Karin Erdmann
email: erdmann@maths.ox.ac.uk

Edward L Green

AND Nicole Snashall

Rachel Taillefer
email: rachel.taillefer@uca.fr

Stable Green ring of the Drinfeld doubles of the generalised Taft algebras (corrections and new results)

Keywords: Mathematics Subject Classification. 17B37, 81R50, 16E30, 06B15, 16T05 Hopf algebra, Drinfeld double, fusion rules, tensor product, indecomposable module, stable Green ring

INTRODUCTION

Fusion rules, that is, the decomposition of tensor products of modules as a direct sum of indecomposable modules over a Hopf algebra, have been studied in several contexts, such as quantum groups or conformal field theory. In the case of quasi-triangular Hopf algebras, the tensor product of modules over the base field is commutative. Examples of quasi-triangular Hopf algebras are given by Drinfeld doubles of finite dimensional algebras. We are interested here in the Drinfeld doubles D(Λ n,d ) of the duals Λ n,d of the extended Taft algebras over an arbitrary field k whose characteristic does not divide d, where n and d are positive integers with n a multiple of d. Such quantum doubles were originally defined by Drinfeld in order to provide solutions to the quantum Yang-Baxter equation arising from statistical mechanics. The algebra D(Λ n,d ) has the advantage of being relatively small, and the Hopf subalgebra Λ n,d is finite-dimensional and basic, but the representations of D(Λ n,d ) share properties with finite-dimensional representations of U(sl 2 ) and variations. As an algebra, D(Λ n,d ) is tame, and a parametrisation of the indecomposable modules is known. Therefore we studied direct sum decompositions for tensor products of indecomposable modules in [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]. This has also been done for the Drinfeld double of the Taft algebras, that is, the case n = d, in [START_REF] Chen | Indecomposable decomposition of tensor products of modules over Drinfeld Doubles of Taft algebras[END_REF], using different methods.

We returned to this problem because H.-X. Chen (one of the authors of [START_REF] Chen | Indecomposable decomposition of tensor products of modules over Drinfeld Doubles of Taft algebras[END_REF]) asked about the proof of [9, Theorem 4.18] (which is wrong) and the proof and statement of [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Theorem 4.22] (they are both wrong). We are grateful to H.-X. Chen for drawing our attention to these problems. Moreover, in looking at the details, we also noticed that [9, Proposition 3.2] is incorrect, the proof of [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Proposition 4.17] is not quite complete, and there are redundancies in our classification of D(Λ n,d )-modules. We also realised that many of the proofs can be simplified by working over a specific block of D(Λ n,d ).

In this paper we present a new and more homological approach to these tensor product calculations, which is based on exploiting stable module homomorphisms. This enables us to provide corrections to the proofs and statements mentioned above (except [9, Proposition 3.2], which was just a tool we used for some of our results in [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]). Using this new method, we are also able to give general formulas for the decompositions of tensor products involving the remaining modules of even length, the band modules, for which we had only given some examples in [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]. As a consequence, we now have a complete description of the stable Green ring of D(Λ n,d ), which we give in Section 5 (Table 1

on page 17).

We also include new results, classifying endotrivial and algebraic D(Λ n,d )-modules. If H is a finitedimensional ribbon Hopf algebra (see for instance [5, Section 4.2.C]), then a finite-dimensional H-module M is endotrivial if there is an isomorphism M ⊗ k M * ∼ = k ⊕ P where k is the trivial H-module and P is projective. Tensoring with an endotrivial module induces an equivalence of the stable module category, and such equivalences form a subgroup of the auto-equivalences of the stable module category. When H = kG is the group algebra of a finite group G, endotrivial modules have been studied extensively. They have also been studied for finite group schemes in [START_REF] Carlson | Endotrivial modules for finite group schemes[END_REF][START_REF] Carlson | Endotrivial modules for finite group schemes II[END_REF]. However, we have not seen any results on endotrivial modules for other Hopf algebras. When the Hopf algebra H is our Drinfeld double D(Λ n,d ), we show that the indecomposable endotrivial modules are precisely the syzygies of the simple modules of dimension 1 or d -1, see Proposition 6.3.

The concept of an algebraic module is quite natural; it was introduced as a kG-module satisfying a polynomial equation with coefficients in Z in the Green ring of kG, for a finite group G. We shall use the following equivalent definition: a kG-module M is called algebraic if the number of non-isomorphic indecomposable summands of the set of modules M ⊗ k t , when t 1 varies, is finite. Such modules occur in particular in the study of the Auslander-Reiten quiver of kG. For a study of algebraic kG-modules, see for instance [START_REF] Craven | Algebraic modules for finite groups[END_REF] and the references there. Here, we replace the Hopf algebra kG with D(Λ n,d ), and we classify the indecomposable algebraic D(Λ n,d )-modules.

The paper is organised as follows. In Section 1, we describe the quiver and the representations of D(Λ n,d ), in particular removing the redundancies mentioned above, and we recall the decomposition of the tensor product of two simple modules. Section 2 contains our new proof of [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Theorem 4.18] in Theorem 2.5. Proposition 2.4 is a special case of [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Proposition 4.17] (enough for our purposes) whose proof is now complete. Section 3 contains a corrected statement and our new proof of [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Theorem 4.22], this is Theorem 3.4. The proof is by induction, the initial step being Proposition 3.3. In Section 4, we use these methods to determine the decomposition of the tensor product of a band module with any other indecomposable module in Propositions 4.4 and 4.6 and in Theorem 4.9. We conclude this section with a result that gives a more conceptual view of the approach we have used above. In Section 5, we combine all our results on tensor products of representations of D(Λ n,d ) in order to describe the stable Green ring of D(Λ n,d ). Finally, in Section 6, we present the classification of endotrivial and algebraic D(Λ n,d )-modules.

Throughout the paper, k is an algebraically closed field and q is a fixed primitive d-th root of unity in k (in particular, the characteristic of k does not divide d). All modules are left modules. The cyclic group of order n is denoted by Z n and ⊗ denotes the tensor product over k.

We refer to [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF] for all the background on the representation theory of D(Λ n,d ), where n and d are integers such that d divides n. However, we recall (mainly in Section 1) the definitions and results from [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF] that are required for the understanding of this paper, so that the reader does not need to refer to our original paper.

PARAMETRIZATION OF MODULES OVER D(Λ n,d )

In this section, we recall briefly a description of the algebras Λ n,d , Λ * cop n,d and D(Λ n,d ), as well as the isomorphism classes of representations of D(Λ n,d ). We refer to [9, Sections 2 and 3] for more details. Notation 1.1. As in [9, Notation 2.5 and Definition 2.11], we denote by r the residue of an integer r modulo d taken in the set {1, 2, . . . , d} and, for any u ∈ Z n , we define a permutation σ u of Z n by

σ u (j) = d + j -2j + u -1 (recall that d divides n). If 2j + u -
(i) = i + td and σ u (i) 2t+1 (i) = σ u (i) + td in Z n . 1.1.
The original algebras. The algebra Λ n,d is described by quiver and relations. Its quiver is the cyclic quiver with vertices e 1 , . . . , e n-1 and arrows a 0 , . . . , a n-1 , where the indices are viewed in Z n and each arrow a i goes from e i to e i+1 . The ideal of relations of Λ n,d is generated by the paths of length d. We also denote by γ m i = a i+m-1 • • • a i+1 a i the path of length m that starts at e i . Since we assume that d divides n, this algebra is a Hopf algebra by [START_REF] Cibils | A quiver quantum group[END_REF]. We shall only need the antipode here, which is determined by S(e i ) = e -i and S(a i ) = -q i+1 a -i-1 for all i ∈ Z n .

The co-opposite of the dual Hopf algebra, Λ * cop n,d , is the extended Taft algebra, and it is presented by generators and relations:

Λ * cop n,d = G, X | G n = 1, X d = 0, GX = q -1 XG . Its antipode is determined by S(G) = G -1 and S(X) = -XG -1 .
These algebras are both Hopf subalgebras of the Drinfeld double

D(Λ n,d ), that is equal to Λ n,d ⊗ Λ * cop
n,d as a vector space, and has a basis given by the set of G i X j γ m ℓ with i, ℓ in Z n and 0 j, m d -1. In this paper we shall not need the relations between the generators.

+ u -1 is divisible by d, in which case dim L(u, i) = d. When L(u, i) is not projective, then dim L(u, i) = d -2i + u -1 = σ u (i) -i and L(u, i) contains two distinguished
vectors Hu,i and Fu,i , with the following properties: (a) Hu,i spans the kernel of the action of X on L(u, i) as a vector space and e j Hu,i = δ ij Hu,i , (b) e j Fu,i = δ j,σ u (i)-1 Fu,i and the element Fu,i is annihilated by all the arrows in the quiver of Λ n,d , where δ is the Kronecker symbol. The module L(u, i) has basis {X t Fu,i : 0

t < dim L(u, i) = N} and X N-1
Fu,i is a non-zero scalar multiple of Hu,i . Moreover, the action of G on these basis elements is given by

GX t

Fu,i = q -i-t Fu,i . In [9, Proposition 2.21], we characterised the simple modules as follows.

Proposition 1.2. Let S be a simple module. Set E u = 1 n ∑ i,j∈Z n q -i(u+j) G i e j for u ∈ Z n . Then S is isomorphic to L(u, i) if and only if the three following properties hold:

(a) dim S = dim L(u,

i). (b) E u acts as identity on S, and E v acts as zero on S if v = u. (c) Let Y be the generator of S which is in the kernel of the action of X (this is well-defined up to a non-zero scalar and corresponds to Hu,i ). Then the vertex e i acts as identity on Y, and the other vertices act as zero.

The projective cover P(u, i) of L(u, i) has four composition factors. The socle and the top are isomorphic to L(u, i), and

rad P(u, i)/ soc P(u, i) ∼ = L(u, σ u (i)) ⊕ L(u, σ -1 u (i)).
The indecomposable modules of odd length are precisely the syzygies of the non-projective simple modules, that is, the

Ω m (L(u, i)) for m ∈ Z and (u, i) ∈ Z 2 n where d does not divide 2i + u -1.
It follows from the structure of the projective modules that the simple modules in the block of L(u, i) are precisely the simple modules L(u, σ t u (i)) for all t, and there are 2 n d of them. Theorem 2.26 of [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF] completely describes the basic algebra of a non-simple block. Each block is symmetric and special biserial with radical cube zero.

More precisely, as an algebra, D(Λ n,d ) is the direct sum of n 2 d simple blocks and of

n(d-1) 2 blocks B u,i for (u, i) ∈ Z 2 n such that 2i + u -1 is not a multiple of d. The quiver of B u,i is • b b l l • b , , b q q • b b •b 1 1 • b V V • with 2n
d vertices and 4n d arrows. The relations on this quiver are bb, bb and b bbb (there are 6n d relations on each of these quivers). The vertices in this quiver correspond to the simple modules

L(u, i), L(u, σ u (i)), L(u, σ 2 u (i)), . . . , L(u, σ 2n d -1 u (i)). Hence B u,i = B v,j if and only if u = v and j = σ t u (i) for some t ∈ Z.
Moreover, the arrows b and b are described as follows. For each p, there are basis elements in L(u, σ p (i)) that, following the notation in [9, Propostion 2.17], we denote by Du,σ p+1 u (i)

and

F u,σ p-1 u (i)
, such that:

• the action of b p on L(u, σ p (i)) is given by multiplication by

γ dim L(u,σ p u (i)) σ p u (i) and γ σ p u (i) Du,σ p+1 u (i) is a non- zero scalar multiple of Hu,σ p+1 u (i) ,
• the action of bp-1 on L(u, σ p (i)) is given by multiplication by

X dim L(u,σ p u (i)) and X dim L(u,σ p u (i)) F u,σ p-1 u (i)
is equal to Fu,σ

L(u,i) ◆ ◆ ◆ L(u,σ 2 u (i)) ❏ ❏ ❏ ♠ ♠ ♠ L(u,σ 2(ℓ-1) u (i)) ❱ ❱ ♦ ♦ ♦ ♦ L(u,σ u (i)) ••• L(u,σ 2(ℓ-1)+1 u (i))
The lines joining the simple modules are given by multiplication by the appropriate b-arrow or b-arrow (in the case n = d, when there is an ambiguity, the first line is multiplication by γ dim L(u,i) , the next one is multiplication by a non-zero scalar multiple of X d-dim L(u,i) , and so on, up to non-zero scalars).

