Simple modules of small quantum groups at dihedral groups

Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over the dihedral groups $\mathbb{D}_{4t}$ with $t\geq 3$. To this end, we develop new techniques that can be applied to Nichols algebras over any Hopf algebra. Namely, we explain how to construct recursively irreducible representations when the Nichols algebra is generated by a decomposable module, and show that the highest-weight of minimum degree in a Verma module determines its socle. We also prove that tensoring a simple module by a rigid simple module gives a semisimple module.


Introduction
This paper is devoted to study the representations of certain families of Hopf algebras D(V, D m ), which are given by Drinfeld doubles of bosonizations of finitedimensional Nichols algebras B(V ) over dihedral groups D m of order 2m with m = 4t ≥ 12.The Hopf algebras D(V, D m ) might be considered as analogs of small quantum groups but with non-abelian torus.This election is based on the classification result in [FG] of all finite-dimensional Nichols algebras over D m .In particular, V belongs to an infinite family of reducible Yetter-Drinfeld modules over D m and B(V ) ≃ V .
The small quantum groups or Frobenius-Lusztig kernels u q (g) are finite dimensional quotients of quantum universal enveloping algebras U q (g) at a root of unity q for g a semisimple complex Lie algebra [L], with some restrictions on the order ℓ of q depending on the type of g.As it is well-known, U q (g) can be described as a quotient of the Drinfeld double of the quantum group U q (b) associated with a standard Borel subalgebra b of g.Consequently, u q (g) can also be described as a quotient of the Drinfeld double of the small quantum group u q (b).The latter is a pointed Hopf algebra over the abelian group Z n ℓ with n = rk g.As such, it is isomorphic to the smash product or bosonization u q (b) ≃ u q (n)#CZ n ℓ and u q (n) is a Nichols algebra of diagonal type [Ro], [AS], [A].In fact, as a consequence of the classification theorem due to Andruskiewitsch and Schneider [AS2], under mild conditions on the order of the groups, all complex finite-dimensional pointed Hopf algebras over abelian groups are variations of u q (b).In these notes, we consider objects analogous to the small quantum groups u q (b) but containing a non-abelian torus.These pointed Hopf algebras are given by bosonizations of finite-dimensional Nichols algebras over the infinite family of dihedral groups D m classified in [FG].These Nichols algebras actually turn out to be exterior algebras over semisimple objects in the braided category Dm Dm YD.See Section 4 for more details.More generally, one may consider the Drinfeld double D(V, H) := D(B(V )#H) of the bosonization of a finite-dimensional Nichols algebra B(V ) over a finitedimensional Hopf algebra H.These kind of generalized small quantum groups admits a triangular decomposition B(V )⊗D(H)⊗B (V ) in the sense of Holmes and Nakano [HN], [BT].Thus, as in the classical context, the simple modules can be obtained as quotients of generalizations of Verma modules and consequently are classified by their highest-weights.Here, the weights are the simple representations of the Drinfeld double D(H), which plays the role of the Cartan subalgebra.In case H = kΓ is a group algebra of an abelian group, the weights are one-dimensional and there are several results known, see for example [AAMR], [Ch], [KR1,KR2], [HY].However, in case Γ is not an abelian group, the weights are not necessarily one-dimensional and the computations turn out to be more involved.In this case, the description of the simple modules is only known for B(V ) = E 3 being the Fomin-Kirillov algebra over the symmetric group S 3 [PV1].It is worth pointing out that the category of graded modules can be endowed with an structure of highest-weight category when H is semisimple.
In conclusion, with the classification of the irreducible representations at hand, the central problem to address is to compute their characters, i.e. their weight decomposition.Our main contribution is the solution to this problem for small quantum groups at the dihedral groups D m , see Theorem 5.2.On the way, we develop new techniques and provide results that can be applied to Nichols algebras over any Hopf algebra.
Specifically, we develop a recursive process based on different triangular decompositions and bosonizations when V is reducible: If V = U ⊕ W , then the Nichols algebra B(V ) can be described as a braided bosonization B(Z)#B(U ) for certain module Z associated with the adjoint action of U on W , and B(V )#H ≃ B(Z)# B(U )#H as Hopf algebras, see [AA], [AHS], [HS].In this situation, we construct first the simple modules of D(U, H) = D(B(U )#H from the simple D(H)-modules -the weights-and then use the same proceeding to construct those of D(V, H) from the former.To use this tool, we need first to prove some technical results on composition of certain functors that assure that our recursive process gives the desired answer.
We would like to emphasize other two new results which might help to describe the simple modules of generalized small quantum groups.First, it is known that the simple modules can also be obtained as the socle of the Verma modules [PV1,Theorem 2]; we show in Corollary 3.6 that the socle of a Verma module is generated by its highest-weight of minimum degree.Second, a simple D(V, H)-module is said to be rigid if it is also simple as D(H)-module; we prove in Theorem 3.8 that the tensor product between a simple module and a rigid simple module is semisimple.
As the category D(V,Dm) M is non-semisimple, this is a first step towards a complete description of it.Nevertheless, taking into account that the main results needed lots of computations, we prefer to present the result on simple modules first and leave the study of the indecomposable modules, tensor products and extensions for future work.Finally, we point out that the description of the simple modules H (λ). The (big) dots represent their (highest-)weights.The shadow regions indicate submodules generated by highest-weights.In particular, the region on the bottom is the socle S V H (λ) which is generated by the highest-weight of minimum degree.The white region on the top depicts its unique simple quotient L V H (λ).
over D(V, H) can also be seen as a first step for finding new finite-dimensional Hopf algebras by using the generalized lifting method, see for instance [AA].
The paper is mostly self-contained and includes figures to lighten the reading.It is organized as follows.In the preliminaries we collect definitions, notation and basic facts that are used along the paper.In Section 3 we recall the general framework of Hopf algebras with triangular decomposition and present the new results mentioned above, cf.Corollary 3.6 and Theorems 3.8 and 3.11.In Section 4 we summarize some facts about the category of D(D m )-modules.We list the simple D(D m )-modules and compute some tensor products between them.Dealing with these tensor products is one of the main issues when the corresponding weights are not one-dimensional.We also recall the classification due to [FG] of the finitedimensional Nichols algebras B(V ) over D m , cf.Theorem 4.5.We fix a lighter notation to work with their Drinfeld doubles D(V, D m ) in the last section and characterize those which are spherical, see Theorem 4.8.Finally, we compute in Section 5 the weight decomposition of the simple D(V, D m )-modules for all V in the classification of [FG], see Theorem 5.2; we first consider the case when V is simple in §5.1, and then prove the recursive step in §5.2.
We work over an algebraically closed field k of characteristic zero.All vector spaces are considered over k and ⊗ = ⊗ k .For n ∈ N, we denote by Z n the ring of integers module n.We use the same letter to indicate an integer and its class in Z n .Through the work graded means Z-graded.Let N = ⊕ n∈Z N n be a graded vector space.For i ∈ Z, the shift N[i] of N is the same vector space N but with shifted grading, where N[i] n = N n−i as homogeneous component of degree n.
We work with Hopf algebras H over k.As usual, we denote the comultiplication by ∆, the antipode by S and the counit by ε.The comultiplication is written using Sweedler's sigma notation without the summation symbol, i.e. ∆(h) = h (1) ⊗h (2) for all h ∈ H. Analogously, for a left H-comodule (V, λ) we write λ(v) = v (−1) ⊗v (0) ∈ H⊗V for all v ∈ V to denote its coaction.We refer to [R2] for basic and well-known results in the theory.
Throughout these notes, we make use of the triangular decomposition associated with finite-dimensional graded algebras as introduced by Holmes and Nakano [HN].
Here we apply it to Drinfeld doubles of finite-dimensional Hopf algebras.A graded algebra A = n∈Z A n admits a triangular decomposition if there exist graded subalgebras A − , T and A + such that the multiplication m : A − ⊗T ⊗A + → A gives a linear isomorphism and (td1) In our situation, this coincides with [BT,Definition 3.1] as T is a split k-algebra because of our assumptions on k.We denote by A M the category of finite-dimensional left A-modules and by A G the category of the graded ones with morphisms preserving the grading.We write A C to refer to either one of these categories.We denote by Irr A C a complete set of non-isomorphic simple objects in A C.

