On the Representation Theory of the Drinfeld Double of the Fomin-Kirillov Algebra FK3$\mathcal {FK}_{3}$

Abstract

Let D be the Drinfeld double of F K 3 #kS 3 . The simple D-modules were described in [24]. In the present work, we describe the indecomposable summands of the tensor products between them. We classify the extensions of the simple modules and show that D is of wild representation type. We also investigate the projective modules and their tensor products. 2000 Mathematics Subject Classification.16W30. C. V. was partially supported by CONICET, Secyt (UNC), FONCyT PICT 2016-3957, Programa de Cooperación MINCyT-FWO, MathAmSud project GR2HOPF and ESCALA Docente AUGM.The remainder simple modules generate a single block of the category of D-modules because they are composition factors of an indecomposable module, the Verma module of (σ, −) [24, Theorem 7]. In Section 3, we compute the extensions between these simple modules and show that D is of wild representation type. We draw the separated quiver of D in Figure 1.The major effort of our work is in describing the indecomposable summands of the tensor products of simple modules.Theorem 1.1. Let D be the Drinfeld double of FK 3 #kS 3 . Given λ, µ ∈ Λ, the indecomposable summands of the tensor product L(λ)⊗L(µ) are described in Propositions 4.1, 4.3, 4.7, 4.9, 4.10, 5.5 and 5.6.The outcome of the above is resumed in Table 2. We find out new indecomposable modules A, B and C which are not either simple or projective. We schematize them in Figures 2, 3 and 4, respectively. If one of the factors is projective, the tensor product is also projective and then we can use results from [26] in order to describe its direct summands. However, we do not have enough space in the table to write them except when both factors are projective. In this case, Ind(λ · µ) is the induced module D⊗ D(S3) (λ⊗µ) which is not necessary indecomposable. The cells under the diagonal are empty because D is quasitriangular and hence the tensor product is commutative. We do not include L(ε) in the table because it is the unit object.

On Projective Modules Over Finite Quantum Groups

“…Proof. (vi) follows from Lemma 2.6 (i) and (27). The remaining items are deduced from (i) and Lemma 2.6 (iii).…”

“…. Together with Barbara Pogorelsky, we study these projective modules more in detail [27]. We also discuss the tensor product between the simple and projective modules.…”

“…Moreover, we can infer the tensor product between simple and projective modules from the multiplication of their graded characters in the ring ZΛ[t, t −1 ]. We carry out this plan for B(V ) = F K 3 and G = S 3 jointly with Barbara Pogorelsky in [27].…”

Simple modules of small quantum groups at dihedral groups

“….1] or[22, Proposition 4.3].Simple modules of small quantum groups at dihedral groups 153.3. A recursive strategy for V decomposableWe assume here that V D W ˚U is decomposable as D.H /-module with W ¤ 0 ¤ U .…”