• The module M - 2ℓ (u, i) has top composition factors L(u, i), L(u, σ -2 u (i)), . . . , L(u, σ -2(ℓ-1) u (i)) and socle com- position factors L(u, σ -1 u (i)), L(u, σ -3 u (i)), . . . , L(u, σ -2(ℓ-1)+1 u (i)): L(u,σ -2(ℓ-1) u (i)) P P P P ❤ ❤ L(u,σ -2 u (i)) ❙ ❙ r r r r L(u,i) ♦ ♦ ♦ L(u,σ -2(ℓ-1)+1 u (i)) ... L(u,σ -1 u (i))
As for the other string modules, the lines represent multiplication by an appropriate b or b arrow, and in the case n = d the first one from the left is multiplication by a power of X and so on.

In both cases, indices are taken modulo 2n d . These modules are periodic of period 2n d . Moreover, the Auslander-Reiten sequences of the string modules

M ± 2ℓ (u, i) are given by A(M + 2ℓ (u, i)) : 0 → M + 2ℓ (u, i -d) → M + 2ℓ+2 (u, i -d) ⊕ M + 2ℓ-2 (u, i) → M + 2ℓ (u, i) → 0, A(M - 2ℓ (u, i)) : 0 → M - 2ℓ (u, i + d) → M - 2ℓ+2 (u, i + d) ⊕ M - 2ℓ-2 (u, i) → M - 2ℓ (u, i) → 0 (where M ± 0 (u, i) = 0).
1.3.2. Band modules. For each λ = 0 in k, and for each ℓ 1, there is an indecomposable module of length 2n d ℓ, which we denote by C ℓ λ (u, i). It is defined as follows. Denote by ǫ p , for p ∈ Z 2n/d , the vertices in the quiver of B u,i , with ǫ 0 corresponding to L(u, i). The arrow b p goes from ǫ p to ǫ p+1 and the arrow bp goes from ǫ p+1 to ǫ p . Let V be an ℓ-dimensional vector space. Then C ℓ λ (u, i) has underlying space C = 2n d -1 p=0 C p with C p = V for all p. The action of the idempotents ǫ p is such that ǫ p C = C p . The action of the arrows b2p and b 2p+1 is zero. The action of the arrows b2p+1 is the identity of V. The action of the arrows b 2p with p = 0 is also the identity. Finally, the action of b 0 is given by the indecomposable Jordan matrix J ℓ (λ).

It is periodic of period 2 and its Auslander-Reiten sequence is given by

A(C ℓ λ (u, i)) : 0 → C ℓ λ (u, i -d) → C ℓ+1 λ (u, i) ⊕ C ℓ-1 λ (u, i) → C ℓ λ (u, i) → 0 (where C 0 λ (u, i) = 0). Note that soc(C) = rad(C) = p ǫ 2p+1 C and that C/ rad(C) = p ǫ 2p C.
Remark 1.3. In [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF], we denoted C ℓ λ (u, i) by C ℓ+ λ (u, i), and we had more band modules. First note that

C ℓ λ (u, i) is isomorphic to C ℓ λ (u, σ 2 u (i))
, so that in (c) above we have removed some repetitions that occurred in [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]. In addition, in [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF] we also had the modules C ℓ- λ (u, i), defined in a similar way by interchanging b's and b's. However, we should have noted that for each λ

∈ k \ {0}, there exists µ ∈ k \ {0} and (v, j) ∈ Z 2 n such that C ℓ- λ (u, i) ∼ = C ℓ µ (v, j)
. Indeed, by definition of the module C ℓ- λ (u, i), the action of b0 is given by J ℓ (λ), the action of bi for i even, i = 0, is given by id, the action of b i for i odd is given by id and the action of the other arrows is given by 0. Changing bases of the vector spaces C - p , we can ensure that the action of b -1 is given by J ℓ 1 λ , the action of b0 is given by id and the rest is unchanged, thus obtaining the module C ℓ

1 λ (u, σ -1 u (i)). Hence C ℓ- λ (u, i) ∼ = C ℓ 1 λ (u, σ -1 u (i)
). Therefore we need only consider one family of band modules, which we denote by C ℓ λ (u, i) for λ ∈ k \ {0}, and we need only one of these for each orbit of σ 2 u in Z n . However, it should be noted that this parameter µ such that C ℓ- λ (u, i) ∼ = C ℓ µ (v, j) is not well defined, unless we have fixed the block B u,i (including which vertex is labelled ǫ 0 ): using the definition in [9, Section A.2.2], where the representations of the basic algebra of a given block are described, the module C ℓ- λ (u, i) is defined by considering that the block B u,i is equal to B u,σ u (i) with a shift in the indices of the vertices and arrows by 1;

here we get immediately that Proof. We use the notation from the last part of Subsection 1.2. It follows from [START_REF] Cibils | A quiver quantum group[END_REF] and [START_REF] Li | The Green rings of the generalized Taft Hopf algebras[END_REF] that the indecomposable projective modules over

C ℓ- λ (u, i) = C ℓ λ (u, i). As we mentioned above, there is a difficulty in differentiating modules of length 2 n d , since M + 2 n d (u, i), M - 2 n d (u, i) and C 1 λ (u, i)
Λ n,d or Λ * cop n,d are precisely the indecomposable modules of dimension d. Therefore M + 2 (u, i), which is equal to Λ n,d Du,σ u (i) as a Λ n,d -module, is indecomposable of dimension d, hence projective. Moreover, as a Λ n,d -module, M + 2ℓ (u, i) ∼ = d-1 t=0 M + 2 (u, i + td) is also projective. Similarly, the Λ * cop n,d -module M - 2 (u, i) = Λ * cop n,d F u,σ -1 u (j) is projective, and M - 2ℓ (u, i) ∼ = ℓ-1 t=0 M - 2 (u, i -td) is a projective Λ * cop n,d -module. As a Λ * cop n,d -module, M + 2ℓ (u, i) decomposes as L(u, i) ⊕ ℓ-1 t=1 M - 2 (u, i + td) ⊕ L(u, σ u (i) + (ℓ -1)d)
, therefore it has a summand L(u, i) whose dimension is strictly less than d, hence that is not projective. Therefore In this paper, we consider tensor products of D(Λ n,d )-modules over the base field. The tensor products of Λ n,d -modules and of Λ * cop n,d -modules have been studied in [START_REF] Cibils | A quiver quantum group[END_REF] and [START_REF] Li | The Green rings of the generalized Taft Hopf algebras[END_REF]. It follows in particular from their results that if M and N are two indecomposable Λ n,d -modules (respectively Λ * cop n,d -modules), then M ⊗ N is projective if and only if M or N is projective. Definition 1.6. We shall say that two string modules M + 2ℓ (u, i) and M + 2t (u, j) (respectively two string modules M - 2ℓ (u, i) and M - 2t (u, j), respectively two band modules C ℓ λ (u, i) and C t µ (u, j)) have the same parity if i ≡ j (mod d) and that they have different parities

M + 2ℓ (u, i) is not projective. Similarly, M - 2ℓ (u, i) is not projective as a Λ n,d -module. Finally, as a Λ n,d -module, C ℓ λ (u, i) ∼ = n d -1 t=0 M + 2 (u, i + td) ℓ is projective, and as a Λ * cop n,d -module, C ℓ λ (u, i) ∼ = n d -1 t=0 M - 2 (u, i -td) ℓ is projective.
if j ≡ σ u (i) (mod d).
This definition is consistent with [9, Definition A.3].

Definition 1.7. We shall write core(M) for the direct sum of the non-projective indecomposable summands in M.

In [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Theorem 4.1], we determined the decomposition of the tensor product of two simple modules. Here we shall need the core of this module.

Proposition 1.8. (cf. [9, Theorem 4.1]) Let L(u, i) and L(v, j) be two non-projective simple D(Λ n,d )-modules. Then core(L(u, i) ⊗ L(v, j)) ∼ = θ∈I L(u + v, i + j + θ) where I = {θ ; 0 θ min(dim L(u, i), dim L(v, j)) -1} if dim L(u, i) + dim L(v, j) d {θ ; dim L(u, i) + dim L(v, j) -d θ min(dim L(u, i), dim L(v, j)) -1} otherwise. Moreover, if dim L(u, i) + dim L(v, j) d, we have L(u, i) ⊗ L(v, j) = core(L(u, i) ⊗ L(v, j)).
Proof. Set N = dim L(u, i) and N ′ = dim L(v, j). Without loss of generality, we can assume that N N ′ . We proved in [9, Theorem 4.1] that

L(u, i) ⊗ L(v, j) ∼ = N-1 θ=0 L(u + v, i + j + θ) if N + N ′ d + 1 N-1 θ=N+N ′ -d L(u + v, i + j + θ) ⊕ projective if N + N ′ d + 1
so we need only determine which of the simple modules are projective. Note that 2i + u -1 ≡ -N (mod d)

and 2j + v -1 ≡ -N ′ (mod d), therefore 2(i + j + θ) + (u + v) -1 ≡ -N -N ′ + 2θ + 1 (mod d). Moreover, since θ N -1 N ′ -1, we have -N -N ′ + 2θ + 1 -1. If N + N ′ d we have -N -N ′ + 2θ + 1 -d + 1, and if N + N ′ > d + 1 we have -N -N ′ + 2θ + 1 -N -N ′ + 2(N + N ′ -d) + 1 = N + N ′ -2d + 1 > d + 1 -2d + 1 = 2 -d, therefore in both cases 2(i + j + θ) + (u + v) -1 ≡ 0 (mod d) and L(u + v, i + j + θ) is not projective. If N + N ′ = d + 1, we have -N -N ′ + 2θ + 1 2θ -d ≡ 2θ (mod d) with 0 2θ 2(N -1) N + N ′ -2 = d -1, therefore 2(i + j + θ) + (u + v) -1 ≡ 0 (mod d) if and only if θ = 0.
Finally, the only projective that can occur is L(u + v, i + j) when N + N ′ = d + 1. The result then follows.

DECOMPOSITION OF THE TENSOR PRODUCT OF A SIMPLE MODULE WITH A STRING MODULE OF EVEN

LENGTH

The stable Green ring r st (H) of a finite dimensional Hopf algebra H is the ring whose Z-basis is the set of isomorphism classes of indecomposable modules in the stable category H-mod, that is, the non-projective indecomposable H-modules, and whose addition and multiplication are induced respectively by the direct sum and the tensor product.

The aim of this section is to determine the tensor product of a string module of even length with a simple module up to projectives, from which we can deduce the tensor product of a string module of even length with a string module of odd length in the stable Green ring r st (D(Λ n,d )). This will provide a correction to the proof of [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Theorem 4.18].

For the proof, we will use a general result involving splitting trace modules. We recall that a module M is a splitting trace module if the trivial module L(0, 0) is a direct summand in End k (M). Moreover, for any ribbon Hopf algebra such as D(Λ n,d ), there are isomorphisms End

k (M) ∼ = M * ⊗ M ∼ = M ⊗ M * of D(Λ n,d )-modules for any module M (where M * is the k-dual of M), see for instance [5, Section 4.2.C].
The indecomposable splitting trace modules for D(Λ n,d ) were given in [9, Subsection 3.3]: they are precisely the non-projective indecomposable modules of odd length.

(i) C is a direct summand in End k (M) ⊗ C. (ii) C is a direct summand in M * ⊗ M ⊗ C. (iii) The sequence A(C) ⊗ M does not split. (iv) If 0 → A → B → C → 0 is any non-split exact sequence of D(Λ n,d )-modules, then the exact sequence 0 → A ⊗ M → B ⊗ M → C ⊗ M → 0 does not split.
If moreover M is a splitting trace module, then all these properties hold.

Proof. The equivalence of the first two statements follows from the isomorphism End k (M) ∼ = M * ⊗ M of D(Λ n,d )-modules. The equivalence between statements (ii), (iii) and (iv) is proved in the same way as [

1, Proposition 2.3]. Finally, if M is a splitting trace module, then L(0, 0) is a summand in End k (M) so that C ∼ = L(0, 0) ⊗ C is a direct summand in End k (M) ⊗ C and (i) holds.
We start with the special cases of the tensor product of M ± 2 (0, td) with a simple module in Proposition 2.2 and of the tensor product of M ± 2ℓ (0, td) with a simple module in Proposition 2.4, from which we deduce the general case in Theorem 2.5.