The general framework
We outline in this section the general framework of our study.Eventually, we will be more explicit in the successive subsections according to our convenience.For more details we refer the reader to [AS], [BT] and [V].

Nichols algebras. A left Yetter
The finite-dimensional left Yetter-Drinfeld modules over H together with morphisms of left H-modules and left H-comodules form a braided rigid tensor category denoted by H H YD. The braiding is given by c Then, the tensor algebra T (V ) is a graded braided Hopf algebra in H H YD. Roughly speaking, it satisfies the axioms of a Hopf algebra but the structural maps are morphism in the category.For instance, its comultiplication is determined by setting ∆(v) = v⊗1 + 1⊗v for all v ∈ V .
The Nichols algebra ), where J (V ) is the largest Hopf ideal of T (V ) generated as an ideal by homogeneous elements of degree greater than or equal to 2. By definition, we have that B 0 (V ) = k and B 1 (V ) = V .
In case B(V ) is finite-dimensional, we denote by n top its maximum degree.It is well-known that λ V := B ntop (V ) is one-dimensional (in fact, B(V ) satisfies the Poincaré duality); in particular, it is a simple H-module and H-comodule.A fixed linear generator v top ∈ B ntop (V ) is usually called a volume element.We refer to [A] for more details on Nichols algebras.
3.1.2.The bosonization of B(V ) over H is a usual Hopf algebra whose underlying vector space is and it is endowed with a Hopf algebra structure which is a sort of semidirect product.It is generated by V and H as an algebra, whereas its multiplication and comultiplication are completely determined by for all v ∈ V and h ∈ H.In particular, H = 1#H is a Hopf subalgebra and B(V ) = B(V )#1 is a subalgebra which coincides with the subalgebra of left coinvariants associated with the projection and h ∈ H.Note that the adjoint action of H on V coincides with the action as Yetter-Drinfeld module.That is, for all h ∈ H and v ∈ V we have that .
By convention we write hf = h⊗f for all h ∈ H, f ∈ H * , cf. [M,Theorem 7.1.1].
The Drinfeld double D(H) is a quasitriangular Hopf algebra with R-matrix given by R = i f i ⊗h i ∈ D(H)⊗D(H), where {h i } ⊂ H and {f i } ⊂ H * are dual bases of H and H * , respectively.Also, it holds that R −1 = i S(f i )⊗h i .Thus, the category The functors F R and F R −1 are both fully faithful and are defined as follows: For M ∈ D(H) M, the object F R (M ) (resp.F R −1 (M )) coincides with M as D(H)module, whereas the D(H)-coaction is provided by the action of the R-matrix R (resp.R −1 ) as follows for all m ∈ M .Note that the braidings of V and F R • F (V ) = V coincide as linear maps.This implies that the Nichols algebras B(V ) and B(F R •F (V )) are isomorphic as braided Hopf algebras [T].In particular, they are isomorphic as algebras and coalgebras.In the sequel, we identify both Nichols algebras and consider B(V ) ∈ D(H) D(H) YD.Let V * be the D(H)-module dual to V .We define As above, the Nichols algebras B(V ) in ) are isomorphic as braided Hopf algebras.Besides, there is an isomorphism of D(H)module algebras where the algebra on the right-hand side is the Nichols algebra of F R (V * ) in  c).This follows from [AHS,Lemma 1.11], see also [V, (4.9)].
Actually, one may consider V * ∈ H * op H * op YD with action and coaction defined by The Nichols algebras B(V ) and B(V ) in D(H) D(H) YD play a central role in the rest of the paper.In case they are finite-dimensional, they are related by the fact that there is an isomorphism of Hopf algebras (B(V )#H) * op ≃ B(V )#H * op (adapt the proof of [PV1, Lemma 5]).
3.1.4.A generalized quantum group.From now on, we denote by D(V, H) the Drinfeld double of B(V )#H with B(V ) a finite-dimensional Nichols algebra in H H YD. We describe its Hopf algebra structure following [V,Lemma 4.3].
From the very definition of the Drinfeld double, it is possible to show that as vector spaces.Via this isomorphism, we may assume that D(V, H) is generated as an algebra by the elements of V , V , H and H * .Henceforth, the Hopf algebra structure of D(V, H) is completely determined by the following features: (a) The subalgebra generated by H and H * is a Hopf subalgebra isomorphic to D(H).
(b) The subalgebra generated by V (resp.V ) and D(H) is isomorphic to the bosonization In particular, both D ≤0 (V, H) and D ≥0 (V, H) are Hopf subalgebras. where It turns out that D(V, H) is a graded Hopf algebra with grading determined by deg V = −1, deg V = 1 and deg D(H) = 0.Moreover, there is an isomorphism of graded vector spaces induced by the multiplication: In conclusion, D(V, H) admits a triangular decomposition with T = D(H), A − = B(V ) and A + = B(V ).

