The Brauer group of Sweedler’s Hopf algebra 𝐻₄

Abstract

We calculate the Brauer group of the four dimensional Hopf algebra H 4 H_4 introduced by M. E. Sweedler. This Brauer group B M ( k , H 4 , R 0 ) {\mathrm {BM}}(k,H_4,R_0) is defined with respect to a (quasi-) triangular structure on H 4 H_4 , given by an element R 0 ∈ H 4 ⊗ H 4 R_0\in H_4\otimes H_4 . In this paper k k is a field . The additive group ( k , + ) (k,+) of k k is embedded in the Brauer group and it fits in the exact and split sequence of groups: 1 ⟶ ( k , + ) ⟶ B M ( k , H 4 , R 0 ) ⟶ B W ( k ) ⟶ 1 \begin{equation*} 1\longrightarrow (k,+)\longrightarrow {\mathrm {BM}}(k,H_4,R_0)\longrightarrow {\mathrm {BW}}(k)\longrightarrow 1 \end{equation*} where B W ( k ) {\mathrm {BW}(k)} is the well-known Brauer-Wall group of k k . The techniques involved are close to the Clifford algebra theory for quaternion or generalized quaternion algebras.