 2q+1 ; ψ 1 ; (1 -c)ω; (1 -c)ψ 0 ; 0 p k 2 -1, 0 q s 2 -1 and
its dimension is k+s 2 + 3 -2c, and L 2 (g ′ ) is the span of the set ϕ ′ 2p+1 ; θ ′ 2q+1 ; c ′ ω ′ ; 0 p k ′ 2 -1, 0 q s ′ 2 -1 and it has dimension k+s 2 + c ′ . These dimensions are different when cc ′ = 0, therefore g and g ′ are not isomorphic.

In case (ii), we may assume that k = k ′ is odd and s = s ′ is even.

Here, L 1 (g) is the span of

ϕ 2p+1 ; θ 2q+1 ; ψ 1 ; (1 -c)ϕ k-1 ; (1 -c)ω; (1 -c)ψ 0 ; 1 p k-3 2 , 0 q s 2 -1 so its dimension is k+s-1 2 + 3 -3c + δ k,1 , and L 1 (g ′ ) is the span of ϕ ′ 2p+1 ; θ ′ 2q+1 ; ω ′ ; 1 p k-3 2 , 0 q s 2 -1 and has dimension k+s-1 2 + δ k,1 ,
and these dimensions are different when (c, c ′ ) = (1, 0). Moreover, if i > k-3 2 , we have dim L i (g) = s 2 + 4 -3c and dim L i (g ′ ) = s 2 + c ′ , which are different when c = 1 and c ′ = 0. Therefore g and g ′ are not isomorphic.

Finally, in case (iii), again assume that k = k ′ and s = s ′ for the proof. In this case, L 1 (g) is the span of

ϕ 2p+1 ; θ 2q+1 ; (1 -c)ω; (1 -c)ψ 0 ; (1 -c)ϕ 1 ; (1 -c)θ 1 ; 1 p k-3 2 , 1 q s-3 2 and L 1 (g ′ ) is spanned by ϕ ′ 2p+1 ; θ ′ 2q+1 ; ω ′ ; (1 -c ′ )χ ′ ; (1 -c)ϕ ′ 1 ; (1 -c ′ )θ ′ 1 ; 1 p k-3 2 , 1 q s-3 2 , therefore dim L 1 (g) -dim L 1 (g ′ ) = (c ′ -c)δ k,1 + 3c ′ -4c
, which is non-zero when (c, c ′ ) = (0, 0). Therefore g and g ′ are not isomorphic.

Remark. If ks is odd, then the Lie algebras HH 1 (SD(2B) k,s 1 (0)) and HH 1 (SD(2B) k,s 2 (0)) are isomorphic, so that the Lie algebra structure of the first Hochschild cohomology group does not bring anything new. Indeed, if {ϕ t ; θ r ; ψ; χ; ω; ζ 0 ; 1 t k -1; 1 r s -1} is our basis of HH 1 (SD(2B) k,s 1 (0)) and ϕ ′ t ; θ ′ r ; ψ ′ 0 ; ψ ′ 1 ; ω ′ ; χ ′ ; 1 t k -1, 1 r s -1 is our basis of HH 1 (SD(2B) k,s 2 (0)), then the isomorphism is determined by

ϕ t → ϕ ′ t , θ r → θ ′ r , ω → ω ′ , ψ → ψ ′ 1 , ζ 0 → χ ′ , χ → ψ ′ 0 .
We now assume that char(K) = 2. In order to differentiate the algebras of type SD(2B) 1 and SD(2B) 2 up to stable equivalence of Morita type, we give the Lie algebra structure when char(K) = 2. Lemma 4.26. We define cocycles in Hom KΓ 0 -KΓ 0 (KΓ 1 , Λ) as follows.

