The Brauer group of Sweedler’s Hopf algebra 𝐻₄

Van Oystaeyen, Fred; Zhang, Yinhuo

doi:10.1090/S0002-9939-00-05628-8

The Brauer group of Sweedler’s Hopf algebra $H_4$
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by Fred Van Oystaeyen and Yinhuo Zhang PDF

Proc. Amer. Math. Soc. 129 (2001), 371-380 Request permission

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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