An exact sequence for the Brauer group of a finite quantum group.
The Brauer group \(BQ(k,H)\) of a Hopf algebra \(H\) over a field \(k\) with bijective antipode was introduced by \textit{S. Caenepeel, F. Van Oystaeyen} and \textit{Y.-H. Zhang} [K-theory 8, No. 3, 231-255 (1994; Zbl 0829.16032)]. It is the Brauer group of the braided monoidal category of the representations of the Drinfeld double of \(H\) (see \textit{F. Van Oystaeyen} and \textit{Y.-H. Zhang} [J. Algebra 202, No. 1, 96-128 (1998; Zbl 0909.18005)]). If \(H\) is coquasitriangular with coquasitriangular structure \(R\), the Brauer group \(BC(k,H,R)\) of the braided monoidal category of right \(H\)-comodules \(M^H_R\) is a subgroup of \(BQ(k,H)\). The computation of such groups can be nontrivial and a general method for determining them is investigated. Let \({\mathcal H}_R\) be the braided Hopf algebra in the category \(M^H_R\) constructed by \textit{S. Majid} [J. Pure Appl. Algebra 86, No. 2, 187-221 (1993; Zbl 0797.17004)] and let \({\mathcal H}_R^*\) be its dual. The author defines quantum commutative \({\mathcal H}_R^*\)-biGalois objects and shows that the set \(\text{Gal}({\mathcal H}_R)\) of isomorphism classes of such objects forms a group with respect to a generalization of the cotensor product. In analogy to the classical case, the author defines a group homomorphism from \(BC(k,H,R)\) to \(\text{Gal}({\mathcal H}_R)\) whose kernel is the classical Brauer group of the base field \(k\). This reduces the computation of \(BC(k,H,R)\) to the analysis of \(\text{Gal}({\mathcal H}_R)\). This result is illustrated with the explicit description of the group \(\text{Gal}({\mathcal H}_R)\) when \(H\) is Sweedler's Hopf algebra \(H_4\) and \(R\) is \(R_0\), its coquasitriangular structure which is trivial outside the group algebra of the grouplike elements. In this case \(\text{Gal}({\mathcal H}_R)\cong ((k,+)\times k^\cdot/k^{\cdot 2})\rtimes\mathbb{Z}_2\) and the above mentioned group homomorphism is surjective. This description is compared with the previous computation of \(BC(k,H_4,R_0)\) by \textit{F. Van Oystaeyen} and \textit{Y.-H. Zhang} [Proc. Am. Math. Soc. 129, No. 2, 371-380 (2001; Zbl 0964.16036)].
scientific article; zbMATH DE number 2057676
scientific article