ϕ t = (α(βγα) t , 0, 0, 0) if 1 t k -1 θ r = (0, 0, 0, η r+1 ) if 2 r s -1 θ 1 = (0, (s -1)(αβγ) k-1 αβ, 0, η 2 ) ψ = ((αβγ) k , 0, 0, 0) ϕ 0 = (α -c(βγα) k-1 βγ, 0, 0, 0) and θ 0 = (0, β, 0, η) -ϕ 0 if k = 0 and s = 0 in K, ω = (2(k + s -ks)α + c(3ks -2k -2s)(βγα) k-1 βγ, 2k(s -1)β, 0, 2kη) if k = 0 or s = 0 in K. Set B = {ϕ t ; θ r ; ψ; 1 t k -1; 1 r s -1} and B ′ = {ϕ 0 ; θ 0 } if k and s are both zero in K, {ω} if k = 0 or s = 0 in K.
Then the cohomology classes of the elements in B ∪ B ′ form a basis of HH 1 (SD(2B) k,s 2 (c)). The (potentially) non-zero brackets are given by

[ϕ t , ϕ t ′ ] = (t ′ -t)ϕ t+t ′ if 1 t, t ′ k -1, t + t ′ k -1, [θ r , θ r ′ ] = (r ′ -r)θ r+r ′ if 1 r, r ′ s -1, r + r ′ s -1, [ϕ 0 , ϕ t ] = tϕ t , [θ 0 , θ r ] = rθ r , [ψ, ϕ 0 ] = ψ, [ψ, θ 0 ] = -ψ, [ω, ϕ t ] = 2stϕ t , [ω, θ r ] = 2krθ r , [ω, ψ] = 2(2ks -k -s)ψ.
Remark. Here again, if we specialise to s = 2, a basis of HH 1 (SD(2B) k,2 2 (c)) is given by the non-zero cohomology classes of the following elements of Hom K (KΓ 1 , Λ), written as (g(α), g(β), g(γ)):

ϕ t = (α(βγα) t , 0, 0, 0) if 1 t k -1, θ 1 = (0, (αβγ) k-1 αβ, 0), ω = ((2 -k)α + 2c(k -1)(βγα) k-1 βγ, kβ, 0), ψ = ((αβγ) k , 0, 0)
and the brackets are the same as those given in Lemma 4.26 above.

It is easy to check that the Lie algebras HH 1 (SD(2B) k,s 2 (c)) and HH 1 (SD(2B) k ′ ,s ′ 2 (c ′ )) are isomorphic if {k, s} = {k ′ , s ′ } and c, c ′ ∈ {0, 1}. The Lie algebra structure does not provide any new information within the family SD(2B) k,s 2 (c), but we have the following result. Corollary 4.27. Assume that char(K) = 3, that {k, s} = {k ′ , s ′ } and that c, c ′ ∈ {0, 1} . Then there is no stable equivalence of Morita type between SD(2B) k,s 2 (c) and SD(2B) k ′ ,s ′ 1 (c ′ ).

Assume that char(K) = 2, 3 and that either ks = 0 in K, or that ks = 0 and 2ksks = 0 in K. Then there is no stable equivalence of Morita type between SD(2B) k,s 2 (c) and SD(2B) k ′ ,s ′ 1 (c ′ ).

Proof. Set g = HH 1 (SD(2B) k,s 2 (c)) and g ′ = HH 1 (SD(2B) k ′ ,s ′ 1 (c ′ )). Let L i (g) (respectively L i (g ′ )) denote the i th term in the lower central series of g (respectively g ′ ).

• First assume that char(K) = 3. We already know from Corollary 4.21 that there is no stable equivalence of Morita type between SD(2B) k,s 2 (c) and SD(2B) k ′ ,s ′ 1 (c ′ ) when ks = 0 in K. Therefore, assume that that ks = 0 in K. Then the centre of the Lie algebra g is spanned by ψ if (k, s) = (1, 1) in K 2 and vanishes otherwise, so its dimension is at most 1, whereas the centre of the Lie algebra g ′ is spanned by {ϕ k-1 , θ s-1 } so its dimension is at least 2. Therefore the algebras g and g ′ are not isomorphic and the first part of the corollary follows.

