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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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.


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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: http://dx.doi.org/10.1090/S0002-9939-00-05628-8
PII: S 0002-9939(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