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The Brauer group of Sweedler's Hopf algebra $H_4$


Authors: Fred Van Oystaeyen and Yinhuo Zhang
Journal: Proc. Amer. Math. Soc. 129 (2001), 371-380
MSC (1991): Primary 16W30, 16H05, 16K50
DOI: https://doi.org/10.1090/S0002-9939-00-05628-8
Published electronically: September 19, 2000
MathSciNet review: 1706961
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Abstract:

We calculate the Brauer group of the four dimensional Hopf algebra $H_4$ introduced by M. E. Sweedler. This Brauer group ${\mathrm{BM}}(k,H_4,R_0)$ is defined with respect to a (quasi-) triangular structure on $H_4$, given by an element $R_0\in H_4\otimes H_4$. In this paper $k$ is a field . The additive group $(k,+)$ of $k$ is embedded in the Brauer group and it fits in the exact and split sequence of groups: \begin{equation*}1\longrightarrow (k,+)\longrightarrow {\mathrm{BM}}(k,H_4,R_0)\longrightarrow {\mathrm{BW}}(k)\longrightarrow 1 \end{equation*}where ${\mathrm{BW}(k)}$ is the well-known Brauer-Wall group of $k$. The techniques involved are close to the Clifford algebra theory for quaternion or generalized quaternion algebras.


References [Enhancements On Off] (What's this?)

  • 1. R.J. Blattner, M. Cohen, and S. Montgomery, Crossed Products and Inner Actions of Hopf Algebras, Trans. AMS 298(1986), 672-711. MR 87k:16012
  • 2. S. Caenepeel, F. Van Oystaeyen and Y.H. Zhang, Quantum Yang-Baxter Module Algebras, K-Theory 8(1994), 231-255. MR 95e:16031
  • 3. S. Caenepeel, F. Van Oystaeyen and Y.H. Zhang, The Brauer Group of Yetter-Drinfel'd Module Algebras, Trans. Amer. Math. Soc. 349(1997), 3737-3771. MR 98c:16047
  • 4. Y. Doi and M. Takeuchi, Quaternion Algebras and Hopf Crossed Products, Comm. Alg. 23(9)(1995), 3291-3325. MR 96d:16049
  • 5. M. Koppinen, A Skolem-Noether Theorem for Hopf Algebra Measurings, Arch. Math. 13(1981), 353-361.
  • 6. L.A. Lambe, D.E. Radford, Algebraic Aspects of the Quantum Yang-Baxter Equation, J. Alg. 154(1992), 228-288. MR 94b:17026
  • 7. S. Majid, Algebras and Hopf Algebras in Braided Categories, Advances in Hopf Algebras, Lecture Notes Pure and Applied Math. 158(1994), 55-105. MR 95d:18004
  • 8. A. Masuoka, Coalgebra Actions on Azumaya algebras, Tsukuba J. Math. 14(1990), 107-112. MR 91d:16026
  • 9. F. Van Oystaeyen and Y.H. Zhang, The Embedding of Automorphism Group into The Brauer Group, Canadian Math. Bull. 41(1998), 359-367. MR 99h:16066
  • 10. F. Van Oystaeyen and Y.H. Zhang, The Brauer group of a Braided Monoidal Category, J. Alg. 202(1998), 96-128. MR 99c:18006
  • 11. M.E. Sweedler, Hopf Algebras, Benjamin, 1969. MR 40:5705
  • 12. C.T.C. Wall, Graded Brauer Groups, J. Reine Angew. Math. 213(1964), 187-199. MR 29:4771
  • 13. D.N. Yetter, Quantum Groups and Representations of Monoidal Categories, Math. Proc. Cambridge Philos. Soc. 108(1990), 261-290. MR 91k:16028

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Additional Information

Fred Van Oystaeyen
Affiliation: Department of Mathematics, University of Antwerp (UIA), B-2610 Wilryck, Belgium

Yinhuo Zhang
Affiliation: Department of Mathematics, University of Antwerp (UIA), B-2610 Wilryck, Belgium
Email: zhang@uia.ua.ac.be

DOI: https://doi.org/10.1090/S0002-9939-00-05628-8
Received by editor(s): February 22, 1999
Received by editor(s) in revised form: May 4, 1999
Published electronically: September 19, 2000
Communicated by: Ken Goodearl
Article copyright: © Copyright 2000 American Mathematical Society

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