• If char(K) = 2, 3 and ks = 0, then dim L 1 g = dim L 1 g ′ + 1 (the extra element is ψ), hence the Lie algebras g and g ′ are not isomorphic.

• If char(K) = 2, 3 and ks = 0 and k + s -2ks = 0 in K, then the centre of g ′ is zero, whereas that of g is spanned by ψ and has dimension 1, hence the Lie algebras g and g ′ are not isomorphic.

Proposition 4.28. Assume that char(K) = 2, 3 and that {k, s} = {k ′ , s ′ }. Put λ = 3 -1 2ks = 0 and µ = 2ksks and assume that λµ = 0 in K and that the following subsets of K are not equal:

E λ = sλ -1 , 2sλ -1 , kλ -1 , 2kλ -1 , (sλ -1 ) -1 , (2sλ -1 ) -1 , (kλ -1 ) -1 , (2kλ -1 ) -1
and E µ = sµ -1 , 2sµ -1 , kµ -1 , 2kµ -1 , (sµ -1 ) -1 , (2sµ -1 ) -1 , (kµ -1 ) -1 , (2kµ -1 ) -1 .

Then there is no stable equivalence of Morita type between SD(2B) k,s 2 (c) and SD(2B) k ′ ,s ′ 1 (c ′ ).

Proof. For λ ∈ K * , let g λ be the 6-dimensional Lie algebra with basis {e 0 , . . . , e 5 } and whose bracket is determined by [e 0 , e i ] = ν i e i with (ν 1 , ν 2 , ν 3 , ν 4 , ν 5 ) = (s, 2s, k, 2k, λ). Now consider the algebra HH 1 (SD(2B) k,s 2 (c))/D 2 (HH 1 (SD(2B) k,s 2 (c))) where D 2 (g) is the second term in the derived series of g. Since D 2 (HH 1 (SD(2B) k,s 2 (c))) is spanned by the ϕ t and θ r for t 3 and r 3, this is a Lie algebra that is isomorphic to g λ with λ = 2ksks, via the isomorphism given by e 0 = 1 2 ω, e 1 = ϕ 1 , e 2 = ϕ 2 , e 3 = θ 1 , e 4 = θ 2 and e 5 = ψ (recall that 2 and 3 are invertible in K). Similarly, the Lie algebra HH 1

(SD(2B) k,s 1 (c ′ ))/D 2 (HH 1 (SD(2B) k,s 1 (c ′ ))) is isomorphic to g µ with µ = 2ks
3 (the isomorphism sends ω to 1 3 ω in this case). If HH 1 (SD(2B) k,s 2 (c)) and HH 1 (SD(2B) k,s 1 (c ′ )) are isomorphic, then so are g λ and g µ .

We now prove that if the sets E λ and E µ are distinct, then the Lie algebras g λ and g µ are not isomorphic, using generalised derivations.

For ρ ∈ K * , we consider der g λ (ρ, 1, 1). Let D be a (ρ, 1, 1)-derivation of g λ . Set D(e j ) = ∑ 5 i=0 a ij e i for i = 0, 1, . . . , 5. Then, for i = 1, 2, . . . , 5, we have ρD([e 0 , e i ]) = [D(e 0 ), e i ] + [e 0 , D(e i )],

which is equivalent to the set of equations

     ρν i a 0i = 0 for 1 i 5, (ρν i -ν j )a ji = 0 for 1 i = j 5, (ρ -1)a ii = a 00 for 1 i 5 that is equivalent to      a 0i = 0 for 1 i 5, (ρν i -ν j )a ji = 0 for 1 i = j 5, (ρ -1)a ii = a 00 for 1 i 5.
Note that the equations that come from the identities ρD([e j , e i ]) = [D(e j ), e i ] + [e j , D(e i )] for 1 i = j 5 are a consequence of the first five equations above. Therefore these equations characterise D. The subset of the equations above that involve the parameter λ is

