The Brauer group of some quasitriangular Hopf algebras

Carnovale, Giovanna; Cuadra, Juan

doi:10.1016/s0021-8693(02)00563-x

Cited by 8 publications

(19 citation statements)

References 26 publications

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“…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.…”

Section: 3mentioning

confidence: 99%

“…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.…”

Section: Beattie's Sequence As the Root Of The Known Computationsmentioning

confidence: 99%

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

Cuadra

Femić

2011

Appl Categor Struct

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A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra B in a braided monoidal category C, and under certain assumptions on the braiding (fulfilled if C is symmetric), we construct a sequence for the Brauer group BM(C; B) of B-module algebras, generalizing Beattie's one. It allows one to prove that BM(C; B) ∼ = Br(C) × Gal(C; B), where Br(C) is the Brauer group of C and Gal(C; B) the group of B-Galois objects. We also show that BM(C; B) contains a subgroup isomorphic to Br(C) × H 2 (C; B, I), where H 2 (C; B, I) is the second Sweedler cohomology group of B with values in the unit object I of C. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct B × H, where H is a usual Hopf algebra over a field K, the Hopf subalgebra generated by the quasi-triangular structure R is contained in H and B is a Hopf algebra in the category H M of left H-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that BM(K, H, R)×H 2 ( H M; B, K) is a subgroup of the Brauer group BM(K, B ×H, R), confirming the suspicion that a certain cohomology group of B × H (second lazy cohomology group was conjectured) embeds into BM(K, B × H, R). New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.

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Section: 3mentioning

confidence: 99%

Section: Beattie's Sequence As the Root Of The Known Computationsmentioning

confidence: 99%

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

Cuadra

Femić

2011

Appl Categor Struct

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“…Theorem 5.2 shows that one should understand BM (k, E(2), R N ) in order to compute BQ(k, H 4 ). In view of Proposition 5.3, BM (k, E(2), R N ) seems to be much more complex than the groups of type BM treated in [10,11,20].…”

Section: The Subgroup Bq Grad (K H 4 )mentioning

confidence: 99%

“…We conclude by showing that, contrarily to the cases treated in the literature ( [10,11,20]), a Skolem-Noether-like approach is probably not appropriate for the computation of BM (k, E(2), R N ) because the set of classes admitting a representative with inner action is not a subgroup. Proof.…”

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confidence: 96%

On the subgroup structure of the full Brauer group of Sweedler Hopf algebra

Carnovale

Cuadra

2011

Isr. J. Math.

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We introduce a three-parameter family of two-dimensional algebras representing elements in the Brauer group BQ(k,H _4) of Sweedler Hopf algebra H_4 over a field k. They allow us to describe the mutual intersection of the subgroups arising from a quasitriangular or coquasitriangular structure. We also define a new subgroup of BQ(k,H _4) and construct an exact sequence relating it to the Brauer group of Nichols 8-dimensional Hopf algebra with respect to the quasitriangular structure attached to the 2 x 2-matrix with 1 in the (1, 2)-entry and zero elsewhere

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“…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.…”

Section: Cleft Extensions and H-azumaya Algebrasmentioning

confidence: 99%

When is a Cleft Extension H-Azumaya?

Carnovale¹

2006

Algebr Represent Theor

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We show which H op -cleft extensions k# ' H op of k for a dual quasi-triangular Hopf algebra (H, r) are H-Azumaya. The result is given in terms of bijectivity of a map ' : H ! H* defined in terms of the universal r-form r and the 2-cocycle ', generalizing a well-known result for the commutative and co-commutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n). (2000): 16W30, 16H05, 16K50. Mathematics Subject Classifications

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The Brauer group of some quasitriangular Hopf algebras

Cited by 8 publications

References 26 publications

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

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

On the subgroup structure of the full Brauer group of Sweedler Hopf algebra

When is a Cleft Extension H-Azumaya?

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