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

• 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