Simple D(V, H)-modules.
In this subsection we explain how to compute the simple modules of D(V, H) by exploiting its triangular decomposition.For more details we refer to [BT,PV1,V].
We begin by constructing the proper standard modules which are the images of the composition of the functors where Inf is given by the canonical (graded) Hopf algebra epimorphism Notice that V acts by zero on H with the endofunctor given by taking the head, one has the functor whose image is the maximal semisimple quotient of M V H (N ).Note that both M V H and L V H commute with the shift-of-grading functors.The proper standard modules of simple objects in D(H) C will play a special role in the classification of those in D(V,H) C. For that reason, we introduce a particular terminology.We call weights the elements in C generated by a highest-weight is called a highestweight module.We define lowest-weight (modules) analogously using V instead of V .Given λ ∈ Irr D(H) C, we call Verma module 2 the proper standard module Thus, any highest-weight module is a quotient of a Verma module.
A classification of the simple modules over algebras with triangular decomposition is well-known, see for instance [HN, BT].In the case of D(V, H) this is given as follows.
Theorem 3.1. (a) We list some general remarks about simple modules that will be useful later.
. This is a direct consequence of the description above.
Remark 3.3.Let B and B be linear bases of V and V , respectively.As the elements of V and V act nilpotently, it holds that L V H (λ) = λ, with V and V acting trivially, if and only if Φ α,v acts by zero on λ for all v ∈ B and α ∈ B. These simple modules are called rigid [BT].
Remark 3.4.Let us fix non-zero homogeneous elements v top ∈ B ntop (V ) and α top ∈ B ntop (V ); note the maximum degree of these Nichols algebras is the same.By the triangular decomposition of D(V, H), there exists Θ ∈ D(H) such that In such case, these Verma modules are also projective, see [V,Corollary 5.12].

The character of the simple modules.
For any N ∈ D(H) M, the proper standard module M V H (N ) in the category D(V,H) M inherits the grading afforded by D(V, H).Explicitly, It follows from [GG,Proposition 3.5] that the head of H commutes with the grading-forgetful functors.In particular, for any λ ∈ Irr D(H) M we have a decomposition 2 We change slightly the notation of the Verma modules with respect to [V] to put the emphasis on V and H as this will be useful for our recursive argument for V decomposable.
3 This follows from the proof of [PV1,Corollary 15], as it holds for any Hopf algebra H.
On the other hand, since D(H) is concentrated in degree 0, we can consider each λ ∈ Irr D(H) M inside D(H) G as an object concentrated in degree 0. Thus, Assume now H is semisimple; hence D(H) also is.Then for each n, This gives us good information about the simple modules as it is a complete invariant, since Res L V H (λ) = λ ⊕ weights in degree <0.

The action on the Verma modules.
By the paragraphs above, one immediately realizes that to describe a simple module L V H (λ) one has to deal with the submodules of the Verma module M V H (λ). For explicit computations, it is convenient to keep in mind the following key facts.
(a) The action of B(V ) and is an isomorphism and a decomposition as D(H)-modules, respectively.

The simple D(V, H)-modules as socles of the Verma modules.
We introduce now the functor As in the case of the head, it follows from [GG,Proposition 3.5] that the socle of a standard module is a graded submodule even if N ∈ D(H) M, considered as a graded module concentrated in degree 0. Also, S V H commutes with the grading-forgetful functors and the shiftof-grading functors.
The socles of the Verma modules give us another classification of the simple modules over D(V, H).The next result for H being a group algebra is in [PV1].)⊗λ and this is simple and isomorphic to Inf This implies (a).For (b), we first consider the category D(V,H) M in which the number of nonisomorphic simple modules is # Irr D(H) M by Theorem 3.1.In (a) we have found the same number of non-isomorphic simple modules as tensoring by λ V gives a bijection on Irr D(H) M. Therefore any simple module in Let now S ∈ D(V,H) G be a simple object and F the grading-forgetful functor.Then F S ≃ S V H (λ) for a unique λ ∈ Irr D(H) M by the above paragraph and hence ) for some n ∈ Z; the first isomorphism is consequence of [GG,Theorem 4.1].This proves (b) for D(V,H) G and completes the proof.
Naturally, the socle of a Verma module is isomorphic to a simple highest-weight module.We can determine its highest-weight as follows.The next result is very useful to compute the simple modules.
Corollary 3.6.Let λ ∈ Irr D(H) C be a weight and Proof.As we mentioned, the socle of λ) j be another homogeneous highest-weight.The submodule generated by ν contains the socle.In particular, µ ⊂ D(V, H)ν.Also, by the triangular decomposition, D(V, H)ν = B(V )ν = ν + i>0 B i (V )ν.Then §3.2.2(b) implies that either j > k or j = k and ν = µ, and the corollary follows.

