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

POGORELSKY, BARBARA; VAY, CRISTIAN

ADVANCES IN MATHEMATICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2016 vol. 301 p. 423 - 457

0001-8708

The Drinfeld double $D$ of the bosonization of a finite-dimensional Nichols algebra $BV(V)$ over a finite non-abelian group $G$ is called a {it 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 $Sn_3$ attached to the 12-dimensional Fomin-Kirillov algebra, computing all the simple modules and calculating their dimensions.