Quaternion algebra

over a field $F$

An associative algebra over a field $F$ that generalises the construction of the quaternions over the field of real numbers.

The quaternion algebra $(a,b{)}_F$ is the four-dimensional vector space over $F$with basis $\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k}$ and multiplication defined by $${\mathbf{i}}^2=a\mathbf{1},{\mathbf{j}}^2=b\mathbf{1},\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k}.$$ It follows that ${\mathbf{k}}^2=-ab\mathbf{1}$ and that any two of $\mathbf{i},\mathbf{j},\mathbf{k}$ anti-commute.

The construction is symmetric: $(a,b{)}_F=(b,a{)}_F$.

The algebra $(1,1{)}_F$ is isomorphic to the $2\times 2$ matrix ring $M_2(F)$. A quaternion algebra isomorphic to a matrix ring is termed split; otherwise the quaternion algebra is a division algebra.

The algebra so constructed is a central simple algebra over $F$. As elements of the Brauer group, it has order 2 or 1 (if split).

References[edit]

• Tsit-Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67 , American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929