Example.
Exterior algebras are examples of Nichols algebras.They arise when the braiding of V is −f lip, that is c V,V (v⊗w) = −w⊗v for all v, w ∈ V .We explain here a general strategy that applies to exterior algebras of two-dimensional vector spaces.In §3.3.1, we consider any even dimensional vector space, as the Nichols algebras appearing in the context of the dihedral groups D m are all exterior algebras of vector spaces of even dimension.
Fix V ∈ H H YD a two-dimensional module with basis {v + , v − } and braiding −f lip.
Let λ be a weight with V ⊗λ semisimple and let µ be the highest-weight of minimum degree in M V H (λ), recall Corollary 3.6.We have the following three possibilities: is simple projective.This occurs when Θ acts non-trivially on λ, see Remark 3.4.One can compute Θ using v top and We depict this situation in Figure 2. To find µ, first one has to decompose V ⊗λ as a direct sum of weights and then determine which is annihilated by V .
is a rigid module.To find the rigid module, one can use Remark 3.3.For that, one should compute the four elements Φ ±,± associated with the elements v ± and α ± .

Tensoring by rigids.
We observe that any semisimple object in H (N t ).Proof.We have that V • N t = 0 by the grading assumption on N. Then there is an epimorphism p : Then p induces a projection p : L V H (N t ) −→ N/p(R) and hence N/p(R) is semisimple.Also, the homogeneous component of R of degree t is zero.Then p |Nt is injective and therefore p so is.This implies Thus we obtain the desired isomorphism.
We now prove that tensoring a simple module by a rigid module yields a semisimple module when H is semisimple.
Moreover, it is a direct sum of simple rigid modules if λ is also rigid.
Proof.We prove only the last isomorphism, for the others follow from the fact that D(H) M and D(V,H) M are braided categories.
We start by pointing out that, as H (ν) is a graded submodule by [GG,Proposition 3.5].Without loss of generality, we assume that λ and µ are concentrated in degree 0, since the functors involved commute with the shift of grading.
Then the homogeneous components are L V H (λ) n ⊗µ for all n ≤ 0; in particular, its homogeneous component of degree 0 is λ⊗µ.
We prove first that all highest-weights are in degree 0. Let L V H (ν) be a simple graded D(V, H)-submodule of L V H (λ)⊗µ and assume ν ⊂ L V H (λ) n ⊗µ.We claim that n = 0. Indeed, we pick k u k ⊗n k ∈ ν with {n k } k∈K linearly independent.For α ∈ V , we have where the first and the last equality hold because ν and µ are highest-weights, respectively.Hence there should be a highest-weight in degree < 0 which is not possible by the paragraph above.
In conclusion, the D(V, H)-submodule generated by λ⊗µ is semisimple and isomorphic to L V H (λ⊗µ).We prove next that λ⊗µ actually generates the whole module as we wanted.For that, we fix bases {m j } j∈J and {n k } k∈K of the weights λ and µ, respectively.Since µ is rigid, we have that x • n k = 0 for any homogeneous element x ∈ B(V ) of degree ≥ 1.Also, because B(V )#H is a graded coalgebra, for such an element x one may write its comultiplication by ∆(x) = x⊗1 + t y t ⊗z t , where the elements z t are homogeneous of degree ≥ 1 for all t.Hence, for all j ∈ J and k ∈ K we have that H (λ) is generated as a D(V, H)-module by the action of B(V ) on {m j } j∈J , our assertion follows.
Lastly, if λ is also rigid, then L V H (λ⊗µ) is concentrated in degree 0. This implies that L V H (ν) = ν for every weight ν of λ⊗µ and hence ν is rigid.
Remark 3.9.The hypothesis of L V H (µ) being rigid is necessary.Otherwise, the tensor product might neither be generated in degree zero nor all its highest-weights be in degree zero.See for instance [EGST,Theorem 4.1] or [PV2, Proposition 4.3].

A recursive strategy for V decomposable.
We assume here that V = W ⊕ U is decomposable as D(H)-module with W = 0 = U .This situation arises when H is the group algebra of the dihedral group D m .In particular, B(W ) and B(U ) are braided graded Hopf subalgebras of B(V ).Following [AA,§2.3],we set where ad c is the braided adjoint action of a Hopf algebra in D(H) M. Notice that W ⊆ Z.It holds that Z is a Yetter-Drinfeld module over B(U )#H via the adjoint action and the coaction (π as Hopf algebras, see loc.cit.or [HS,§8] for details and references.Naturally, we can apply the techniques described in the previous sections to the bosonization on the right hand side of (9), i.e.B(U )#H and Z playing the role of H and V , respectively.This gives us a new description of D(V, H) and its simple modules in terms of those over D(U, H), the Drinfeld double of B(U )#H.In this sense, we have Namely, one may consider another Z-grading on the Drinfeld double D(Z, U, H) given by − deg yields a new triangular decomposition on D(V, H).Hence the simple D(Z, U, H)modules can be constructed from the simple D(U, H)-modules as before.Of course, the latter can also be described by the same proceeding.Then, we have the functors For instance, the Verma module associated with the simple module where we consider Z acting by zero on L U H (λ). Observe that M Z B(U)#H (L U H (λ)) is not necessarily isomorphic to the Verma module M V H (λ) defined in (5).Nevertheless, we show that their heads are isomorphic.This allows us to construct the simple modules in a recursive way.
Lemma 3.10.Keeping the notation above, we have , which is the identity on N .By §3.2.2 (a) and ( 9), we have that , that is both objects have the same dimension.This implies that η N is in fact an isomorphism.Moreover, as η| N = id N , we see that Hence, η defines a natural isomorphism between both functors.
Since D(Z, U, H) ≃ D(V, H) as Hopf algebras, we may consider the Verma module with the unique grading satisfying that deg λ = 0 thanks to [GG].We prove next that there is a commutative diagram ≃ whose arrows are epimorphisms of D(V, H)-modules.
and the neous components of L V H (λ) and L U H (λ) satisfy as D(H)-modules Proof.By definition, we know that W ⊆ Z and U act by zero on λ = 1⊗λ inside , and hence also ).Then, by the characterization of the highest-weight modules, λ) by Remark 3.2.Finally, by looking at the associated gradation, we deduce the second assertion.
We have an analogous result for the socle.
Proof.Let λ Z = B nZ (Z) and λ U = B nU (U ) be the homogeneous components of maximum degree of B(Z) and B(U ), respectively.These are one-dimensional and simple as modules over D(U, H) and D(H), respectively, recall §3.1.1.In particular, Using the triangular decomposition (10), Theorem 3.5 says that for the first equality recall ( 7) and the second one follows from the first paragraph.
In conclusion, S Z B(U)#H (S U H (λ)) is a simple D(V, H)-module with lowest-weight λ Z ⊗λ U λ ≃ λ V λ.Again by Theorem 3.5 it should be isomorphic to S V H (λ) as desired.
We stress that Theorems 3.1 and 3.5, Corollary 3.6, Remarks 3.2, 3.3 and 3.4 and §3.2.2 also apply to , since one may take B(U )#H and Z to play the role of H and V , respectively.We will make use of these remarks under these generalized hypotheses when we consider Nichols algebras over the dihedral groups D m .In such a case we refer to them as the recursive version.
Remark 3.13.The coaction [AA,Remark 2.5].In this case, we have H (W ) is simple and rigid; e.g. when c U,W • c W,U = id W ⊗U by Remark 3.13.The Nichols algebras appearing in the present work satisfy this property.By applying Theorem 3.8, we have that W ⊗k is a direct sum of simples rigid modules and hence so is ) and its head are semisimple as D(U, H)-modules for any λ ∈ Irr D(H) M. The homogeneous components of its head satisfy for all n ≤ 0 that Finally, we point out that the Hopf subalgebra generated by W and H is D(W, H).