Proposition 2.2. (cf. [9, Proposition 4.17]) For all t ∈ Z, we have core(M + 2 (0, td) ⊗ L(v, j)) ∼ = M + 2 (v, j + td) and core(M - 2 (0, td) ⊗ L(v, j)) ∼ = M - 2 (v, j + td). Proof. Set N = dim L(v, j). We tensor the non-split exact sequence 0 → L(0, 1 + td) → M + 2 (0, td) → L(0, td) → 0 by L(v, j).
Using Proposition 1.8 as well as dim L(0, td) = 1 and dim L(0, 1 + td) = d -1 (so that #I = 1), we obtain an exact sequence 0 → L(v, j + td + N) ⊕ P → M + 2 (0, td) ⊗ L(v, j) → L(v, j + td) → 0 with P a projective module. Since P is injective, we have M + 2 (0, td) ⊗ L(v, j) ∼ = U ⊕ P for some module U and we obtain an exact sequence 0 → L(v, j + td + N) → U → L(v, j + td) → 0. The sequence cannot split by Lemma 2.1 because L(u, j) is a splitting trace module, so U is an indecomposable module of length 2 with top U = L(v, j + td) and soc U = L(v, j + td + N) = L(v, σ v (j + td)). Moreover, it follows from Property 1.4 and Remark 1.5 that M + 2 (0, td) ⊗ L(v, j), and hence U, is projective as a Λ n,d -module but not as a Λ * cop n,d -module, therefore we must have U = M + 2 (v, j + td). The proof of the second part is similar.

In the next lemma as well as in Proposition 3.3, we shall use properties of homomorphisms between string modules of even length over a given block; these properties can easily be seen by working over the basic algebra of the block. Lemma 2.3. Let t and ℓ be integers with 1 t ℓ. Then the spaces Ext 1

D(Λ n,d ) (M + 2ℓ (u, i + d), M + 2t (u, i)) and Ext 1 D(Λ n,d ) (M - 2ℓ (u, i -d), M - 2t (u, i)) have dimension #{0 y t -1 | y ≡ 0 (mod n d )} -#{0 y t -1 | y ≡ ℓ (mod n d )} + #{0 y t -1 | y ≡ t -ℓ -1 (mod n d )}.
In particular, Ext 1

D(Λ n,d ) (M + 2ℓ (u, i + d), M + 2 (u, i)) and Ext 1 D(Λ n,d ) (M - 2ℓ (u, i -d), M - 2 (u, i)) have dimension 1. A basis of Ext 1 D(Λ n,d ) (M + 2ℓ (u, i + d), M + 2 (u, i)) (respectively Ext 1 D(Λ n,d ) (M - 2ℓ (u, i -d), M - 2 (u, i))
) is given by the equivalence class of the exact sequence

0 → M + 2 (u, i) → M + 2ℓ+2 (u, i) → M + 2ℓ (u, i + d) → 0 (respectively 0 → M - 2 (u, i) → M - 2ℓ+2 (u, i) → M - 2ℓ (u, i -d) → 0). Proof.
We prove the first statement, the proof of the second is similar.

Applying Hom

D(Λ n,d ) (-, M + 2t (u, i)) to the exact sequence 0 → Ω(M + 2ℓ (u, i + d)) ∼ = M + 2ℓ (u, σ u (i)) → Q := ℓ x=1 P(u, i + xd) → M + 2ℓ (u, i + d) → 0 gives an exact sequence 0 → Hom D(Λ n,d ) (M + 2ℓ (u, i + d), M + 2t (u, i)) → Hom D(Λ n,d ) (Q, M + 2t (u, i)) → Hom D(Λ n,d ) (M + 2ℓ (u, σ u (i)), M + 2t (u, i)) → Ext 1 D(Λ n,d ) (M + 2ℓ (u, i + d), M + 2t (u, i)) → 0 so that dim Ext 1 D(Λ n,d ) (M + 2ℓ (u, i + d), M + 2t (u, i)) = dim Hom D(Λ n,d ) (M + 2ℓ (u, i + d), M + 2t (u, i)) -dim Hom D(Λ n,d ) (Q, M + 2t (u, i)) + dim Hom D(Λ n,d ) (M + 2ℓ (u, σ u (i)), M + 2t (u, i)).
We determine each of these dimensions. We have dim Hom

D(Λ n,d ) (Q, M + 2t (u, i)) = ∑ ℓ x=1 #{y ; 0 y t -1, y ≡ x (mod n d )}. Since M + 2ℓ (u, σ u (i)) and M + 2t (u, i) have different parities, any non-zero map in Hom D(Λ n,d ) (M + 2ℓ (u, σ u (i)), M + 2t (u, i)) must send the top of M + 2ℓ (u, σ u (i)) into the socle of M + 2t (u, i). Therefore dim Hom D(Λ n,d ) (M + 2ℓ (u, σ u (i)), M + 2t (u, i)) = ℓ-1 ∑ x=0 #{y ; 0 y t -1, y ≡ x (mod n d )} = dim Hom D(Λ n,d ) (Q, M + 2t (u, i)) + #{y ; 0 y t -1, y ≡ 0 (mod n d )} -#{y ; 0 y t -1, y ≡ ℓ (mod n d )}.
Since M + 2ℓ (u, i + d) and M + 2t (u, i) have the same parity, the image of any non-zero map in

Hom D(Λ n,d ) (M + 2ℓ (u, i + d), M + 2t (u, i)) is a quotient of M + 2ℓ (u, i + d) and a string submodule of M + 2t (u, i) that starts at L(u, i). Therefore, since ℓ t, dim Hom D(Λ n,d ) (M + 2ℓ (u, i + d), M + 2t (u, i)) = #{x ; ℓ -t + 1 x ℓ, x ≡ 0 (mod n d )} = #{y ; 0 y t -1, y ≡ t -ℓ -1 (mod n d )}.
The general formula follows from there and the case t = 1 is then clear.

Proposition 2.4. For any integer ℓ 1 and all t ∈ Z, we have

core(M + 2ℓ (0, td) ⊗ L(v, j)) ∼ = M + 2ℓ (v, j + td) and core(M - 2ℓ (0, td) ⊗ L(v, j)) ∼ = M - 2ℓ (v, j + td). Proof.
The proof is by induction on ℓ. Proposition 2.2 shows that the result is true for ℓ = 1. Now assume that core(M + 2ℓ (0, td) ⊗ L(v, j)) ∼ = M + 2ℓ (v, j + td) for all t ∈ Z and for a given ℓ 1. There is a non-split exact sequence

0 → M + 2 (0, td) → M + 2ℓ+2 (0, td) → M + 2ℓ (0, (t + 1)d) → 0. Tensoring with L(v, j) gives an exact sequence 0 → M + 2 (v, j + td) ⊕ P 1 → M + 2ℓ+2 (0, td) ⊗ L(v, j) → M + 2ℓ (v, j + (t + 1)d) ⊕ P 2 → 0
with P 1 and P 2 projective-injective modules, so that, factoring out the split exact sequence 0 → P 1 → P 1 ⊕ P 2 → P 2 → 0, we have an exact sequence

0 → M + 2 (v, j + td) → core(M + 2ℓ+2 (0, td) ⊗ L(v, j)) ⊕ P → M + 2ℓ (v, j + (t + 1)d)
→ 0 for some projective module P. Moreover, since L(v, j) is a splitting trace module, by Lemma 2.1 this sequence is not split. By Lemma 2.3, we have P = 0 and core(M + 2ℓ+2 (0, td) ⊗ L(v, j)) ∼ = M + 2ℓ+2 (v, j + td) thus proving the induction.

The proof of the isomorphism core(M - 2ℓ (0, td)

⊗ L(v, j)) ∼ = M - 2ℓ (v, j + td) is similar, starting with the exact sequence 0 → M - 2 (0, td) → M - 2ℓ+2 (0, td) → M - 2ℓ (0, (t -1)d) → 0.
Theorem 2.5. (cf. [9, Theorem 4.18]) Fix an integer ℓ 1. For any (u, i) and (v, j) in Z 2 n with 2i + u -1 ≡ 0 (mod d) and 2j + v -1 ≡ 0 (mod d), we have

core(M + 2ℓ (u, i) ⊗ L(v, j)) ∼ = θ∈I M + 2ℓ (u + v, i + j + θ) and core(M - 2ℓ (u, i) ⊗ L(v, j)) ∼ = θ∈I M - 2ℓ (u + v, i + j + θ)
where I is defined in Proposition 1.8.

Proof. Using Propositions 2.4 and 1.8 repeatedly, we have, on the one hand,

M + 2ℓ (0, 0) ⊗ L(u, i) ⊗ L(v, j) ∼ = M + 2ℓ (0, 0) ⊗ (L(u, i) ⊗ L(v, j)) ∼ = M + 2ℓ (0, 0) ⊗ θ∈I L(u + v, i + j + θ) ⊕ P 1 ∼ = θ∈I M + 2ℓ (0, 0) ⊗ L(u + v, i + j + θ) ⊕ P 2 ∼ = θ∈I M + 2ℓ (u + v, i + j + θ) ⊕ P 3
and on the other hand

M + 2ℓ (0, 0) ⊗ L(u, i) ⊗ L(v, j) ∼ = (M + 2ℓ (0, 0) ⊗ L(u, i)) ⊗ L(v, j) ∼ = (M + 2ℓ (u, i) ⊕ P 4 ) ⊗ L(v, j) ∼ = (M + 2ℓ (u, i) ⊗ L(v, j
)) ⊕ P 5 for some projective modules P 1 , . . . , P 5 . The projective P 5 is necessarily a direct summand in P 3 , therefore the result follows.

The proof of the second isomorphism is similar.

DECOMPOSITION OF THE TENSOR PRODUCT OF TWO STRING MODULES OF EVEN LENGTH

In this section, we determine the tensor products of two string modules of even length, in order to provide a correction to the statement and proof of [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Theorem 4.22].

We proved in [9, Proposition 4.21] that M + 2 (u, i) ⊗ M - 2 (v, j) is projective for any (u, i) and (v, j) in Z 2 n . It then follows, using induction on ℓ and t and the Auslander-Reiten sequences for the string modules of even length, that M + 2ℓ (u, i) ⊗ M - 2t (v, j) is projective for any (u, i) and (v, j) in Z 2 n and any positive integers ℓ and t. We denote by Hom D(Λ n,d ) (M, N) the space of stable homomorphisms from M to N. We shall need the following isomorphisms. Lemma 3.1. Let U, V and W be any D(Λ n,d )-modules. Then there are isomorphisms

Hom D(Λ n,d ) (U ⊗ V, W) ∼ = Hom D(Λ n,d ) (U, V * ⊗ W) Hom D(Λ n,d ) (U, V ⊗ W) ∼ = Hom D(Λ n,d ) (U ⊗ V * , W).
Proof. The adjoint functors -⊗ V and Hom k (V, -) take projective modules to projective modules, therefore by the Auslander-Kleiner theorem (see [11, 5.3.4]), there is an isomorphism Hom [START_REF] Chari | A Guide to Quantum Groups[END_REF]Section 4.2.C]), and the first isomorphism follows.

D(Λ n,d ) (U ⊗ V, W) ∼ = Hom D(Λ n,d ) (U, Hom k (V, W)). Moreover, there is an isomorphism Hom k (V, W) ∼ = V * ⊗ W of D(Λ n,d )-modules (see
The second isomorphism is a consequence of the first isomorphism and of the isomorphism

V * * ∼ = V of D(Λ n,d )-modules (see [5, Section 4.2.C]).
We now concentrate on tensor products of the form

M + 2ℓ (u, i) ⊗ M + 2t (v, j) and M - 2ℓ (u, i) ⊗ M - 2t (v, j).
In order to prove our results, we shall need the duals of some modules. Lemma 3.2. Assume that (u, i) ∈ Z 2 n and 2i + u -1 ≡ 0 (mod d). Then the following isomorphisms hold.

(i) L(u, i) * ∼ = L(-u, 1 -σ u (i)). (ii) M + 2ℓ (u, i) * ∼ = M + 2ℓ (-u, 1 -i -ℓd). (iii) M - 2ℓ (u, i) * ∼ = M - 2ℓ (-u, 1 -i -(ℓ -1)d).
Proof. We start with the dual of a simple module. It is also a simple module, therefore there exists (v, j) such that L(u, i) * ∼ = L(v, j). Set N = dim L(u, i) = dim L(v, j) < d. We must determine v and j.

