Cayley-Dickson form of a Quaternion - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:40:12Z http://mathoverflow.net/feeds/question/13998 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13998/cayley-dickson-form-of-a-quaternion Cayley-Dickson form of a Quaternion Marios 2010-02-03T16:47:36Z 2011-05-05T09:33:48Z <p>It is known that using the Cayley-Dickson construction a quaternion $q$ can be written in a symplectic form as $q=x+\mathbf{i}y$ with $x,y \in \mathbb{C}$.</p> <p>I read in a couple of references that $x$ is called the <i>simplex</i>-part whereas $y$ is called the <i>perplex</i>-part of the quaternion. Is this widely accepted and if not what is the proper onomatology?</p> <p>On the same topic is there a standard name for the two Quaternion parts comprising an Octonion?</p> http://mathoverflow.net/questions/13998/cayley-dickson-form-of-a-quaternion/62462#62462 Answer by Ted Shaneyfelt for Cayley-Dickson form of a Quaternion Ted Shaneyfelt 2011-04-20T21:34:50Z 2011-04-20T21:34:50Z <p>[1] T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier Transforms of Color Images,” IEEE Transactions on Image Processing, vol. 16, no. 1, pp. 22-35, Jan. 2007.</p> http://mathoverflow.net/questions/13998/cayley-dickson-form-of-a-quaternion/63981#63981 Answer by Tom De Medts for Cayley-Dickson form of a Quaternion Tom De Medts 2011-05-05T09:33:48Z 2011-05-05T09:33:48Z <p>I believe there is a good reason why mathematicians don't use the terminology "simplex-part" and "perplex-part": they are not canonical! Indeed, algebraically there is no way to distinguish the elements $i$, $j$ and $k$ in the quaternion algebra $\mathbb{H}$ (and there are in fact many more elements playing the same rôle).</p> <p>On the other hand, there is a canonical <em>standard involution</em> on $\mathbb{H}$, namely $$\sigma \colon x = a + bi + cj + dk \mapsto \overline{x} := a - bi - cj - dk,$$ and therefore the decomposition of $a + bi + cj + dk$ into the two parts $a$ and $bi + cj + dk$ <em>is</em> canonical. The part $bi + cj + dk$ is often called the <em>pure part</em> of the element $x$.</p> <p>(This terminology is also used for octonions, and also for (generalized) quaternion and octonion algebras over arbitrary fields of characteristic different from $2$.)</p>