A recursive example.
Keep the notation and the assumptions of Remark 3.14.Assume further that W ∈ H H YD is a simple two-dimensional module with basis {w + , w − } and braiding −f lip.For instance, these hypotheses are satisfied by exterior algebras of vector spaces of even dimension; such is the case for H = kD m .Then we have that Let λ ∈ Irr D(H) M be such that W ⊗λ is semisimple.We explain below how to describe the socle of M W B(U)#H (L U H (λ)) using the recursive version of Corollary 3.6.See Figure 3.
First, we observe that W ⊗L U H (λ) ≃ L U H (W ⊗λ) by Theorem 3.8.Hence, as Note that it is enough to check for which homogeneous summand L U H (µ) it holds that W • µ = 0, since the Verma module is semisimple and the action of W is a morphism of D(U, H)-modules, by the recursive version of §3.2.2.A similar reasoning can be made using the recursive versions of Remarks 3.3 and 3.4.That is, it is enough to check that the elements Φ act trivially on λ (resp., Θ acts non trivially on λ) to conclude that they also act trivially (resp., non-trivially) on all L U H (λ). Thus, as in §3.2.4,we have three possibilities: To find µ, one has to decompose W ⊗λ as a direct sum of weights and then determine the one that is annihilated by W .In this case, one may deduce that W ⊗λ = µ ⊕ λ for some weight λ and hence as D(U, H)-modules.We leave the computation for the interested reader.
) and hence it is a simple D(V, H)-module over which W and W act trivially.

The dihedral groups framework
From now on, we fix a natural number m ≥ 12 divisible by 4 and an m-th primitive root of unity ω.We also set n = m 2 .The dihedral group of order 2m is presented by generators and relations by The algebra of functions k Dm is the dual Hopf algebra of kD m .We denote by {δ t } t∈Dm the dual basis of the basis of kD m given by the group-like elements, i.e. δ t (s) = δ t,s for all t, s ∈ D m .The comultiplication and the counit of these elements are ∆(δ t ) = s∈Dm δ s ⊗δ s −1 t and ε(δ t ) = δ t,e for all t ∈ D m , respectively.
We denote by DD m the Drinfeld double of kD m .Since k Dm is a commutative algebra, k Dm = k Dm op and consequently k Dm and kD m are Hopf subalgebras of DD m .Thus, the algebra structure of DD m is completely determined by the equality In this case, the R-matrix reads R = t∈Dm δ t ⊗t ∈ DD m ⊗DD m .
It is well-known that the simple modules over the Drinfeld double of a group algebra are classified by the conjugacy classes of the group and irreducible representations of their centralizers, c.f. [AG] and references therein.Namely, for g ∈ D m , write O g for its conjugacy class and C g for its centralizer in D m .Let (U, ̺) be an irreducible representation of C g .The kD m -module induced by (U, ̺), is a DD m -module with the k Dm -action defined by Then the set Λ consisting of the modules M (g, ̺)'s is a set of representative of simple DD m -modules up to isomorphism, that is It is worth noting that a k Dm -action on a vector space V is the same as a D mgrading.In this sense, a left DD m -module V (or equivalently a left Yetter-Drinfeld module over D m ) is a kD m -module with a D m -grading that is compatible with the conjugation in D m .In our example, we have that U is concentrated in degree g and the D m -degree of t ⊗ kCg u in M (g, ̺) is tgt −1 .The action of f ∈ k Dm is performed via the evaluation on the degree.We will denote by M [s] the homogeneous component of degree s ∈ D m of a DD m -module.Although this notation coincides with the shift of a grading, we believe that this would not confuse the reader since in the latter case s is an integer and here is an element of D m .
In the following, we recall the description of the simple DD m -modules according to the set of conjugacy classes.We present them by fixing a basis and by describing the action of x, y and the D m -grading.We use symbols like |w to denote elements of a particular basis for each simple module.For more details, see [FG].According to the amount of conjugacy classes of D m , these are 4 one-dimensional, say χ 1 , χ 2 , χ 3 , χ 4 , and n − 1 two-dimensional, which we denote by ρ ℓ , 1 ≤ ℓ < n.For 1 ≤ i ≤ 4 and 1 ≤ ℓ ≤ n − 1, the simple DD m -modules are: Note that M (e, χ 1 ) is given by the counit of DD m .To shorten notation, we write |± = |±, ℓ when the parameter ℓ is clear from the context.
4.1.2.The modules M (y n , ̺).Since m = 2n, the element y n is central in D m .Therefore the simple DD m -modules associated with y n are given by the simple representations of D m .As D m -modules they coincide with the ones given in §4.1.1,but these are concentrated in degree y n instead of e. Explicitly, for 1 ≤ i ≤ 4 and 1 ≤ ℓ ≤ n − 1 these are Again, we write |± = |±, n, ℓ when both the parameters ℓ and n are clear from the context.The latter are the modules M ℓ of [FG,§2A1].
The conjugacy class of y i is {y i , y −i } and its centralizer C y i is the subgroup y ≃ Z m whose simple representations are given by the characters Note that here the simple module is not concentrated in a single degree, in fact . These are the modules M i,k of [FG,§2A2].As before, we simply write |± = |±, i, k when the context allows us to simplify notation.Note that the simple modules M ℓ can be describe as M n,ℓ , where the elements are concentrated in degree 4.1.4.The modules M (x, ̺).The conjugacy class of x is {xy 2j | j ∈ Z n } and its centralizer is given by the subgroup x ⊕ y n ≃ Z 2 ⊕ Z 2 .The irreducible representations are given by the characters sgn s ⊗ sgn t , s, t ∈ Z 2 , where sgn(x) = sgn(y n ) = −1 are the corresponding sgn representation of the Z 2 summand.Hence, the simple DD m -modules are and |j, 0, s, t ∈ M (x, sgn s ⊗ sgn t )[xy 2j ], for all j ∈ Z n .
In particular, dim M 0,s,t = n and M 0,s,t = j∈Zn M 0,s,t [xy 2j ] as D m -graded module.We write |j = |j, 0, s, t when the notation is clear from the context.

