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

• A quantum group at a non-abelian group is the Drinfeld double of the bosonization of a finite-dimensional Nichols algebra over a finite non-abelian group.
• The head of a Verma module of a quantum group at a non-abelian group is simple.
• The socle of a Verma module of a quantum group at a non-abelian group is simple.
• A quantum group at a symmetric group attached to a Fomin–Kirillov algebra is analyzed.

The Drinfeld double DD of the bosonization of a finite-dimensional Nichols algebra B(V)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 DD and prove that a Verma module has simple head and simple socle. This provides two bijective correspondences between the set of simple modules over DD and the set of simple modules over the Drinfeld double D(G)D(G). As an example, we describe the lattice of submodules of the Verma modules over the quantum group at the symmetric group S3S3 attached to the 12-dimensional Fomin–Kirillov algebra, computing all the simple modules and calculating their dimensions.