The Brauer Group of the Dihedral Group

Abstract

Abstract. Let p m be a power of a prime number p, ‫ބ‬ p m be the dihedral group of order 2p m and k be a field where p is invertible and containing a primitive 2p m -th root of unity. The aim of this paper is computing the Brauer group BM(k, ‫ބ‬ p m , R z ) of the group Hopf algebra of ‫ބ‬ p m with respect to the quasi-triangular structure R z arising from the group Hopf algebra of the cyclic group ‫ޚ‬ p m of order p m , for z coprime with p.

When is a Cleft Extension H-Azumaya?

“…In this particular setting, an algebra in the category M H is an H op -comodule algebra A. In particular, if P is a H-comodule then End(P) with the usual composition of endomorphisms and with comodule structure given by [22] and for the remaining R-matrices in [6]; for the Hopf algebras of type H # and all R-matrices in [7], for the group algebra of the dihedral group in [8] and for the Hopf algebras of type E(n) and all triangular R-matrices in [9]. A key role in these computations was played by H-cleft extensions of the base field k.…”

“…Computations of BM(k, H, R) have been carried out only in a few cases, namely: for Sweedler's Hopf algebra H 4 with respect to the R-matrix R 0 in [22] and for the remaining R-matrices in [6]; for the Hopf algebras of type H ν and all R-matrices in [7], for the group algebra of the dihedral group in [8] and for the Hopf algebras of type E(n) and all triangular R-matrices in [9]. A key role in these computations was played by H-cleft extensions of the base field k.…”

The Brauer group of modified supergroup algebras

“…In [9] the computation of BM is generalized to the case in which G is the cyclic group g of order 2ν for ν an odd integer and V is the irreducible representation on which g again acts as −1. In [10] the first case of a non-abelian group was considered and in [11] the case in which G = Z 2 and V is given by n-copies of the non-trivial irreducible representation of G was treated, for all triangular structures. In all those cases important roles were played by the classical Brauer group, by the Brauer-Wall group, and by lazy cohomology as defined in [35] (with the name central cohomology) and studied in [3] and [16].…”