The module L(u, i) has a basis {ε t ; 0 t < N} given by ε t = X t Fu,i , using Proposition 1.2 and the description of L(u, i) preceding it. Let {ε * t ; 0 t < N} be the dual basis. We have

• (E w • ε * t )(ε s ) = ε * t (S(E w )ε s ) = ε * t (E -w ε s ) = ε * t (ε s ) if -w = u 0 otherwise so that E -u • ε * t = ε * t and E w • ε * t = 0 for w = -u. By Proposition 1.2 we must have v = -u. • Similarly, we have X • ε * t = -q -i-t+1 ε * t-1 if t > 0 0 if t = 0.
In particular, ε * 0 spans the kernel of the action of X. Then, by Proposition 1.2, the idempotent e j is the only one which does not annihilate ε * 0 . We can easily check that e 1-σ u (i) • ε * 0 = ε * 0 , therefore j = 1σ u (i). We now consider the dual of an indecomposable string module M of even length. Since M ∼ = M * * as D(Λ n,d )-modules, the dual module M * is also indecomposable. Moreover, if M has radical length 2, then so has M * and top(M * ) ∼ = (soc M) * and soc(M * ) ∼ = (top M) * . We also note that if x ∈ Z n with x ≡ 0 (mod d),

then x + -x = d. First consider M + 2 (u, i) * . We have soc(M + 2 (u, i) * ) ∼ = top(M + 2 (u, i)) * ∼ = L(u, i) * ∼ = L(-u, 1 -σ u (i)) and top(M + 2 (u, i) * ) ∼ = soc(M + 2 (u, i)) * ∼ = L(u, σ u (i)) * ∼ = L(-u, σ 2 u (i)) = L(-u, 1 -i -d). Note that σ -u (1 -i - d) = 1 -σ u (i). Moreover, M + 2 (u, i) is
+ 2 (u, i) * is not projective as a Λ * cop n,d -module. Therefore M + 2 (u, i) * ∼ = M + 2 (-u, 1 -i -d).
The dual M + 2ℓ (u, i) * is then obtained by induction, using the Auslander-Reiten sequences and the fact that dualising an Auslander-Reiten sequence gives an Auslander-Reiten sequence.

The proof of (iii) is similar.

We shall now study the tensor product of string modules of even length, and we start with some special cases, which constitute the initial step in the proof of Theorem 3.4.

Proposition 3.3. We have

core(M + 2t (u, i) ⊗ M + 2t (v, j)) ∼ = θ∈I M + 2t (u + v, i + j + θ) ⊕ M + 2t (u + v, σ 2t-1 u+v (u + v + θ)) core(M + 2t (u, i) ⊗ M + 2(t+1) (v, j)) ∼ = θ∈I M + 2t (u + v, i + j + θ) ⊕ M + 2t (u + v, σ 2t+1 u+v (u + v + θ)) core(M - 2t (u, i) ⊗ M - 2t (v, j)) ∼ = θ∈I M - 2t (u + v, i + j + θ) ⊕ M - 2t (u + v, σ -(2t-1) u+v (u + v + θ)) core(M - 2t (u, i) ⊗ M - 2(t+1) (v, j)) ∼ = θ∈I M - 2t (u + v, i + j + θ) ⊕ M - 2t (u + v, σ -(2t+1) u+v (u + v + θ))
where I is defined in Proposition 1.8.

Proof. We start by proving the first two isomorphisms in the case (u, i) = (v, j) = (0, 0), that is,

core(M + 2t (0, 0) ⊗ M + 2t (0, 0)) ∼ = M + 2t (0, 0) ⊕ M + 2t (0, σ 2t-1 0 (0)) core(M + 2t (0, 0) ⊗ M + 2(t+1) (0, 0)) ∼ = M + 2t (0, 0) ⊕ M + 2t (0, σ 2t+1 0 (0)) Set ℓ = t or ℓ = t + 1.
We have an exact sequence

0 → L(0, 1 + (ℓ -1)d) → M + 2ℓ (0, 0) → Ω ℓ-1 (L(0, σ ℓ-1 0 (0))) → 0. Tensoring on the left with M + 2t (0, 0) gives an exact sequence 0 → M + 2t (0, 1 + (ℓ -1)d) ⊕ P 1 → M + 2t (0, 0) ⊗ M + 2ℓ (0, 0) → Ω ℓ-1 (M + 2t (0, σ ℓ-1 0 (0)
)) ⊕ P 2 → 0 for some projective-injective modules P 1 and P 2 . It follows that there is an exact sequence

(3.1) 0 → M + 2t (0, 1 + (ℓ -1)d) → core(M + 2t (0, 0) ⊗ M + 2ℓ (0, 0)) ⊕ P 3 → M + 2t (0, 0) → 0
for some projective module P 3 (obtained by factoring out the split exact sequence 0 → P 1 → P 1 ⊕ P 2 → P 2 → 0). The exact sequence (3.1) is isomorphic to the following pullback

0 / / M + 2t (0, 1 + (ℓ -1)d) / / E / / M + 2t (0, 0) / / ϕ 0 0 / / M + 2t (0, 1 + (ℓ -1)d) / / P := ℓ+t-1 y=ℓ P(0, yd) π / / Ω -1 (M + 2t (0, 1 + (ℓ -1)d)) ∼ = M + 2t (0, ℓd) / / 0 where E = {(m, p) ∈ M + 2t (0, 0) × P ; ϕ(m) = π(p)} ∼ = core(M + 2t (0, 0) ⊗ M + 2ℓ (0, 0)) ⊕ P.
Assume for a contradiction that the sequence (3.1) is not split. Then ϕ = 0. The modules M + 2t (0, 0) and M + 2t (0, ℓd) have the same parity, therefore the image of ϕ must contain a submodule of M + 2t (0, ℓd) of length 2, that is, M + 2 (0, ℓd). The module M + 2 (0, ℓd) has a generator of the form ϕ(m) = π(e) for some (m, e) ∈ M + 2t (0, 0) × P(0, ℓd) ⊂ E. The submodule of E generated by (m, e) is projective, isomorphic to P(0, ℓd). Since P(0, ℓd) is injective, it is isomorphic to a direct summand of E. Therefore P(0, ℓd), which is a summand in P, is also a summand in M + 2t (0, 0) ⊗ M + 2ℓ (0, 0). Denote by [M : L] the multiplicity of a simple module L as a summand in a semisimple module M. The simple module L(0, ℓd) is in the socle of the projective summand P(0, ℓd) of M + 2t (0, 0) ⊗ M + 2ℓ (0, 0) (but does not occur in the socle of M + 2t (0, 0)), hence it follows from the above that [soc(core(M + 2t (0,

0) ⊗ M + 2ℓ (0, 0))) : L(0, ℓd)] < [soc(M + 2t (0, 1 + (ℓ -1)d)) : L(0, ℓd)]
. We now compute these multiplicities. Set t = r n d + s with 0 s < n d . Recall that ℓ t. We have

[soc(M + 2t (0, 1 + (ℓ -1)d)) : L(0, ℓd)] = [ ℓ+t-1 x=ℓ L(0, xd) : L(0, ℓd)] = [ t-1 x=0 L(0, (x + ℓ)d) : L(0, ℓd)] = #{x ; 0 x t -1, x ≡ 0 (mod n d )}.
Using Lemmas 3.1 and 3.2 and Theorem 2.5, we have

[soc(core(M + 2t (0, 0) ⊗ M + 2ℓ (0, 0))) : L(0, ℓd)] = dim Hom D(Λ n,d ) (L(0, ℓd), core(M + 2t (0, 0) ⊗ M + 2ℓ (0, 0))) = dim Hom D(Λ n,d ) (L(0, ℓd), M + 2t (0, 0) ⊗ M + 2ℓ (0, 0)) = dim Hom D(Λ n,d ) (L(0, ℓd) ⊗ M + 2ℓ (0, 0) * , M + 2t (0, 0)) = dim Hom D(Λ n,d ) (L(0, ℓd) ⊗ M + 2ℓ (0, 1 -ℓd), M + 2t (0, 0)) = dim Hom D(Λ n,d ) (M + 2ℓ (0, 1), M + 2t (0, 0)) = dim Ext 1 D(Λ n,d ) (M + 2ℓ (0, d), M + 2t (0, 0))
which is known by Lemma 2.3. Finally,

δ := [soc(core(M + 2t (0, 0) ⊗ M + 2ℓ (0, 0))) : L(0, ℓd)] -[soc(M + 2t (0, 1 + (ℓ -1)d)) : L(0, ℓd)] = #{y ; 0 y t -1, y ≡ 0 (mod n d )} -#{y ; 0 y t -1, y ≡ ℓ (mod n d )} + #{y ; 0 y t -1, y ≡ t -ℓ -1 (mod n d )} -#{x ; 0 x t -1, x ≡ 0 (mod n d )} = #{x ; 0 x t -1, x ≡ t -ℓ -1 (mod n d )} -#{x ; 0 x t -1, x ≡ ℓ (mod n d )}.
We start with the case ℓ = t. Then

0 > δ = #{x ; 0 x r n d + s -1, x ≡ -1 (mod n d )} -#{x ; 0 x r n d + s -1, x ≡ s (mod n d )}
= rr = 0 so that we have a contradiction.

In the case ℓ = t + 1,

0 > δ = #{x ; 0 x r n d + s -1, x ≡ -2 (mod n d )} -#{x ; 0 x r n d + s -1, x ≡ s + 1 (mod n d )} = (r + 1) -(r + 1) = 0 if s = n d -1 r -r = 0 if s < n d -1 so that we have a contradiction.
Therefore the sequence (3.1) is split in both cases and we have

core(M + 2t (0, 0) ⊗ M + 2t (0, 0)) ∼ = M + 2t (0, 0) ⊕ M + 2t (0, σ 2t-1 0 (0)) (when ℓ = t) core(M + 2t (0, 0) ⊗ M + 2(t+1) (0, 0)) ∼ = M + 2t (0, 0) ⊕ M + 2t (0, σ 2t+1 0 (0)) (when ℓ = t + 1).
The result then follows, tensoring these isomorphisms with L(u, i) on the left and L(v, j) on the right, and using the commutativity of the tensor product and Proposition 1.8 on the tensor product of simple modules.

The other two isomorphisms are proved in the same way: there is an exact sequence

0 → M + 2t (0, 1 -ℓd) → core(M - 2t (0, 0) ⊗ M - 2ℓ (0, 0)) ⊕ P 3 → M - 2t (
0, 0) → 0 and if the middle term has a projective summand, then it must contain P(0, -ℓd) as a summand. We then compare the multiplicities of the simple module L(0, -ℓd) in soc(core(M - 2t (0, 0) ⊗ M - 2ℓ (0, 0))) and in soc(M - 2t (0, 1ℓd)). We may now rectify [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Theorem 4.22]. Theorem 3.4. For any positive integer t and any integer ℓ with ℓ t, we have

core(M + 2ℓ (v, j) ⊗ M + 2t (u, i)) ∼ = θ∈I M + 2t (u + v, i + j + θ) ⊕ M + 2t (u + v, σ 2ℓ-1 u+v (i + j + θ)) and core(M - 2ℓ (v, j) ⊗ M - 2t (u, i)) ∼ = θ∈I M - 2t (u + v, i + j + θ) ⊕ M - 2t (u + v, σ -(2ℓ-1) u+v (i + j + θ)) ,
where I is defined in Proposition 1.8.

Proof. We prove the first isomorphism by induction on ℓ, the proof of the second isomorphism is similar. By Proposition 3.3, we already know that the result is true for ℓ = t and for ℓ = t + 1. Now take ℓ t + 1 and assume that the decomposition holds for any ℓ ′ with t ℓ ′ ℓ. We apply Lemma 2.1 with

M = M + 2t (u, i) and C = M + 2ℓ (v, j + d). The module M * ⊗ M ⊗ C ∼ = M + 2t (-u, 1 -i -td) ⊗ M + 2t (u, i) ⊗ M + 2ℓ (v, j + d)
is the direct sum of a projective module and of indecomposable modules of length 2t < 2ℓ by the induction hypothesis and Proposition 3.3, therefore

M + 2ℓ (v, j + d) is not a summand in M + 2t (-u, 1 -i -td) ⊗ M + 2t (u, i) ⊗ M + 2ℓ (v, j + d). It follows that A(M + 2ℓ (v, j + d)) ⊗ M + 2t (u, i) splits, so that core((M + 2(ℓ+1) (v, j) ⊗ M + 2t (u, i)) ⊕ (M + 2(ℓ-1) (v, j + d) ⊗ M + 2t (u, i))) ∼ = core((M + 2ℓ (v, j) ⊗ M + 2t (u, i)) ⊕ (M + 2ℓ (v, j + d) ⊗ M + 2t (u, i))
). Since t ℓ -1, using the induction hypothesis we have

core(M + 2(ℓ-1) (v, j + d) ⊗ M + 2t (u, i)) ∼ = θ∈I M + 2t (u + v, i + j + θ + d) ⊕ M + 2t (u + v, σ 2ℓ-1 u+v (i + j + θ)) core(M + 2ℓ (v, j) ⊗ M + 2t (u, i)) ∼ = θ∈I M + 2t (u + v, i + j + θ) ⊕ M + 2t (u + v, σ 2ℓ-1 u+v (i + j + θ)) core(M + 2ℓ (v, j + d) ⊗ M + 2t (u, i)) ∼ = θ∈I M + 2t (u + v, i + j + θ + d) ⊕ M + 2t (u + v, σ 2ℓ+1 u+v (i + j + θ)) therefore core(M + 2(ℓ+1) (v, j) ⊗ M + 2t (u, i)) ∼ = θ∈I M + 2t (u + v, i + j + θ) ⊕ M + 2t (u + v, σ 2ℓ+1 u+v (i + j + θ)) as re- quired.
This concludes the induction.

