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

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: 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.