(ρλ -s)a 15 = 0 (ρs -λ)a 51 = 0 (ρλ -2s)a 25 = 0 (2ρs -λ)a 52 = 0 (ρλ -k)a 35 = 0 (ρk -λ)a 53 = 0 (ρλ -2k)a 45 = 0 (2ρk -λ)a 54 = 0.
Therefore, if µ ∈ K * is another parameter, and if ρ ∈ E µ and ρ ∈ E λ , then there are strictly fewer equations characterising der g µ (ρ, 1, 1) than those characterising der g λ (ρ, 1, 1). It follows that dim der g λ (ρ, 1, 1) < dim der g µ (ρ, 1, 1) and hence that g λ and g µ are not isomorphic Lie algebras.

Finally, Theorem 4.10 is obtained by combining Propositions 4.16 and 4.28 and Corollaries 4.13, 4.21, 4.25 and 4.27.

ALGEBRAS OF QUATERNION TYPE

As we mentioned in Subsection 1.4, we shall only consider the local tame symmetric algebras of quaternion type. Using a result of Erdmann and Skowro ński, in this case we can compute the dimensions of all the Hochschild cohomology groups. The dimension of the first Hochschild cohomology group, as well as the Lie algebra structure of the first cohomology group HH 1 (Λ), give new information on stable equivalence of Morita type, but we are not able to distinguish all the algebras. The main result of this subsection is Corollary 5.6.

Once more, we assume that the field K has characteristic 2. We have defined the algebras Q(1A) k 1 and Q(1A) k 2 (c, d) in Subsection 1.1. In these algebras, the following relations hold: x 3 = (xy) k = (yx) k = y 3 and x 4 = 0. The element z := (xy) k-1 + (yx) k-1 is central in these algebras by [2], therefore from the equalities y 2 z = yzy = zy 2 , using the other relations, we obtain x 2 y = 0 = yx 2 . It then follows that xy 2 = 0 = y 2 x, and that y(xy) k = y 4 = 0, even in Q(1A) k 1 . We may therefore view Q(1A) 5.1. Dimensions of the Hochschild cohomology groups. Erdmann and Skowro ński have shown in [START_REF] Erdmann | The stable Calabi-Yau dimension of tame symmetric algebras[END_REF] that Λ is periodic of period 4 and they give explicitly a minimal projective resolution of Λ as a Λ-Λ-bimodule in [3, Theorem 5.9]:

(1) We shall now give the dimensions of all the Hochschild cohomology groups for Λ. Note that for Λ = Q(1A) k 1 , these were already given in [START_REF] Generalov | The Hochschild cohomology of quaternion-type algebras. I. Generalized quaternion groups (Russian)[END_REF]. Proposition 5.1. We have the following dimensions Λ e (Λ) ∼ = Λ. In particular, HH i+4 (Λ) = HH i (Λ) for all i 1. Moreover, Λ is periodic Frobenius of period π 4 and dimension π -1 in the sense of [START_REF] Eu | Calabi-Yau Frobenius algebras[END_REF]. We also deduce that Ω π Λ ∼ = id modΛ , so that if we assume π 3, then the stable Calabi- Yau dimension of Λ in the sense of [START_REF] Erdmann | The stable Calabi-Yau dimension of tame symmetric algebras[END_REF] is at most 2. However, by [3, Proposition 5.8 and Corollary 5.10], this last stable dimension is equal to 3. Therefore π = 4.