DECOMPOSITION OF TENSOR PRODUCTS INVOLVING BAND MODULES

In [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF], we were unable to give a general expression of the decomposition of the tensor product of a band module with another module. We can do this now, using the methods in the previous two sections.

We consider first the tensor product of a band module of shortest length with a simple module.

Proposition 4.1. (cf. [9, Proposition 4.17]) For any λ ∈ k\{0}, there exists µ ∈ k\{0} such that core(C 1

λ (0, 0) ⊗ L(v, j)) ∼ = C 1 µ (v, j). Proof.
The proof is similar to that of Proposition 2.2. Set N = dim L(v, j). We tensor the non-split exact se-

quence 0 → Ω 1-n d (L(0, σ n d 0 (0))) → C 1 λ (0, 0) → L(0, 0) → 0 by L(v, j).
Using Proposition 1.8 as well as dim L(0, 0) = 1 and dim L(0, σ n d 0 (0)) ∈ {1, d -1} depending on the parity of n d (so that #I = 1), we obtain an exact sequence 0

→ Ω 1-n d (L(v, σ n d v (j))) ⊕ P → C 1 λ (0, 0) ⊗ L(v, j) → L(v, j)
→ 0 with P a projective module or 0. Since P is injective, we have C 1 λ (0, 0) ⊗ L(v, j) ∼ = U ⊕ P for some module U and we obtain an exact sequence 0

→ Ω 1-n d (L(v, σ n d v (j))) → U → L(v, j) → 0.
The sequence cannot split by Lemma 2.1, so U is an indecom- posable module of length 2 n d , such that L(v, j) is contained in the top of U. Moreover, it follows from Property 1.4 and Remark 1.5 that C 1 λ (0, 0) ⊗ L(v, j), and hence U, is projective as a Λ n,d -module and as a Λ * cop n,d -module, therefore we must have U = C 1 µ (v, j) for some parameter µ ∈ k\{0}. As in the case of string modules of even length, this proposition constitutes the initial step in the proof that core(C ℓ λ (0, 0) ⊗ L(v, j)) is a band module of the form C ℓ µ (v, j). In order to continue, we shall need the extensions between band modules. Lemma 4.2. Let ℓ and t be positive integers. For any (u, i) ∈ Z 2 n with 2i + u -1 ≡ 0 (mod d) and any parameters λ and µ in k \ {0},

dim Ext 1 D(Λ n,d ) (C t λ (u, i), C ℓ µ (u, i)) = min(t, ℓ) if λ = µ 0 if λ = µ. Moreover, if t = 1 and λ = µ, the equivalence class of the exact sequence 0 → C ℓ λ (u, i) → C ℓ+1 λ (u, i) → C 1 λ (u, i) → 0 is a basis of Ext 1 D(Λ n,d ) (C 1 λ (u, i), C ℓ µ (u, i)) Proof. There is a parameter ω(λ) ∈ k\{0} such that Ω(C t λ (u, i)) = C t ω(λ) (u, σ u (i)). Apply Hom D(Λ n,d ) (-, C ℓ µ (u, i)) to the exact sequence 0 → C t ω(λ) (u, σ u (i)) → n d r=1 P(u, i + rd) t → C t λ (u, i) → 0 to get the exact sequence 0 → Hom D(Λ n,d ) (C t λ (u, i), C ℓ µ (u, i)) → n d r=1 Hom D(Λ n,d ) (P(u, i + rd), C ℓ µ (u, i)) t → Hom D(Λ n,d ) (C t ω(λ) (u, σ u (i)), C ℓ µ (u, i)) → Ext 1 D(Λ n,d ) (C t λ (u, i), C ℓ µ (u, i)) → 0. • Let f be a non-zero map in Hom D(Λ n,d ) (C t ω(λ) (u, σ u (i)), C ℓ µ (u, i)). Since the modules have different parities, f must map the top n d -1 x=0 L(u, σ 2x+1 u (i)) t of C t ω(λ) (u, σ u (i)) into the socle n d -1 x=0 L(u, σ 2x+1 u (i)) ℓ of C ℓ µ (u, i), therefore dim Hom D(Λ n,d ) (C t ω(λ) (u, σ u (i)), C ℓ µ (u, i)) = tℓ n d . • dim n d r=1 Hom D(Λ n,d ) (P(u, i + rd), C ℓ µ (u, i)) t = tℓ n d . • Therefore dim Ext 1 D(Λ n,d ) (C t λ (u, i), C ℓ µ (u, i)) = dim Hom D(Λ n,d ) (C t λ (u, i), C ℓ µ (u, i)) and we must prove that dim Hom D(Λ n,d ) (C t λ (u, i), C ℓ µ (u, i)) = 0 if λ = µ min(t, ℓ) if λ = µ.
Both C t λ (u, i) and C ℓ µ (u, i) are modules over the same block B u,i , therefore we may work over the basic algebra B u,i with band modules such that b 0 is the arrow acting as J t (λ) and J ℓ (µ

) respectively. A morphism f ∈ Hom D(Λ n,d ) (C t λ (u, i), C ℓ µ (u, i)) is then completely determined by a linear map f 0 ∈ Hom k (k t , k ℓ ) such that J ℓ (µ) f 0 = f 0 J t (λ). Such a map is zero if λ = µ, and if λ = µ its matrix is of the form A 0 ℓ-t if t ℓ and A 0 t-ℓ if t ℓ,
where A is an upper triangular Toeplitz matrix, that is, of

the form A =       a 1 a 2 • • • a m 0 a 1 . . . . . . . . . . . . a 2 0 • • • • • • a 1      
, with m = min(ℓ, t) and a 1 , . . . , a m in k.

We can now determine the tensor product of a longer band module with a simple module.

Proposition 4.3. Let ℓ be a positive integer and let λ be in k\{0}. Let µ ∈ k\{0} be as in Proposition 4.1. Then

core(C ℓ λ (0, 0) ⊗ L(v, j)) ∼ = C ℓ µ (v, j). Proof.
The proof is by induction on ℓ. Proposition 4.1 shows that the result is true for ℓ = 1. Now assume that core(C ℓ λ (0, 0) ⊗ L(v, j)) ∼ = C ℓ µ (v, j) for a given ℓ 1. There is an exact sequence 0 → C ℓ λ (0, 0) → C ℓ+1 λ (0, 0) → C 1 λ (0, 0) → 0. Tensoring with L(v, j) gives an exact sequence

0 → C ℓ µ (v, j) ⊕ P 1 → C ℓ+1 λ (0, 0) ⊗ L(v, j) → C 1 µ (v, j) ⊕ P 2 → 0
with P 1 and P 2 projective-injective modules, so that, factoring out the split exact sequence 0 → P 1 → P 1 ⊕ P 2 → P 2 → 0, we have an exact sequence 0 → C ℓ µ (v, j) → core(C ℓ+1 λ (0, 0) ⊗ L(v, j)) ⊕ P → C 1 µ (v, j) → 0 for some projective module P. Moreover, since L(v, j) is a splitting trace module, by Lemma 2.1 this sequence is not split. By Lemma 4.2, we have P = 0 and core(C ℓ+1 λ (0, 0) ⊗ L(v, j)) = C ℓ+1 µ (v, j) thus proving the induction.

As we mentioned in Section 1, the parameter λ of the module C ℓ λ (u, i) is not well defined, due to the fact that defining it requires a fixed labelling of the vertices and arrows of the basic algebra of D(Λ n,d ), which we do not have. Moreover, when computing tensor products, we work with D(Λ n,d )-modules (and not over the basic algebra). The method in [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Example 4.23] allows us to determine the parameter µ in specific examples, once the labelling choices are made, but it does not give a general rule.

However, Proposition 4.3 allows us to fix the parameters coherently once and for all in the following way.

Convention on parameters. Define C ℓ λ (0, 0) as in Subsection 1.3, with the vertex ǫ 0 of B 0,0 corresponding to the trivial module L(0, 0). We then set

C ℓ λ (v, j) = core(C ℓ λ (0, 0) ⊗ L(v, j)). With the definitions recalled in Subsection 1.3 we have C ℓ λ (u, i) ∼ = C ℓ λ (u, σ 2 u (i)) = C ℓ λ (u, i + d) = Ω 2 (C ℓ λ (u, i))
, and this is compatible with the convention above. The tensor product of any band module with a simple module is an immediate consequence of this convention.

Proposition 4.4. Fix an integer ℓ 1 and a parameter λ ∈ k\{0}. For any (u, i) and (v, j) in Z 2 n with 2i + u -1 ≡ 0 (mod d) and 2j + v -1 ≡ 0 (mod d), we have

core(C ℓ λ (u, i) ⊗ L(v, j)) ∼ = θ∈I C ℓ λ (u + v, i + j + θ)
where I is defined in Proposition 1.8.

Remark 4.5. As a consequence, we know the tensor product of band modules with string modules of odd length up to projectives.

We can now consider the tensor product of band modules with modules of even length. The first result concerns tensor products of band modules with different parameters and tensor products of band modules with string modules of even length. Proposition 4.6. Let (u, i) and (v, j) be in Z 2 n such that d does not divide 2i + u -1 or 2j + v -1, let ℓ and t be any positive integers.

For any λ, µ in k\{0} such that λ = µ, the module C ℓ λ (u, i) ⊗ C t µ (v, j) is projective. For any λ in k\{0}, the module C ℓ λ (u, i) ⊗ M ± 2t (v, j) is projective. Proof. We start by proving that C 1 λ (0, 0) ⊗ C 1 µ (0, 0) is projective. First note that for any parameter λ ∈ k\{0} there exist parameters ω(λ), ω ′ (λ) ∈ k\{0} such that

Ω -1 (C 1 λ (0, 0)) = C 1 ω(λ) (0, 1) and Ω -1 (C 1 λ (0, 1)) = C 1
ω ′ (λ) (0, 0), and moreover that ω ′ (ω(λ)) = λ so that ω is an injective map. Assume first that n d is odd. • There is an exact sequence 0 → L(0, σ

n d 0 (0)) → Ω -n d (L(0, 0)) → C 1 λ (0, 0) → 0.
We tensor this with C 1 µ (0, 0) on the right and factor out projective-injectives. Using the fact that

C 1 µ (0, σ n d 0 (0)) ∼ = C 1 µ (0, 1), the resulting exact sequence is 0 → C 1 µ (0, 1) ---→ ( f 1 , f 2 ) Ω -n d (C 1 µ (0, 0)) ⊕ projective ∼ = C 1 ω(µ) (0, 1) ⊕ P → C 1 λ (0, 0) ⊗ C 1 µ (0, 0) → 0. If f 1 = 0, then ω(µ) = µ and f 1 is an isomorphism, therefore C 1 λ (0, 0) ⊗ C 1 µ (0, 0) ∼ = P. If f 1 = 0, then C 1 λ (0, 0) ⊗ C 1 µ (0, 0) ∼ = C 1 ω(µ) (0, 1) ⊕ Ω -1 (C 1 µ (0, 1) ⊕ projective. So either C 1 λ (0, 0) ⊗ C 1 µ (0, 0) is projective or C 1 λ (0, 0) ⊗ C 1 µ (0, 0) ∼ = C 1 ω(µ) (0, 1) ⊕ C 1 ω ′ (µ) (0, 0) ⊕ projective.
• Exchanging λ and µ shows that either

C 1 λ (0, 0) ⊗ C 1 µ (0, 0) is projective or C 1 λ (0, 0) ⊗ C 1 µ (0, 0) ∼ = C 1 ω(λ) (0, 1) ⊕ C 1 ω ′ (λ) (0, 0) ⊕ projective. • Therefore, since ω is injective, either λ = µ or C 1 λ (0, 0) ⊗ C 1 µ (0, 0) is projective.
If n d is even, the same arguments work if we start with the exact sequence

0 → C 1 λ (0, 0) → Ω n d (L(0, 0)) → L(0, σ n d 0 (0)) → 0 The proof that C 1 λ (0, 0) ⊗ M ± 2 (0, xd) for x ∈ Z is projective is similar, using the exact sequences 0 → L(0, 1 + (x -1)d) → Ω -1 (L(0, xd)) → M + 2 (0, xd) → 0 and 0 → L(0, 1 + xd) → Ω -1 (L(0, xd)) → M - 2 (0, xd) → 0.
Then, using induction, the Auslander-Reiten sequences for string and band modules, and the fact that the tensor product of any module by a projective module is projective, it follows that C ℓ λ (0, 0) ⊗ C t µ (0, 0) for λ = µ and C ℓ λ (0, 0) ⊗ M ± 2t (0, xd) are projective. The general case follows by tensoring with L(u, i) ⊗ L(v, j).