Some tensor products of weights.
The category of DD m -modules is semisimple.As such, any tensor product of two weights can be written as a direct sum of weights.In order to perform our study on simple modules over doubles of bosonizations of Nichols algebras, which is carried out in §5, by dealing with Verma modules, we need to know the direct summands of the following products of simple DD m -modules.
In the following two lemmata, we decompose the tensor product of the simple modules M r,s,t with r, s, t ∈ Z 2 as in §4.1.4,§4.1.5with some other families of simple modules.
as DD m -modules; here we write r + i, s + 1 + δ i,n t and t + k for their classes in Z 2 .Moreover, the simple submodules inside the tensor product are given by Proof.We prove first the case i < n and r = 0, i.e.M 0,s,t is as in §4.1.4.Let us show that the subspaces N ± are simple DD m -modules.The elements n ± are eigenvectors of y n with eingenvalue sgn t+k (y n ) because and This implies that Moreover, the elements n ± are homogeneous of the same degree for Hence deg y a • n ± = y a xy i y −a = xy i−2a for all 0 ≤ a ≤ n − 1.This implies that in fact N ± are DD m -modules with dim N ± = n.Moreover, a direct check shows that they are isomorphic to the simple DD m -modules displayed in §4.1.4and §4.1.5,depending on the parity of i.One way to distinguish these modules is by looking at the eigenvalues of the action of y n and x on the homogeneous component of D m -degree x if i = 2z is even, or the action of y n and xy on the homogeneous component of D m -degree xy if i = 2z + 1 is odd.In the case of N ± , these homogeneous components are spanned by 13).Hence M 1,s,t+k ≃ N − and M 1,s+1,t+k ≃ N + .In both cases the submodules are simple and non-isomorphic.Therefore The strategy to prove the case i < n and r = 1 is similar.We still have that y n •n ± = sgn t+k (y n ) n ± and, instead of (13), we have that x•n ± = ∓ sgn s (xy) y i+1 • n ± .In this case, both n ± are homogeneous of degree xy i+1 .For i even, we have N − ≃ M 1,s,t+k and N + ≃ M 1,s+1,t+k , meanwhile for i odd, we have N − ≃ M 0,s,t+k and N + ≃ M 0,s+1,t+k .We leave the details for the reader.
The proof for i = n follows mutatis mutandis from the paragraphs above.
We end this subsection with the following lemma.
Lemma 4.3.Let M (e, χ 2 ) be a simple DD m -modules as in §4.1.1.Then as DD m -modules, where we write s + 1 for its class in Z 2 .
Proof.Straightforward.For instance, since M (e, χ 2 ) = M (e, χ 2 )[e] we have that Also, as the action on M (e, χ 2 ) is given by x • |u 2 = −|u 2 and y • |u 2 = |u 2 , the lemma follows easily by the definition of the action on the tensor product.
Here we recall the classification of finite-dimensional Nichols algebras in kDm kDm YD, or equivalently in DDm M. Roughly speaking, they are all given by exterior algebras of direct sums of some families of simple DD m -modules M i,k , recall Notation 4.1.
The classification in [FG, Theorem A] is given in terms of direct sums of three families of simple modules.To shorten notation, we present them below in just one family by changing slightly the description.Notation 4.4.Let I be the family of all finite multisets {(i 1 , k 1 ), ..., (i r , k r )} of pairs such that 1 ≤ i s ≤ n, 0 ≤ k s ≤ m − 1 and ω iskt = −1 for all 1 ≤ s, t ≤ r.For I ∈ I, we define Observe that the families I, L and K defined in [FG] fit in the description above.Indeed, if there is a pair (n, ℓ) in a sequence I ∈ I and (i, k) ∈ I, then ℓ and k must be odd because Theorem 4.5.[FG] Let B(M ) be a finite-dimensional Nichols algebra in DDm M. Then M ≃ M I for some I ∈ I and B(M ) ≃ M .Remark 4.6.We stress that if V = M I = W ⊕ U is decomposable, then it satisfies Remarks 3.13 and 3.14.In particular, B(V Here we present by generators and relations the Drinfeld double of the bosonization of a finite-dimensional Nichols algebra over D m by specifying the recipe given in §3.1.4.For that purpose, we need to set up some notation. Let V be a DD m -module with dim B(V ) < ∞.We write and V for the dual object of V as in §3.1.3.By Theorem 4.5, we can fix a decomposition V = (i,k)∈I M i,k and the orthogonal decomposition V = (i,k)∈I M i,k .Given a (two-dimensional) direct summand M i,k , we write v + and v − the elements of the basis {|+ , |− } given in §4.1.2or §4.1.3,as appropriate.So, we have Also, we denote by α + and α − the elements in That is, {α ± } and {v ± } are dual bases.Then the action of DD m on these elements is determined by Thus, M i,k ≃ M i,k as DD m -modules via the assignment α ± → v ∓ .The DD mcoactions defined by the functors F R and F R −1 on M i,k and M i,k , recall (1), are Proposition 4.7.As an algebra, D(V ) is generated by the elements of V , V , D m and k Dm subject to the relations (15)-( 21) below.
• For s, t ∈ D m and z ∈ V ∪ V , Proof.We briefly explain why these relations hold.Relation ( 15) is the commutation rule in DD m .The commutation rules ( 16) are given by the bosonizations B(V )#DD m and B(V )#DD m .The relations ( 17)-( 20) and ( 21) for generators in orthogonal direct summands follow from ( 1) and (3) by using (14).By (2), B(V ) is isomorphic as an algebra to a finite-dimensional Nichols algebra over D m .Then it as an exterior algebra like B(V ) and hence (21) holds.
Later on, in the upcoming section, we describe the simple D(V )-modules using the strategy developed in §3.2 and §3.3.Among all the relations above, we use only those involving v ± and α ± .Besides, the following elements of D(V ) are going to be useful: For a fixed a summand M i,k , we set Using ( 17)-( 20), a straightforward computation shows that these elements satisfy (6) or its recursive version, as appropriate.Explicitly, where Theorem 4.8.The Drinfeld double D(V ) is spherical if and only if V does not contain a direct summand isomorphic to M i,k with both i and k even.In such a case, we may choose ̟ = y n χ 3 as the involutive spherical element.
Proof.By [R1, Proposition 9], we know that the group of group-like elements of D(V ) equals D m × {χ 1 , χ 2 , χ 3 , χ 4 }.Since S 2 is the identity on DD m , a pivot element has to belong to the subset {y n } × {χ 1 , χ 2 , χ 3 , χ 4 }, which consist only of involutive elements.Then, in order to prove the statement, it is enough to analyse the existence of the pivot for V simple.Assume V = M i,k for some 1 ≤ i ≤ n and 0 ≤ k < m.Then for the generators v ± ∈ M i,k and α ± ∈ M i,k .Indeed, the formulas for the conjugation by χ 3 and y n follow from ( 16).The formulas for S 2 are deduced using ( 14) and the definition of the coaction in a bosonization.Similarly, one can see that χ 1 is central, χ 2 commutes with v ± , and α ± and χ 3 χ 2 = χ 4 .We deduce then that y n χ 3 is a pivot if i + k is odd and that there is no pivot when i and k are even.
The case i and k both odd cannot occur because by assumption ω ik = −1.
Remark 4.9.The quantum dimension of any simple module in D(V,Dm) M is zero, except for those simple modules that are rigid.

