Verma and simple modules for quantum groups at non-abelian groups

Abstract

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 S3 attached to the 12-dimensional Fomin-Kirillov algebra, computing all the simple modules and calculating their dimensions.

29 pages, 4 figures v2: final version. Main changes: Theorem 5 is new and Sections 4.3, 4.4, 4.5 and 4.5 were improved

Cited by in corpus (6)

• Finite-dimensional Nichols algebras over dual Radford algebras
• On projective modules over finite quantum groups
• Simple Modules of the Quantum Double of the Nichols Algebra of Unidentified Diagonal Type $\mathfrak{ufo}(7)$
• Formal multiparameter quantum groups, deformations and specializations
• Simple modules of small quantum groups at dihedral groups
• Simple modules over finite quantum groups and their Drinfel'd doubles