Cocycle Twisting of E(n)-Module Algebras and Applications to the Brauer Group

Abstract

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group BQ(k, E(n)) of E(n) that are isomorphic. For a triangular structure R on E(n) we prove that the subgroup BM (k, E(n), R) of BQ(k, E(n)) arising from R is isomorphic to a direct product of BW (k), the BrauerWall group of the ground field k, and Sym n (k), the group of n × n symmetric matrices under addition. For a general quasi-triangular structure R ′ on E(n) we construct a split short exact sequence having BM (k, E(n), R ′ ) as a middle term and as kernel a central extension of the group of symmetric matrices of order r < n (r depending on R ′ ). We finally describe how the image of the Hopf automorphism group inside BQ(k, E(n)) acts on Sym n (k).

A Sequence to Compute the Brauer Group of Certain Quasi-Triangular Hopf Algebras

“…where BW(K) denotes the Brauer-Wall group of K and (K, +) the additive group of K. Other computations generalizing this one were done later: for Radford Hopf algebra [12], Nichols Hopf algebra [13], and a modified supergroup algebra [11]. In all these computations a direct product decomposition for the Brauer group holds and there was no interpretation for one of the factors ((K, +) in the above case) appearing in it.…”

“…After the group of lazy 2-cocycles was studied in [7], and knowing the computations of Brauer groups done in [45], [12], [13] and [11], it was suspected that the group of lazy 2-cocycles would embed in the Brauer group, as mentioned in the introduction of [7]. In view of the following result, there is an embeding of the second braided cohomology group of the braided Hopf algebra into the Brauer group of the corresponding Radford biproduct.…”

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.…”

“…BMðk; H 4 ; R 0 Þ in [22] and the construction of the map Sym n ðkÞ ! BMðk; EðnÞ; R 0 Þ in [9] for n = 1.…”

Structure theorems of E(n)-Azumaya algebras

“…Its quasitriangular structures are parameterized by matrices in M n (k) (see [14]). Carnovale and Cuadra [6] where Sym n (k) is the additive group of symmetric n × n matrices over k. Armour, Chen and Zhang [1] gave the reverse exact sequence for M = 0,…”