The Brauer Group of a Braided Monoidal Category

Oystaeyen, Freddy Van; Zhang, Yinhuo

doi:10.1006/jabr.1997.7295

Cited by 68 publications

(73 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [44] Van Oystaeyen and Zhang constructed the Brauer group of a braided monoidal category extending a previous construction of Pareigis for symmetric monoidal categories [37]. This categorical construction collects practically all examples of Brauer groups ocurring in the literature.…”

Section: The Brauer Group Of a Braided Monoidal Categorymentioning

confidence: 99%

“…We recall from [44] the construction of the Brauer group of a braided monoidal category C. We give an alternative description of one of the functors associated to an Azumaya algebra (Proposition 3.4). When the braiding is H-linear the Brauer group BM(C; H) is defined as the Brauer group of the category H C of left H-modules.…”

Section: 9mentioning

confidence: 99%

“…We recall a characterization of Azumaya algebras which appears in [44 Analogously, the category of A-B-bimodules is also isomorphic to the category of right A ⊗ B-modules. An algebra A ∈ C is an A-A-bimodule through the multiplication and then it becomes a left A ⊗ A-bimodule and a right A ⊗ A-module.…”

Section: Azumaya Algebrasmentioning

confidence: 99%

See 2 more Smart Citations

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

Cuadra

Femić

2011

Appl Categor Struct

View full text Add to dashboard Cite

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.

show abstract

Section: The Brauer Group Of a Braided Monoidal Categorymentioning

confidence: 99%

Section: 9mentioning

confidence: 99%

See 1 more Smart Citation

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

Cuadra

Femić

2011

Appl Categor Struct

View full text Add to dashboard Cite

show abstract

“…The group obtained in this way is denoted by BC(k, H, r) and it is called the Brauer group of H with respect to the coquasitriangular structure r. For a quasi-triangular Hopf algebra (H, R), H * is a coquasitriangular Hopf algebra with coquasi-triangular structure r induced on H * by R. [19].…”

Section: Preliminariesmentioning

confidence: 99%

The Brauer Group of the Dihedral Group

Carnovale¹,

Cuadra

2004

Glasgow Math. J.

View full text Add to dashboard Cite

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.

show abstract

“…In particular, Caenepeel, Van Oystaeyen and Zhang defined in [1] the Brauer group BQ(k, H) of a Hopf algebra H with bijective antipode. This is a special case of Brauer group of a braided monoidal category (see [18]: the Brauer group of a symmetric monoidal category had been defined by B. Pareigis in [15]). Here the category is that of left modules of the Drinfel'd quantum double (see [5] and [11]) of a finitely generated projective Hopf algebra H over a commutative ring k. BQ(k, H) generalizes the Brauer-Long group of a commutative and cocommutative Hopf algebra over a commutative ring defined by Long in [10].…”

Section: Introductionmentioning

confidence: 99%

Some Isomorphisms for the Brauer Groups of a Hopf Algebra

Carnovale¹

2001

Communications in Algebra

View full text Add to dashboard Cite

Using equivalences of categories we provide isomorphisms between the Brauer groups of different Hopf algebras. As an example, we show that when k is a field of characteristic different from 2 the Brauer groups BC(k, H 4 , r t ) for every dual quasitriangular structure r t on Sweedler's Hopf algebra H 4 are all isomorphic to the direct sum of (k, +) and the Brauer-Wall group of k. We provide an isomorphism between the Brauer group of a Hopf algebra H and the Brauer group of the dual Hopf algebra H * generalizing a result of Tilborghs. Finally we relate the Brauer groups of H and of its opposite and co-opposite Hopf algebras.

show abstract

The Brauer Group of a Braided Monoidal Category

Cited by 68 publications

References 21 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

The Brauer Group of the Dihedral Group

Some Isomorphisms for the Brauer Groups of a Hopf Algebra

Contact Info

Product

Resources

About