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Quaternions in Mathematics and Physics.

3 first steps in the Cayley-Dickson construction : from reals to octonions.

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	e	i
e	e	i
i	i	-e

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	e	i	j	k
e	e	i	j	k
i	i	-e	k	-j
j	j	-k	-e	i
k	k	j	-i	-e

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	e	i	j	k	l	m	n	p
e	e	i	j	k	l	m	n	p
i	i	-e	k	-j	m	-l	-p	n
j	j	-k	-e	i	n	p	-l	-m
k	k	j	-i	-e	p	-n	m	-l
l	l	-m	-n	-p	-e	i	j	k
m	m	l	-p	n	-i	-e	-k	j
n	n	p	l	-m	-j	k	-e	-i
p	p	-n	m	l	-k	-j	i	-e

Quaternions are usually written as hypercomplex numbers of the form, q = e a + i b + j c + k d, with {a,b,c,d} ∈ Reals. The keys, e=1, i, j, k, obey the non commutative multiplication table above. Applications in physics eventually need the extension {a,b,c,d} ∈ Complex (Biquaternions). In this case, because of a possible confusion in the use of the symbol i, imaginary or quaternionic key, it should eventually be convenient to rewrite the quaternionic keys as, ke, ki, kj and kk). Even more generally {a,b,c,d} ≡ (differential) operators should be allowed. Mathematica is not at ease with quaternions. A specific package exists only capable of handling with real numerical quaternions but difficulties arise when the coefficients become complex, litteral or differential (operators).

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A more efficient (and secure) approach utilizes a matrix representation of the quaternion algebra. Here are two of them (i = ) :

1. 2x2 matrix representation of the four quaternionic keys (k2e, k2i, k2j and k2k). Each quaternion is written as : q = a k2e + b k2i + c k2j + d k2k.

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