0 → Λ j → Λ ⊗ Λ ∂ 3 → (Λ ⊗ Λ) 2 = (Λ ⊗ x Λ) ⊕ (Λ ⊗ y Λ) ∂ 2 → ∂ 2 → (Λ ⊗ x Λ) ⊕ (Λ ⊗ y Λ) ∂ 1 → Λ ⊗ Λ ∂ 0 → Λ → 0 where ∂ 0 is multiplication, ∂ 1 (1 ⊗ α 1) = α ⊗ 1 + 1 ⊗ α for α ∈ {x,
dim HH i (Q(1A) k 1 ) = k + 3 if i ≡ 0, 3 (mod 4) k + 5 if i ≡ 1, 2 (mod 4) dim HH i (Q(1A) k 2 (c, d)) =      k + 3 if i ≡ 0,
It now follows from [4, Theorem 2.3.27(ii)], using the fact that Λ is symmetric (hence the K-dual Λ * is isomorphic to Λ as a Λ-Λ-bimodule) and using Corollary 2.1.13 and Definitions 2.1.22 to 2.1.28 in [START_REF] Eu | Calabi-Yau Frobenius algebras[END_REF] as well as the two-sided resolution of Λ obtained from [START_REF] Erdmann | The stable Calabi-Yau dimension of tame symmetric algebras[END_REF], that dim HH 3-i (Λ) = dim HH i (Λ) for i = 0, 1, 2, 3 and therefore that dim HH 2 (Λ) = dim HH 1 (Λ). (A direct computation using the resolution in [START_REF] Erdmann | The stable Calabi-Yau dimension of tame symmetric algebras[END_REF] also gives this last fact.) Moreover, computing the dimensions from the complex obtained from (1), we get dim HH 3 (Λ) = dim HH 4 (Λ) = dim HH 0 (Λ)dim(Im(? • j • ∂ 0 )) = dim HH 0 (Λ). The result follows.

We can therefore resolve some of the classification questions in this case (note that the first Hochschild cohomology group is enough for this).

Corollary 5.2. If k is odd then there is no stable equivalence of Morita type between Q(1A) k 1 and Q(1A) k 2 (c, d). 5.2. Lie algebra structure on HH 1 (Λ). Let Γ be a quiver of type 1A, with arrows x and y. Then the local tame symmetric algebras of quaternion type may be defined as KΓ/I k 2 (c, d) for (c, d) ∈ K 2 . Let Λ be such an algebra. We then have Hom Λ-Λ ((Λ ⊗ We then have the following result.

Lemma 5.3. We keep the notation above.

(1) If k is odd and (c, d) = (0, 0) then a basis for HH 1 (Q(1A) k 2 (c, d)) is given by the cohomology classes of {θ 1 ; θ -1 ; θ 2 ; θ -2 ; ϕ t , 1 t k -1; ψ := dχ + cω} .

  ; x 2 + (yx) k-1 y + c(xy) k ; y 2 + (xy) k-1 x + d(yx) k ; (xy) k x; (yx) k y and let Λ := K x, y /I k (c, d) be one of the algebras Q(1A) k 1 or Q(1A) k 2 (c, d). Clearly, Q(1A) k 2 (c, d) ∼ = Q(1A) k 2 (d, c).

  x Λ) ⊕ (Λ ⊗ y Λ), Λ) ∼ = Hom K (KΓ 1 , Λ) via the correspondence f ↔ g given by f (1 ⊗ x 1) = g(x) and f (1 ⊗ y 1) = g(y). Moreover, if Z = x 2 + (yx) k-1 + c(yx) k , y 2 + (xy) k-1 x + d(xy) k , we can identify Q 2 with Λ ⊗ KΓ 0 KZ ⊗ KΓ 0 Λ.Define the following elements in Hom K (KΓ 1 , Λ) (as pairs (g(x), g(y))):ϕ t = (x(yx) t , 0) for 1 t k -1, θ 1 = (y(xy) k-1 , 0), θ -1 (0, x(yx) k-1 ), χ = (1 + cx, x(yx) k-2 + d(xy) k-1 ), θ -2 = (0, (xy) k ), θ 2 ((xy) k , 0), ω = (y(xy) k-2 + c(yx) k-1 , 1 + dy).

Algebras of semi-dihedral type.

  The remaining question for the local algebras of semi-dihedral type is whether the stable equivalence of Morita type classes for the algebras SD(1A) k 2 (c, d) depend on

	t ,	(βγα) t+1 ,	(αβγ) t α,	(βγα) t βγ,	(βγα) t β,
	(αβγ) t αβ,	(γαβ) t γ,	(γαβ) t γα,	(γαβ) t+1 ,	η r .