Characters of simple D(V )-modules
In this section we follow the strategy summarized in §3. the simple highest-weight module over D(M I ) associated with λ.The appearance of L I (λ) depends on certain subsets of Λ where the weight λ belongs.We present first these subsets and then state the results.First, for (i, k) ∈ I, we fix the partition Λ = Λ r i,k ∪ Λ p i,k ∪ Λ o given in Table 1.The subset Λ r i,k corresponds to the rigid simple modules when I = {(i, k)}, that is, those weights that satisfy L (i,k) (λ) = λ as D(M i,k )-modules, see Lemma 5.4.The subset Λ p i,k corresponds to the simple projective modules, that is, those that satisfy 1. Partition of the sets of weights with respect to M i,k The rigidity or projectivity of L I (λ) when |I| > 1 is determined by the subsets defined below.
Definition 5.1.For each λ ∈ Λ, we define In particular, we are under the hypothesis of §3.3, with U = M I r λ and W = M I p λ .Here is our main result which, in particular, gives the characters of the simple D(M I )-modules.To simplify the notation, we write the associated functors where Proof.With the aim of giving a clear exposition, we prove in detail the case in which I does not contain pairs (n, ℓ).In particular, ℓ J = 0 and ǫ J = 0.The other case follows mutatis mutandis.
We proceed by induction on the cardinal of I.The case |I| = 1 is considered in §5.1, see Lemmata 5.4 and 5.5.The inductive step is then proved in §5.2, see Lemma 5.8 for part (a) and Lemma 5.7 for part (b).
As a direct consequence, one gets the description of the rigid and simple projective modules over D(M I ).5.1.The singleton case.
We assume here that Analogously, through a sheer calculation one can show that Θλ = 0 for λ ∈ Λ p i,k and Θλ = 0 for λ ∈ Λ o .Thus, (b) follows from Remark 3.4 for V = M i,k .
For the remaining simple modules, we proceed as in §3.2.4.
Proof.Taking into account (see Figure 6 M (e, χ 2 ) Figure 6.The big dots represent the weights of B(M i,k ) and M (i,k) (λ).Their degrees are indicated on the right.Those in the shadow region form the socle S (i,k) (λ); the others the head L (i,k) (λ).
We deduce then that M I (M r,s,t ) has exactly two composition factors, each of them has to be the direct sum of two weights.One must be the socle S I (M r,s,t ) with Res S I (M r,s,t ) ≃ µ ⊕ (kv top ⊗M r,s,t ) as DD m -modules, where µ is the unique highest-weight in degree −1 and S I (M r,s,t ) ≃ L I (µ)[−1] by Corollary 3.6.The other composition factor is the head L I (M r,s,t ) with L I (M r,s,t ) ≃ M I (M r,s,t )/S I (M r,s,t ) as D(M i,k )-modules.Also, Res L I (M r,s,t ) ≃ M r,s,t ⊕ λ as DD m -modules, where λ is the complement of µ in degree −1.
Hence, we should determine which weight in degree −1 is annihilated by V = M ik .By Lemma 4.2, we know these weights are generated by n ± = ω rk v − ⊗|0 ± v + ⊗|i .Using ( 17)-( 18), we see that Example 5.6.In Figure 7 below, we depict the simple module L I (M 0,s,t ) over D (M i,k ) for i even.The nodes |j denote the basis elements of the DD m -direct summands M 0,s,t and M 0,s,t+k of Res L I (M 0,s,t ) .In each node, there should be two arrows going in and two arrows going out, but we only draw those corresponding to | i 2 in level −1 to make the diagram easy to read.Keep the notation as in the proof of Lemma 5.5, with n ± = v − ⊗|0 ± v + ⊗|i and |0 , |i ∈ M 0,s,t .Set n ± = v − |0 ±v + |i for the images of these elements in the quotient M I (M 0,s,t )/S I (M 0,s,t ).
Since n + = 0, we have that v − |0 = −v + |i as elements of degree −1.Looking at the DD m -action, one gets that both elements equal (a non-zero scalar multiple of) | i 2 in M 0,s,t+k inside Res L I (M 0,s,t ) .This is depicted by the two arrows arriving  Furthermore, the action of Θ on L J (λ) is non-trivial because, by the proof of Lemma 5.4 (b), the action of Θ on λ ⊂ L J (λ) is not so.Then, by Theorem 3.11 and the recursive version of Remark 3.4 with W = M i,k and U = M J , we have that where the last isomorphism follows by Lemma 3.10.
The following lemma gives the description of L I (λ) for λ ∈ Λ o .With it, we finish the proof of Theorem 5.2.Its proof also relies on the recursive argument in §3.3.More explicitly, this fits in the situation of Remark 3.14 and §3.3.1.where i J = (i,k)∈J i and k J = (i,k)∈J k, with i J = 0 = k J if J = ∅.E (L E (M r,s,t )).Their degrees are indicated on the right.Those in the shadow region form its socle and the others its head which is isomorphic to L (i,k) (M r,s,t ).D(M E )-submodule generated by the former weight, that is L E (M r+i,s+1,t+k ), and hence s = s.
Finally, the lemma follows by ( 27) and the inductive hypothesis.