As a consequence, we can give the dual of a band module.

(u, i) is C ℓ λ (-u, 1 -i).
Proof. First, as in the case of string modules of even length, and using the fact that the dual of a projective Λ n,dmodule (respectively Λ * cop n,d -module) is projective because both algebras are self-injective as finite dimensional Hopf algebras, it can be shown that there exists µ ∈ k\{0} such that

C ℓ λ (u, i) * ∼ = C ℓ µ (-u, 1 -i). Moreover, we have 0 = Hom D(Λ n,d ) (C ℓ λ (u, i), C ℓ λ (u, i)) ∼ = Hom D(Λ n,d ) (C ℓ λ (u, i) ⊗ C ℓ λ (u, i) * , L(0, 0)) ∼ = Hom D(Λ n,d ) (C ℓ λ (u, i) ⊗ C ℓ µ (-u, 1 -i), L(0, 0)) therefore C ℓ λ (u, i) ⊗ C ℓ µ (-u, 1 -i)
is not projective and by Proposition 4.6 it follows that µ = λ.

Another consequence is that we can determine the syzygy of any band module.

Corollary 4.8. Let (u, i) be in Z 2 n with 2i + u -1 ≡ 0 (mod d) and let λ ∈ k\{0} be a parameter. Then for any positive integer ℓ we have

Ω(C ℓ λ (u, i)) ∼ = C ℓ λ (u, σ u (i)) ∼ = Ω -1 (C ℓ λ (u, i)). Proof.
We start with the syzygy of C ℓ λ (0, 0). We know that there exists a parameter µ ∈ k\{0} such that

Ω(C ℓ λ (0, 0)) = C ℓ µ (0, 1). By Lemma 4.2, the space Ext 1 D(Λ n,d ) (C ℓ µ (0, 1), C ℓ µ (0, 1)) is not zero. Moreover, there are isomorphisms Ext 1 D(Λ n,d ) (C ℓ µ (0, 1), C ℓ µ (0, 1)) ∼ = Hom D(Λ n,d ) (Ω(C ℓ µ (0, 1)), C ℓ µ (0, 1) ⊗ L(0, 0)) ∼ = Hom D(Λ n,d ) (Ω 2 (C ℓ λ (0, 0)) ⊗ C ℓ µ (0, 1) * , L(0, 0)) ∼ = Hom D(Λ n,d ) (C ℓ λ (0, 0) ⊗ C ℓ µ (0, 0), L(0, 0))
and therefore C ℓ λ (0, 0) ⊗ C ℓ µ (0, 0) is not projective. It follows from Proposition 4.6 that µ = λ and therefore that Ω(C ℓ λ (0, 0)) = C ℓ λ (0, 1). We then have the following isomorphisms.

Ω(C ℓ λ (u, i)) ∼ = Ω(core(C ℓ λ (0, 0) ⊗ L(u, i))) ∼ = core(Ω(C ℓ λ (0, 0) ⊗ L(u, i))) ∼ = core(Ω(C ℓ λ (0, 0)) ⊗ L(u, i)) ∼ = core(C ℓ λ (0, 1) ⊗ L(u, i)) ∼ = core(C ℓ λ (0, 0) ⊗ L(0, 1) ⊗ L(u, i)) ∼ = core(C ℓ λ (0, 0) ⊗ L(u, σ u (i))) ∼ = C ℓ λ (u, σ u (i)). Finally, since Ω 2 (C ℓ λ (u, i)) ∼ = C ℓ λ (u, i), we also have Ω -1 (C ℓ λ (u, i)) ∼ = Ω(C ℓ λ (u, i)).
The next result gives the tensor product of any two band modules with the same parameter up to projectives. Theorem 4.9. For any positive integer t and any integer ℓ with ℓ t, we have

core(C ℓ λ (v, j) ⊗ C t λ (u, i)) ∼ = θ∈I C t λ (u + v, i + j + θ) ⊕ C t λ (u + v, σ u+v (i + j + θ)) .
where I is defined in Proposition 1.8.

Proof. We prove the isomorphism in the case (u, i) = (v, j) = (0, 0), that is, core(C ℓ λ (0, 0) ⊗ C t λ (0, 0)) ∼ = C t λ (0, 0) ⊕ C t λ (0, 1) and the result follows by taking the tensor product with L(u, i) ⊗ L(v, j).

By working with modules over the basic algebra, we can see that there is an exact sequence

0 → L(0, 1 -d) → C ℓ λ (0, 0) → Ω ℓ n d -1 (L(0, σ ℓ n d -1 0 (0))) → 0.
Tensoring on the right with C t λ (0, 0) gives an exact sequence

0 → C t λ (0, 1) ⊕ P 1 → C ℓ λ (0, 0) ⊗ C t λ (0, 0) → Ω ℓ n d -1 (C t λ (0, σ ℓ n d -1 0 (0))) ⊕ P 2 → 0
for some projective-injective modules P 1 and P 2 . It follows that there is an exact sequence (4.1) 0 → C t λ (0, 1) → core(C ℓ λ (0, 0) ⊗ C t λ (0, 0)) ⊕ P 3 → C t λ (0, 0) → 0 for some projective module P 3 (obtained by factoring out the split exact sequence 0 → P 1 → P 1 ⊕ P 2 → P 2 → 0). The exact sequence (4.1) is isomorphic to the following pullback

0 / / C t λ (0, 1) / / E / / C t λ (0, 0) / / ϕ 0 0 / / C t λ (0, 1) / / P := n d -1 y=0 P(0, yd) t π / / Ω -1 (C t λ (0, 1)) ∼ = C t λ (0, 0) / / 0 where E = {(m, p) ∈ C t λ (0, 0) × P ; ϕ(m) = π(p)} ∼ = core(C ℓ λ (0, 0) ⊗ C t λ (0, 0)) ⊕ P.
Assume for a contradiction that the sequence (4.1) is not split. Then ϕ = 0 so that ϕ is a non-zero endomorphism of C t λ (0, 0). Let m ∈ C t λ (0, 0) be an element that is not in the radical and such that ϕ(m) = 0, we may assume that the submodule generated by m has a simple top. Then ϕ(m) = π(e) where e ∈ P and e generates an indecomposable projective summand. The element (m, e) belongs to E and it generates an indecomposable projective module. It is then a summand of E since projectives are injective. Let this summand be P(0, yd) for some y, it is then also a summand in C ℓ λ (0, 0) ⊗ C t λ (0, 0). Denote by [M : L] the multiplicity of a simple module L as a summand in a semisimple module M. The simple module L(0, yd) is in the socle of the projective summand P(0, yd) of C ℓ λ (0, 0) ⊗ C t λ (0, 0) (but does not occur in the socle of C t λ (0, 0)), hence it follows from the above that

[soc(core In many of our proofs, we have worked with B 0,0 -modules, then tensored with non-projective simple modules to obtain the general result we were seeking. This approach can be formalised as follows.

(C ℓ λ (0, 0) ⊗ C t λ (0, 0))) : L(0, yd)] < [soc C t λ (0,
(0, 1 + yd) ∼ = C t λ (0, 1), we have [soc(core(C ℓ λ (0, 0) ⊗ C t λ (0, 0))) : L(0, yd)] = dim Hom D(Λ n,d ) (L(0, yd), core(C ℓ λ (0, 0) ⊗ C t λ (0, 0))) = dim Hom D(Λ n,d ) (L(0, yd), C ℓ λ (0, 0) ⊗ C t λ (0, 0)) = dim
Theorem 4.10. Let L(u, i) be a non-projective simple module. Then -⊗ L(u, i) induces a stable equivalence between B 0,0 and B u,i .

The proof uses the following lemma. Proof. The isomorphism follows from Lemma 3.2 which states that L(u, i) * ∼ = L(-u, 1σ u (i)), Proposition 1.8, and the fact that 1σ u (i) + i = 1 -N. Since 1min(N, d -N) > -d 2 , we need only prove that the blocks B 0,τ with -d 2 < τ 0 are pairwise different. The block B 0,τ contains precisely the simple modules L(0, j) with j in the σ 0 -orbit of τ, that is, j ∈ {τ + td, σ 0 (τ) + td ; 0 t < n d } (recall that j is taken modulo n). There are precisely two representatives of the σ 0orbit of τ in ]d, 0]. Moreover, if -d < j -d 2 , then -1 -2d < 2j -1 -d -1 so σ 0 (j) = d + j -2j -1 = d + j -(2j -1 + 2d) = -jd + Proof. Let M be a non-projective indecomposable D(Λ n,d )-module. Since there are only finitely many isomorphism classes of projective D(Λ n,d )-modules, we can ignore the projective summands that appear when taking repeated tensor products of M.

• If M = L is a simple module, all the non-projective indecomposable summands in L ⊗t are simple.

Since there are only finitely many non-isomorphic simple modules, L is algebraic. • If M = Ω ℓ (L) for some simple module L and ℓ ∈ Z non-zero, then M is not algebraic. Indeed, we have core(M ⊗t ) ∼ = Ω ℓt (core(L ⊗t )) and when t varies, we get infinitely many non-isomorphic indecomposable summands.

• If M = M +
2ℓ (u, i), all the non-projective indecomposable summands in M ⊗t are of the form M + 2ℓ (v, j) with the same ℓ. There are only finitely many such modules up to isomorphism, therefore M + 2ℓ (u, i) is algebraic. Similarly, M - 2ℓ (u, i) is algebraic.

• If M = C ℓ λ (u, i
), all the non-projective indecomposable summands in M ⊗t are of the form C ℓ λ (v, j) with the same ℓ and λ. There are only finitely many such modules up to isomorphism, therefore C ℓ λ (u, i) is algebraic.

1. 2 .

 2 D(Λ n,d )-modules of odd length, projective D(Λ n,d )-modules and blocks of D(Λ n,d ).

Remark 1 . 5 .

 15 It is clear that the D(Λ n,d )-modules P(u, i) are projective both as Λ n,d -modules and as Λ * cop n,dmodules. Moreover, the non-projective simple D(Λ n,d )-modules L(u, i) have dimension at most d -1 and therefore they are not projective as Λ n,d -modules or as Λ * cop n,d -modules.

Lemma 2 . 1 .

 21 Let C be an indecomposable non-projective D(Λ n,d )-module and let A(C) be its Auslander-Reiten sequence. For any D(Λ n,d )-module M, the following are equivalent.

Corollary 4 . 7 .

 47 The dual of C ℓ λ

  Hom D(Λ n,d ) (L(0, yd) ⊗ C t λ (0, 0) * , C ℓ λ (0, 0)) = dim Hom D(Λ n,d ) (C t λ (0, 1), C ℓ λ (0, 0)) = dim Ext 1 D(Λ n,d ) (C t λ (0, 0), C ℓ λ (0, 0)) = tby Lemma 4.2 since t ℓ. Therefore we have obtained a contradiction, the sequence (4.1) splits, and the result follows.

Lemma 4 . 11 .

 411 Let L(u, i) be a non-projective simple module of dimension N. There is an isomorphismcore(L(u, i) ⊗ L(u, i) * ) ∼ = 0 τ=1-min(N,d-N) L(0, τ),and the blocks B 0,τ , with 1min(N, d -N) τ 0, are pairwise distinct.

  1 is not divisible by d then the orbit of j under σ u has size 2 n d and moreover we have σ 2t u

  ∈ k \ {0}, any positive integer ℓ and (u, i) ∈ Z 2 n with 2i + u -1 ≡ 0 (mod d), taking i modulo d (that is, one for each orbit of the square of σ u ).We now describe the modules of even length in more detail for future use. Fix a block B u,i .

	p-1 u 1.3. D(Λ n,d )-modules of even length. We described all the non-projective indecomposable representations of (i) . D(Λ n,d ) in [9, Section 3 and Appendix A]. These are (a) The modules of odd length described above. (b) The string modules of length 2ℓ, denoted by M + 2ℓ (u, i) and M -2ℓ (u, i), for any positive integer ℓ and (u, i) ∈ Z 2 n with 2i + u -1 ≡ 0 (mod d). (c) The band modules C ℓ λ (u, i) of length 2ℓ n 2n d -1 and for each ℓ 1, there are two indecomposable modules of length 2ℓ which we call M ± 2ℓ (u, σ p u (i)) : d for λ 1.3.1. String modules of even length. For each 0 p • The module M + u (i)), . . . , L(u, σ 2(ℓ-1) u (i)) and socle compo-2ℓ (u, i) has top composition factors L(u, i), L(u, σ 2 sition factors L(u, σ u (i)), L(u, σ 3 u (i)), . . . , L(u, σ 2(ℓ-1)+1 u (i)):

  are modules of length 2 n d with the same top and the same socle. The following property allows us to distinguish them. Recall that Λ n,d and Λ * cop n,d are subalgebras of D(Λ n,d ), so that any D(Λ n,d )module can be viewed as a Λ n,d -module and as a Λ * cop n,d -module. Assume that d does not divide 2i + u -1. Then • the modules M +

	Property 1.4.