Moreover, their centre has dimension k + s + 2.

1.2. Algebras of dihedral type. In the case of algebras of dihedral type, Zhou and Zimmermann proved that the classification up to stable equivalence of Morita type mostly coincides with the classification up to derived equivalence, but a few questions in the classification remain. As they stated in [22, Remark 4.2 and Remark 7.2], in order to complete the classification of the algebras of dihedral type we must determine whether the algebras D(1A) k 2 (0) and D(1A) k 2 (1) are stably equivalent of Morita type or not. We shall prove that they are not in Corollary 3.3.

1.3.

  Lie algebra structure on HH 1 (Λ). We shall now improve on Corollary 4.4 us- ing the Lie algebra structure on HH 1 (Λ) where Λ is one of the algebras SD k 1

	Corollary 4.4. For any k	2, an algebra in the set SD(1A) k 1 ; SD(1A) k 2 (1, 0)
	is not stably equivalent of Morita type to an algebra in the set
	SD(1A) k 2 (0, 1); SD(1A) k 2 (c, 1), c = 0 .
	Moreover, if k is odd there is no stable equivalence of Morita type between the algebras
	SD(1A) k 1 and SD(1A) k 2 (1, 0).
	4.1.2.

k is even and d = 0 or if k is odd and c = 0 = d, k + 5 if k is even and d = 0 or if k is odd, c = 0 and d = 0, k + 4 if k is odd and d = 0.

  and k and s are both even k + s + 2 if char(K) = 2 and k and s are not both even and ksc= 0 in K k + s + 1 if char(K) = 2 and ksc = 0 in K k + s + 2 if char(K) = 3and k and s are both 0 in K k + s + 1 if char(K) = 3 and k and s are not both 0 in K but ks = 0 in K k + s if char(K) = 3 and ks = 0 in K k + s + 1 if char(K) = 2, 3 and k and s are both 0 in K

k + s if char(K) = 2, 3 and k and s are not both 0 in K.

Corollary 4.13. If char(K) = 2 and k and s are both odd and if {k ′ , s ′ } = {k, s}, then there is no stable equivalence of Morita type between SD(2B) k,s 1

  and s are both odd and c = 0.

	The	cohomology	classes	of	the	cocycles	in	B ∪ B ′	with
	B	=	{ϕ

t ; θ r ; ψ; χ; ω; 1 t k -1; 1 r s -1} and

  and k and s are both odd k + s + 1 if char(K) = 2 and k and s are both 0 in K k + s if char(K) = 2 and k and s are not both 0 in K Corollary 4.21. Assume that char(K) = 2 and that {k ′ , s ′ } = {k, s}. Then there is no stable equivalence of Morita type between SD(2B) k,s 2 (0) and SD(2B) k ′ ,s ′ 2

  y} and ∂ 2 , ∂ 3 and

	j are determined by:
	∂ 2 (1 ⊗		x	x +	k-2 ∑ t=0	(yx) t y ⊗ x	y(xy) k-2-t
	+	k-1 ∑	(yx) t ⊗
		t=0
					k-2 ∑ t=0	(xy) t x ⊗ y	x(yx) k-2-t
	+	k-1 ∑
		k-1 ∑

x 1) = x ⊗ x 1 + 1 ⊗ y (xy) k-1-t + c(yx) t y ⊗ x (yx) k-1-t + c(yx) t ⊗ y x(yx) k-1-t ∂ 2 (1 ⊗ y 1) = y ⊗ y 1 + 1 ⊗ y y + t=0 (xy) t ⊗ x (yx) k-1-t + d(xy) t x ⊗ y (xy) k-1-t + d(xy) t ⊗ x y(xy) k-1-t ∂ 3 (1 ⊗ 1) = (x ⊗ x 1 + 1 ⊗ x x)(1 + cx + c 2 x 2 ) + (y ⊗ y 1 + 1 ⊗ y y)(1 + dy + d 2 y 2 ) j(1) = t=0 (xy) t ⊗ (xy) k-t + (yx) t+1 ⊗ (yx) k-t-1 +(xy) t x ⊗ y(xy) k-1-t + (yx) t y ⊗ x(yx) k-1-t .

