The Drinfeld double D of the bosonization of a finite-dimensional Nichols algebra B(V) over a finite non-abelian group G is called a quantum group at a non-abelian group. We introduce Verma modules over such a quantum group D and prove that a Verma module has simple head and simple socle. This provides two bijective correspondences between the set of simple modules over D and the set of simple modules over the Drinfeld double D(G). As an example, we describe the lattice of submodules of the Verma modules over the quantum group at the symmetric group S 3 attached to the 12-dimensional Fomin-Kirillov algebra, computing all the simple modules and calculating their dimensions.
Abstract. We study the copointed Hopf algebras attached to the Nichols algebra of the affine rack Aff(F4, ω), also known as tetrahedron rack, and the 2-cocycle −1. We investigate the so-called Verma modules and classify all the simple modules. We conclude that these algebras are of wild representation type and not quasitriangular, also we analyze when these are spherical.
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.
In this paper we describe the right coideal subalgebras containing all group-like elements of the multiparameter quantum group U + q (g), where g is a simple Lie algebra of type G 2 , while the main parameter of quantization q is not a root of 1. If the multiplicative order t of q is finite, t > 4, t = 6, then the same classification remains valid for homogeneous right coideal subalgebras of the positive part u + q (g) of the multiparameter version of the small Lusztig quantum group.
In this paper we describe the right coideal subalgebras containing all group-like elements of the two-parameter quantum group Uq(g), where g is a simple Lie algebra of type G2, while the main parameter of quantization q is not a root of 1. As a consequence, we determine that there are precisely 60 different right coideal subalgebras containing all group-like elements. If the multiplicative order t of q is finite, t > 4, t = 6, then the same classification remains valid for homogeneous right coideal subalgebras of the two-parameter version of the small Lusztig quantum group uq(g).
In this paper, using a much simpler method than the previous existing ones, we explicitly describe the PBW-generators of the multiparameter quantum groups U + q (g), where g is a simple Lie algebra of small dimension, while the main parameter of quantization q is not a root of the unity.
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