2ℓ (u, i) are projective as Λ n,d -modules but not as Λ * cop n,d -modules, • the modules M - 2ℓ (u, i) are projective as Λ * cop n,d -modules but not as Λ n,d -modules, • the modules C ℓ λ (u, i) are projective both as Λ n,d -modules and as Λ * cop n,d -modules.

  1) : L(0, yd)] = t. Using Lemma 3.1, Proposition 4.3, Corollary 4.7 and the fact that C t λ

TABLE 1 .

 1 1, andd 2 + 1 -jd + 1 < 1 therefore -d 2 < σ 0 (j) 0. Similarly, if -d 2 < j 0 then -d < σ -1 0 (j) = σ 0 (j)d -d 2 .It follows that the σ 0 -orbit of τ has precisely one representative in ]-d 2 , 0], which proves our claim. (L(w, i + j + θ)) Description of the product in the stable Green ring of D(Λ n,d )where, for (u, i) and (v, j) in Z 2 n , we have putI = {θ ; 0 θ min(dim L(u, i), dim L(v, j)) -1} if dim L(u, i) + dim L(v, j) d {θ ; dim L(u, i) + dim L(v, j)d θ min(dim L(u, i), dim L(v, j)) -1} otherwise.and w = u + v. We assume that ℓ t.Set L = L(u, i) and N = dim L(u, i). By Lemma 4.11, core(L ⊗ L * ) ∼ = therefore L is endotrivial if and only if min(N, d -N) = 1, that is, N = 1 or N = d -1.Remark 6.4. A simple module belonging to the block B is endotrivial if, and only if, all the modules of odd length that belong to B are endotrivial.We now classify algebraic modules. An indecomposable module M is algebraic if there are only finitely many non-isomorphic indecomposable summands in T(M) = t 1 M ⊗t . The tensor product of projective modules is again projective and there are only finitely many isomorphism classes of projective D(Λ n,d )-modules, therefore any projective D(Λ n,d )-module is algebraic. Proposition 6.7. A non-projective indecomposable D(Λ n,d )-module is algebraic if and only if it is simple or of even length.

	⊗	Ω m (L(v, j))			M + 2ℓ (v, j)		M -2ℓ (v, j)	C ℓ µ (v, j)
								if n is even:	C ℓ µ (w, i + j + θ)
	Ω n (L(u, i))		θ∈I	M + 2ℓ (w, σ -n w (i + j + θ))	θ∈I	M -2ℓ (w, σ n w (i + j + θ))	if n is odd:	θ∈I C ℓ µ (w, σ w (i + j + θ))
								θ∈I
	[9, Theorem 4.1 and § 4.2]			Theorem 2.5		Theorem 2.5	Proposition 4.4
	M + 2t (u, i)	M + 2t (w, σ -m w (i + j + θ))	θ∈I	M + 2t (w, i + j + θ)⊕		0	0
	θ∈I		M + 2t (w, σ 2ℓ-1 w	(i + j + θ))
		Theorem 2.5			Theorem 3.4	[9, Proposition 4.21]	Proposition 4.6
	M -2t (u, i)							0
							M -2t (w, σ w -(2ℓ-1)	(i + j + θ))
		Theorem 2.5	[9, Proposition 4.21]		Theorem 3.4	Proposition 4.6
	if m is even:						if λ = µ:
	C t λ (u, i)				0			0
								otherwise: 0
		Proposition 4.4			Proposition 4.6		Proposition 4.6	Theorem 4.9

θ∈I

Ω m+n θ∈I M - 2t (w, σ m w (i + j + θ)) 0 θ∈I M - 2t (w, i + j + θ)⊕ θ∈I C t λ (w, i + j + θ) if m is odd: θ∈I C t λ (w, σ w (i + j + θ)) θ∈I C t λ (w, i + j + θ)⊕ C t λ (w, σ w (i + j + θ))

Proof of Theorem 4.10. Define a functor F : B 0,0 -mod → D(Λ n,d )mod by F(M) = core(M ⊗ L(u, i)) on objects and F( f ) = f ⊗ id L(u,i) restricted and co-restricted to the cores of the modules on morphisms.

We first determine the image of a simple module under F. The simple modules in B 0,0 are the modules L(0, td), which have dimension 1, and the modules L(0, 1 + td), which have dimension d -1, for 0 t < n d . By Proposition 1.8, we have

which are simple B u,i -modules. By induction on the length of modules, it follows that F sends any nonprojective B 0,0 -module to a non-projective B u,i -module. Therefore F induces a functor B 0,0 -mod → B u,i -mod which we denote also by F.

We can be more specific. We have the following:

)-mod takes projectives to projectives and is exact, therefore it commutes with Ω. It follows that F commutes with Ω. Therefore we can give the image by F of any non-projective indecomposable module:

Finally, in order to prove that F is indeed an equivalence of categories, we must prove that for any modules M 1 and M 2 in B 0,0 -mod, we have Hom

since M 2 belongs to B 0,0 . However, by Lemma 4.11 and the beginning of this proof, the modules M 1 ⊗ L(0, τ) belong to pairwise distinct blocks, and the only one belonging to B 0,0 is

DESCRIPTION OF THE STABLE GREEN RING OF D(Λ n,d )

Combining the results in this paper with the results in [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF], we now have a complete description of the stable Green ring of D(Λ n,d ), which we give in Table 1 on page 17.

The stable Green ring is commutative so that we may assume for instance that ℓ t in Table 1.

APPLICATION TO ENDOTRIVIAL MODULES AND ALGEBRAIC MODULES

Endotrivial modules have been classified and used in the context of group algebras, see [START_REF] Carlson | Endotrivial modules. Recent developments in Lie algebras, groups and representation theory[END_REF]. Here, we determine all the endotrivial modules over D(Λ n,d ). Proof. As mentioned in the remark above, if M is an endotrivial module, then there exist a non-projective simple module L and an integer m ∈ Z such that M ∼ = Ω m (L). Moreover, core(M ⊗ M * ) ∼ = core(Ω m (L) ⊗ Ω -m (L * )) ∼ = core(Ω m-m (L ⊗ L * )) ∼ = core(L ⊗ L * ) so that M is endotrivial if, and only if, L is endotrivial.


M Nedil

M A Habib

T A Denidni

H Boutayeb

QUASI-METALLIC-WALL TECHNIQUE FOR INCREASING THE EFFICIENCY OF CB-CPW ANTENNAS

This paper presents a new quasi-metallic-wall technique for improving the gain of CB-CPW single antenna and arrays. This technique allows reducing the surface wave losses of the CB-CPW antennas, which decreases the antenna radiation efficiency. It consists on including pins as quasi-metallic wall between the upper and lower ground planes in the CB-CPW antenna structure. To validate the proposed approach, a CB-CPW-slot antenna fed through an inductive coupled CPW-line operating at 5.8 GHz is considered. This approach allows to increase the antenna efficiency from 70% to 95% around the operating frequency. The antenna gain achieves then an improvement of 2 dBi. Also, an antenna array is designed and the pins technique is also applied to prove its applicability for the array case. An efficiency increase from 64% to 95% was achieved. Both single antenna and antenna array with pins were fabricated and measured. A good agreement between numerical and experimental results was obtained.

INTRODUCTION

Recently, wireless communications represent one of the highest growing markets, especially on the development of mobile communications and wireless local area networks (WLANs), where high capacity transmission systems are required. These concern new wideband RF wireless components such as antennas, filters and so on. Especially for antenna issue, planar broadband antennas using coplanar waveguide CPW feed lines have been proposed as an alternative for microstrip feed lines [START_REF] Menzel | A microstrip patch antenna with coplanar feed line[END_REF]. Indeed, CPW feed lines have several useful design characteristics, a low radiation loss, less dissipation, and uniplanar configurations.

Particularly for antenna applications, CPW-fed antennas have long been popular for various applications because of their low cost, low profile, light weight, and compatible with integrated circuits [START_REF] Jiao | A Broadband CPW-FED T-Shape Slot Antenna[END_REF][START_REF] Saed | Broadband CPW-Few Planar Slot Antennas with Various Tuning Stubs[END_REF][START_REF] Rao | On Improving Impedance Matching of a CPW FED Low Permittivity Dielectric Resonator Antenna[END_REF][START_REF] Liu | CPW-FED Notched Monopole Antenna for Umts/imt-2000/WLAN Applications[END_REF][START_REF] Chen | A Novel Small CPW-fed T-shaped Antenna for Mimo System Applications[END_REF][START_REF] Liu | Cpw-Fed Compact Meandered Strip Antenna on a Soft Substrate for Dualband Wlan Communication[END_REF]. In practice, additional conducting planes are often placed below the substrate in order to electromagnetically separate the circuit from its environment [START_REF] Shigesawa | Conductor-backed slot line and coplanar waveguide: Dangers and full-wave analyzes[END_REF]. The main drawback of this approach is the presence of parallel-plate modes, which are considered as unwanted bulk modes [START_REF] Haydl | On the Use of Vias in Conductor-Backed Coplanar Circuits[END_REF]. Therefore, the CB-CPW structure suffers from radiation leakage, which makes it less efficient. This leakage is caused by is the surface wave losses, which decreases considerably the antenna performance. To overcome this problem, various techniques have been proposed. For instance, we can quote: Jahagirdar et al. [START_REF] Jahagirdar | Non-leaky conductorbacked coplanar waveguide-fed microstrip patch antenna[END_REF] have proposed nonleaky conductor backed coplanar waveguide with emphasis on avoiding the leakage of power. Lan et al. [START_REF] Lan | Parallel-plate slot array fed by conductorbacked coplanar waveguide[END_REF] have used parallel-plate mode leakage to radiate through the slot array etched on both sides of the ground plane. In [START_REF] Qiu | Highly efficient unidirectional twin arc-slot antennas on electrically thin substrates[END_REF], twin broadside slots with half guided wavelength are used to reduce the undesired propagating power. However, all these design are more complex and expensive to be implemented. Furthermore, these approaches deal only with a single element, and they have not been investigated yet for antenna arrays. In this paper, a new efficiency-enhancement technique using an appropriate via density to create a quasi metallic wall in CB-CPW technology is proposed. These via connectors join the upper and lower CB-CPW (Conductor Backed CPW) ground planes to eliminate the unwanted parallel plate modes and to create the cavity effect in the antenna by stopping the propagation of surface wave between the upper and lower ground planes. First, a parametric study of the pin inclusion effect in CB-CPW lines was carried out. Second, this technique was applied to a single element antenna case [START_REF] Nedil | Design of a Broadband slot Antenna Fed by CPW for Wireless Application at 5.8 GHz[END_REF]. Third, a serial array of three elements was also designed using this approach. To validate the proposed approach, numerical simulations and experimental measurements were carried out on a single antenna and then on antenna array. The obtained results show a good performance in terms of radiation efficiency. The paper is organized as follows. After this introduction, the quasi metallic wall technique is described in section II. In section III, an antenna element with via shorting pin is presented. Section IV, presents the proposed technique for an antenna array case. Finally, the last section presents the conclusion.

QUASI METALLIC WALL TECHNIQUE

Figure 1 shows the structural view of CB-CPW line, where the two top ground planes and the center conductor form a CPW line. In this structure, a parallel-plate mode is formed between the top and bottom ground planes. The energy will leak along a particular angle once the wave is launched. It is apparent that this leakage can be stopped if a guard ring type method is used along the signal propagation as reported in [START_REF] Haydl | On the Use of Vias in Conductor-Backed Coplanar Circuits[END_REF]. It is well known that periodic structure exhibits a stopband phenomenon, which can be used to stop wave propagation at this band. For our application, we are interested in using via technique in regards to its compactness and its simplicity. To implement this technique, the coupling effect between two CB-CPW lines as illustrated in Fig. 1 was simulated. This CB-CPW lines is built on a substrate with a relative permittivity ε r = 2.2 and a thickness h = 3.175mm. The CPW line is designed in order to achieve a 50 input impedance for matching the characteristic impedance measurement system feed line. Dimensions of the CPW line are calculated using LineCalc software of Agilent [START_REF]Advanced Design System[END_REF]; the parameters at 5.8 GHz are s = 0.4 mm, g = 3.88 mm. First, the case of opposite line disposition as described in Fig. 2 was investigated. The study consists on observing the effect of including pins in the structure. Figure 3 shows simulated results for the transmission coefficients S 12 of the structure with and without pins. Inclusion of pins produces a quasi metallic wall effect that reaches 17 dB of relative stop-band at 5.8 GHz. The optimal value of the pin step distance is 8 mm. The pin diameter is found to be 0.6 mm.