Again, it is straightforward to check that dim HH 1 (Λ) = k + 5 if k is even of if k is odd and (c, d) = (0, 0) k + 4 if k is odd and (c, d) = (0, 0).

  3 (mod 4) k + 5 if i ≡ 1, 2 (mod 4) and k is even k + 4 if i ≡ 1, 2 (mod 4) and k is odd Proof. Let Λ be one of the algebras Q(1A) k 1 or Q(1A) k 2 (c, d). By [3, Theorem 5.9], we have Ω 4

If moreover k or s is even, then for c ∈ {0, 1} there is no stable equivalence of Morita type between SD(2B) k,s 2 [START_REF] Bessenrodt | q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras[END_REF] and SD(2B) k ′ ,s ′ 1 (c). If instead k and s are both odd, then for c, c ′ ∈ {0, 1} not both equal to 0, there is no stable equivalence of Morita type between SD(2B) k,s 2 (c) and SD(2B) k ′ ,s ′ 1 (c ′ ). Finally, if char(K) = 3, {k ′ , s ′ } = {k, s} with ks = 0 in K and if c, c ′ ∈ {0, 1}, then there is no stable equivalence of Morita type between SD(2B) k,s 2 (c) and SD(2B) k ′ ,s ′ 1 (c ′ ). In order to go further, we now consider the Lie algebra structure of HH 1 (Λ). Once more, we identify a morphism f ∈ Hom Λ-Λ (Q 1 , Λ) with g ∈ Hom KΓ 0 -KΓ 0 (KΓ 1 , Λ) and with the quadruple (g(α), g(β), g(γ), g(η)). First assume that char(K) = 2. We start with a basis for HH 1 (SD(2B) k,s 2 (c)). Lemma 4.22. Assume that s > 2. Define the following cochains in Hom K (KΓ 1 , Λ).

if ks is odd and c = 1,

Then the cohomology classes of the elements in B ∪ B ′ ∪ B ′′ form a basis for HH 1 (SD(2B) k,s 2 (c)). Lemma 4.23. When s = 2, a basis is given by the non-zero cohomology classes of the following elements of Hom K (KΓ 1 , Λ), written as (g(α), g(β), g(γ)):

,

We may now compute the brackets of these basis elements.

Lemma 4.24. We use the notation in the previous lemmas. We describe the (potentially) non-zero brackets in HH 1 (SD(2B) k,s 2 (c)). For all k and s we have

Proof. In the basis described in Lemma 5.3, the Killing form of the Lie algebra We then compute the first two terms in the lower central series. In view of Lemma 5.4, we need only consider the cases where cd = 0, that is, (c, d) = (0, 0) and c = 0, d = 0. Proposition 5.5. We keep the notation above. Then L 1 (HH 1 (Λ)) is spanned by:

Moreover, when k is odd or k = 2, L 2 (HH 1 (Λ)) is spanned by: (i) ϕ 2p+1 , 2 p k-3 2 ; ϕ k-1 if k is odd and c = 0 and d = 0; the dimension is k-3

As a consequence of Lemma 5.4 and Proposition 5.5, we get the following result.

Corollary 5.6. Let k 2 be any integer and let c and d be non-zero elements in K. Then Q

Remark. We still do not know whether

with cd = 0 and c ′ d ′ = 0 are stably equivalent of Morita type or not.

In fact, if k is odd and dd ′ = 0, the Lie algebras HH 1 (Q(1A) k 2 (0, d)) and HH 1 (Q(1A) k 2 (0, d ′ )) are isomorphic (in the remaining cases we do not know), and the isomorphism is given by

with the obvious notations for the bases of the two Lie algebras.