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The Brauer group of Sweedler's Hopf algebra

Authors:	Fred Van Oystaeyen Yinhuo Zhang

Institution:	Department of Mathematics, University of Antwerp (UIA), B-2610 Wilryck, Belgium Yinhuo Zhang ; Department of Mathematics, University of Antwerp (UIA), B-2610 Wilryck, Belgium

Abstract:	We calculate the Brauer group of the four dimensional Hopf algebra 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 , given by an element $R_0\in H_4\otimes H_4$ . In this paper is a field . The additive group of 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 . The techniques involved are close to the Clifford algebra theory for quaternion or generalized quaternion algebras.

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