To make our study more complete, the case of parallel line disposition has also been introduced as shown in Fig. 4. All ports are matched. The simulation results shown in Fig. 5 reveal a stop-band effect of 17 dB at maximum. The optimal value of a pin-step distance is 8 mm. From these results, it can be concluded that the pins act like metallic wall, which confirms our approach. This technique is commonly used to forbid high mode propagation. The purpose of this section has been to investigate the proper pin dimension and distance in order to include them in an antenna prototype. In the next section, these pins are added to antenna global structure (antenna + pins), and the overall performance is simulated and measured. Transmission coefficient decrease for parallel lines disposition.

ANTENNA ELEMENT WITH SHORTING VIAS

Design

As mentioned, the proposed technique was first applied to a single CB-CPW antenna. For this issue, a new antenna using inductively fed slot was designed. The inductively fed slot antenna geometry is shown in Fig. 6.

L S L m S a S b W L a G f S f Figure 6
. Configuration of single element CPW inductively fed slot antenna.

This type of feeding technique corresponds to a short-circuited excitation slot [START_REF] Nedil | Design of a Broadband slot Antenna Fed by CPW for Wireless Application at 5.8 GHz[END_REF]. The antenna is composed of one substrate, which has a relative permittivity ε r = 2.2 and a thickness h = 3.175 mm. The parameters of this antenna are the followings: S f = 0.4 mm, G f = 3.88 mm, L a = 13.72 mm, W = 17.74 mm, S a = 3.7 mm, S b = 1 mm, and L s = 17.48 mm. The major problem in this design is the surface wave loss. Even if the structure is optimized to reduce the surface wave loss, there is still some non radiated energy of about 30% of the overall energy. This energy is dissipated as leaky waves due to the parallel plate modes that are inherent and undesired feature of CB-CPW structures. Fig. 7 shows, through simulations carried out with HFSS-Ansoft [START_REF]Ansoft HFSS version 9.0[END_REF], how the electrical field is distributed around the radiating slots.

From this figure, it can be seen that a part of energy is lacked into the substrate between the upper and lower ground planes. In order to recover the lacked energy, the proposed quasi-metallic wall technique was applied to this design. To ensure antenna performance enhancement, the proper dimensions of this antenna were investigated. Figure 8 shows the antenna structure with surrounded pins. Both distances between the upper radiating element and the pins and between the antenna sides and the pins are taken into a count; these parameters affect grandly the antenna performance. The effect of the distance between the upper radiating slot and the pins location (a) and effect of the distance between the antenna sides and the pins location (b) were studied and both are shown in Fig. 9.

It is been found that the first parameter affects the antenna efficiency. It can be observed that this parameter does not affect the resonance frequency (Fig. 9 (a)). The optimum value is obtained for a = 9.94 mm, which corresponds approximately to quarter wavelength. This is due to the fact that the radiating edges are the upper and lower slot edges. The pins row behaves like an electric wall. Combining phase inversion due to the image reflection and the 180 phase introduced by the half wavelength path (go to and come from with quarter wavelength), the sum is then constructive. The effect of the distance between the antenna sides and the pins location affects highly the impedance matching (Fig. 9 (b)). This can be explained by the fact that the current distribution is out of phase for the lateral edges. Thus, if the current is disturbed by the proximity of an electric wall, the overall input impedance is also disturbed. The optimal distance is b = 8.326 mm. Fig. 10 illustrates the behavior of the magnetic current in the radiating slot. The distribution of magnetic current in the slot radiating element without and with pins (case b = 8.326 mm) is almost the same, which confirm that the optimum case does not affect the current distribution of the antenna. However, when, the distance b, for example, changes to b -5, the magnetic current distribution is degraded.

Results and discussion

To validate the proposed design, a prototype was fabricated using above parameters, and measured by HP8772 network analyzer with TRL calibration technique. Also, a prototype of antenna element without pin has been fabricated for comparison purposes. The simulated and measured return loss coefficients for both antennas are shown in Fig. 11. A good agreement between simulation and experimental results was obtained. Furthermore, it can be seen that there is no discrepancy between the return loss of the two antennas. It can be noted that the bandwidth of 300 M Hz ( 6.5%) for a -10 dB return loss level is observed at 5.8 GHz operating frequency. The major difference between the antennas is their efficiency. Fig. 12 describes the efficiency for both cases. Whereas, the simple antenna element shows an efficiency of 75% at 5.8 GHz, the introduction of pins leads to reach 94%.

To verify this approach in terms of propagation of electrical field inside the antenna, the antenna with pins was simulated with HFSS. Simulation results of propagation of electrical field id plotted in Fig. From this figure, it can be shown, that electrical fields are not propagating beyond the pins lattice, compared to the distribution observed in Fig. 7, which confirm that the most part of lacked energy is recovered by using the quasi metallic wall. A measurement of this increase was carried out. For this purpose, a network analyzer HP8772 is used to measure the transmission coefficient (S 12 ) between a reference antenna, which is a wideband horn antenna from Antcom corp., and our prototypes. The distance between the antennas respects the far field condition. It is known that the antenna gain and the

G = ηD (1)
where η is the antenna efficiency, G is the gain and D is the directivity. From Fig. 14, it can be noted that the radiation patterns of both antennas with and without pins are sensitively the same; their directivities are then also similar. This can be justified by the magnetic current distribution within the radiation which remains mainly identical. Since that, the ratio between the gains and the efficiencies is the same, then it is possible to write

G 2 G 1 = η 2 η 1 (2)
G 1 and G 2 are the gains of the antennas without and with pins, respectively. For measured powers, the following formula can be used:

P received ∝ GP transmitted (3) or simply, G ∝ S 2 21 ( 4
)
Comparing the transmission coefficient for both cases of antennas without and with pins permits to write

η 2 η 1 = G 2 G 1 = S 2 21,2 S 2 21,1 (5)
where S 21,i is the transmission coefficient associated to the gain G i . This ratio is illustrated in Fig. 15. This figure shows that the received power, and, de facto, the gain is well increased by including pins in the structure. Around 5.8 GHz, a maximum of 2.3 dB improvement is noted. The measured ratio is then 1.44. The simulation reports a ratio of 1.27. Moreover, the enhancement of efficiency is wideband. The radiation patterns of the antenna were also measured in anechoic chamber with Antcom corp. Figure 16 shows theoretical and measured radiation patterns in principal planes, and a good agreement is observed. The pin introduction does perturb neither the antenna bandwidth nor its radiation pattern. The next section deals with antenna arrays. To verify that the quasimetallic wall technique of pins inclusion is exportable to an array structure, an arrays of three elements are presented.

5, 6 5,7 5,8 5,9 6,0 -28 -27 -26 -25 -24 -23 S 12 (dB) Frequency (GHz)
without Pins (measured)

with Pins (measured)

ANTENNA ARRAY

Usually, the radiation pattern of a single radiating element is relatively wide with low directivity. In many applications, antennas with very directive characteristics are needed and this can be done by enlarging their electric dimensions. A way to do this without increasing the size is to form an array. In most arrays, antenna elements are identical because this convenient, simple, and practical. Few works about improving the performances of antenna array have been reported in literature [START_REF] Yang | Microstrip Antennas Integrated With Electromagnteic Band-Gap (EBG) Structures: A Low Mutual Coupling Design for Array Applications[END_REF][START_REF] Iluz | Microstrip Antenna Phased Array With Electromagnetic Bandgap Substrate[END_REF]. These works were mainly focused on decreasing mutual coupling between array elements. The case of the efficiency enhancement has never been studied yet. To demonstrate, that the proposed enhancement efficiency technique is suitable for series fed arrays, a linear three elements array was designed with this approach. Using the single antenna design presented previously, a three element array was built. The proposed antenna element is an inductively fed one; this allows controlling easily input impedance of the antenna. Thus, serial feeding is a suitable choice for our array design. Figure 17 shows the layout of the antenna array. The elements are identical, excited in phase (broadside array) since the electrical length in the CPW line between exciting slot corresponds to a wavelength at 5.8 GHz. The antenna can be electrically modeled as RLC serial circuit [START_REF] Giauffret | Study of various shapes of the coupling slot in CPW-fed microstrip antennas[END_REF]. To achieve 50Ω as input impedance for the global array, each element should be designed with 50Ω/3 input impedance (Z in /3). For this purpose, the chosen dimensions for the elements are: CPW feed line: S f = 0.4 mm, G f = 3.88 mm; Slot radiating element: L a = 13.94 mm, W = 19.25 mm, S a = 3.7 mm, S b = 1 mm; Exciting slot: L s = 18.38 mm. The distance between the via and the upper radiating slot is 10 mm and 7.33 mm from each radiating slot side to lateral one. First, the prototype of an array without via was fabricated. For comparison purposes, the measured and simulated results are shown in Fig. 18 and19. A good agreement between simulation and measurement is noted. The E-plane radiation pattern is clearly more directive.

As reported in section II, however, the structure has a poor radiating efficiency which is about 64%. To overcome this undesired feature, via are included. The antenna structure with pins was simulated and fabricated. Settings are sensitively the same that those encountered for the single element case. relatively the same, and the radiation pattern does not suffer from any perturbation. Fig. 21 reports the radiating efficiency. It is clear that, at 5.8 GHz, the efficiency climbs form 64% to 95%. This enhancement has been verified by measurements using the same transmission coefficient method described above as shown in Fig. 21. A gain enhancement of 2 dB can be observed at 5.8 GHz.

CONCLUSION

In this paper, a quasi-metallic wall technique has been introduced to CB-CPW antenna technology to suppress surface wave losses and to increase the radiation gain. This technique was originally applied to a CB-CPW single element slot antenna to raise the radiating efficiency from 74% to 94%. In addition, using the same technique, a three-element array has been designed and measured. The radiation efficiency of this array is increased from 60% to 95% which corresponds to 2 dB gain enhancement. With these features, these high gain structures are useful for wireless communication systems as WLAN.

Figure 1 .

 1 Figure 1. CPW electromagnetic field leakage.

Figure 2 .

 2 Figure 2. Opposite CPW line pins disposition.

Figure 3 .

 3 Figure 3.Transmission coefficient decrease for opposite lines disposition.

Figure 4 .

 4 Figure 4. Parallel CPW line pins disposition.

  Figure 5.Transmission coefficient decrease for parallel lines disposition.

Figure 7 .

 7 Figure 7. Electric field distribution in the space around the antenna without pins.

Figure 8 .

 8 Figure 8. Configuration of single element CPW antenna with pins.

Figure 9 .

 9 Figure 9. Return loss versus pins distance effect: (a) frontal distance (b) lateral distance.

Figure 10 .

 10 Figure 10. Current distribution on the radiation slot: (a) without pin (b)optimal value (c) pins shifted by 5 mm.

Figure 11 .

 11 Figure 11. Simulated and measured return loss for single element antennas with and without pins.

Figure 12 .

 12 Figure 12. Comparison of radiating efficiencies for single element antennas with and without pins

Figure 13 .Figure 14 .

 1314 Figure 13. Electric field distribution in the space around the antenna with pins

Figure 15 .Figure 16 .

 1516 Figure 15. Relative received power of single element antennas with and without pins

Figure 17 .

 17 Figure 17. Photograph of three elements CPW array with pins.

Figure 18 .

 18 Figure 18. Simulated and measured return loss coefficients for three elements arrays with and without pins.

Figure 19 .

 19 Figure 19. Simulated and measured radiation pattern of three elements array without pins: (a) E-plane (b) H-plane.

Figure 20 .

 20 Figure 20. Simulated and measured radiation pattern of three elements array with pins: (a) E-plane (b) H-plane.

Figure 21 .

 21 Figure 21. Comparison of radiating efficiencies for three elements arrays with and without pins.

Figure 22 .

 22 Figure 22. Relative received power comparison of three elements arrays with and without pins.

Acknowledgement

This work was supported in part by National Science Engineering Research Council of Canada (NSERC).