Figure 1 .
Figure1.The Verma module M V H (λ). The (big) dots represent their (highest-)weights.The shadow regions indicate submodules generated by highest-weights.In particular, the region on the bottom is the socle S V H (λ) which is generated by the highest-weight of minimum degree.The white region on the top depicts its unique simple quotient L V H (λ).

3. 1
.3.The Drinfeld double of H is the Hopf algebra defined on the vector space D(H) := H⊗H * in such a way that H = H⊗1 and H * op = 1⊗H * op are Hopf subalgebras, and the elements h ∈ H and f ∈ H * obey the multiplication rule YD, or equivalently the Nichols algebra of V * in ( D(H) M,

Figure 2 .
Figure 2. The big dots represent the weights of B(V ) and M V H (λ). Their degrees are indicated on the right.Those in the shadow region form the socle S V H (λ) when deg µ = −1.

Figure 3 .
Figure 3.The dots represent the simple D(U, H)-summands of B(W ) and M W B(U)#H (L U H (λ)). Their degrees are indicated on the right.Those in the shadow region form its socle in the case that deg L U H (µ) = −1.
y m , xyxy .It has n + 3 conjugacy classes: O e = {e} with e the identity, O y n = {y n }, O x = {xy j : j even}, O xy = {xy j : j odd} and 4.1.1.The modules M (e, ̺).Let e be the identity element in D m .Since C e = D m , we use the simple representations of D m to describe the simple DD m -modules.

Figure 4 .
Figure 4.The simple module M 0,s,t associated with the conjugacy class of x.

Figure 5 .
Figure 5.The simple module M 1,s,t associated with the conjugacy class of xy.
,k and V = W ⊕ U , and D(U ) = DD m if V = W . 4.4.1.Spherical.We finish this section by characterizing those Drinfeld doubles D(V ) which are spherical Hopf algebras.This means by [BW, Definition 3.1] that D(V ) has a group-like element ̟ such that S 2 (h) = ̟h̟ −1 and tr N (ϑ̟) = tr N (ϑ̟ −1 )for all h ∈ D(V ), N ∈ D(V M and ϑ ∈ End D(V ) (N).A group-like element satisfying the first condition is called pivot and, if it fulfills both conditions, it is called spherical.An involutive pivot, i.e. ̟ 2 = 1, is clearly an spherical element.The pivot is unique up to multiplication by a central group-like element.
2 and §3.3 to describe the simple modules over D(M I ) for V = M I = ⊕ (i,k)∈I M i,k with I as in Notation 4.4.Recall the set of weight Λ in §4.1.For λ ∈ Λ, we set L I (λ) := L MI kDm (λ),
1 and ω ik = −1.We keep the notation of §4.4.In particular, V = M i,k , v ± and α ± are the generators of D(M i,k ) that belong to M i,k and its dual, respectively.The elements Φ •,• with •, • ∈ {+, −} and Θ defined in (17)-(20) and (22) are instrumental to determine which simple modules are rigid or projective.Lemma 5.4.Let λ ∈ Λ and L I (λ) be a simple D(M i,k )-module.Then (a) L I (λ) = λ if and only if

Proof.MFigure 8 .
Figure 8.The dots represent the simple D(M E )-summands of B(M i,k ) and M(i,k) As the braiding is invertible, one may consider also D(H) M as braided category with braiding c −1 .
kv top and kv top ≃ M (e, χ 2 ) as DD m -modules, we have that Res M I (M r,s,t ) is the direct sum of four weights.Indeed, Res M I (M r,s,t ) 0 = k⊗M r,s,t = M r,s,t , Res M I (M r,s,t ) −1 = M i,k ⊗M r,s,t ≃ M i+r,s+1,t+k ⊕ M i+r,s,t+k by Lemma 4.2, and Res M I (M r,s,t ) −2 = kv top ⊗M r,s,t ≃ M (e, χ 2 )⊗M r,s,t ≃ M r,s+1,t by Lemma 4.3.Note that all these weights are in Λ o .Then M I (M r,s,t ) is not simple and its composition factors are not concentrated in a single degree by Lemma 5.4, as they are not rigid and consist of more than a weight.