In his timeless monograph 'The Skeleton Key Of Mathematics' (CHapter XIII, p. 101, Algebras) D.E. Littlewood gives us one of the best introductions to that domain of mathematics. He starts in a logical place, Hamiton's 19th century development of the quaternions. He begins by looking at Hamilton's approach to vector and scalar products, i.e. in 3 dimensions:
Then noting(p. 102) :
"A vector (e.g. R) with components x, y,z would be denoted by:
xi + yj + zk"
Adding:
"If for the law of multiplication it is assumed (for the unit vectors):
i 2 = - I, j 2 = - I, k 2 = - I,
i j =k, jk = i, ki = j
ji = -k, kj = -i, ik = - j
Then the product of two vectors (xi, yj, zk), (x' i, y'j, z' k) will have a scalar part which is equal to the scalar part with the sign reversed, and a vector part equal to the vector product. Thus:
(xi, yj, zk) (x' i, y'j, z' k) = - (xx' + yy' + zz') + i(y z' - y'z)
+ j (zx' - z' x) + k( xy' - x' y)
The quaternions so formed will obey all the laws of numbers except the commutative law for multiplication. Most importantly, they are associative, i.e. three quaternions A, B, C satisfy:
AB (C) = A (BC)
"If then:
Q = x 0 + ix 1 + jx 2 + kx 3
is any quaternion, the quaternion obtained by changing the sign of the vector part ( Q = x 0 - ix 1 - jx 2 - kx 3 ) is called the conjugate quaternion. The product of a quaternion with its conjugate is a positive non-zero scalar, i.e.
( x 0 + ix 1 + jx 2 + kx 3 ) (x 0 - ix 1 - jx 2 - kx 3 ) =
(x 0 2+ x 1 2 + x 2 2 + x 32)
which is called the norm of the quaternion. It is thus always possible to divide by a quaternion, except of course by zero, e.g.
I/ (x 1 + ix 1 + jx 2 + kx 3 ) =
(x 0 - ix 1 - jx 2 - kx 3 / (x 0 2+ x 1 2 + x 2 2 + x 32)
Quaternions with real coefficients, or real quaternions, are thus said to form a division algebra.
If complex coefficients are allowed, however, division may fail. This is given that for complex numbers it is possible that:
x 0 2+ x 1 2 + x 2 2 + x 32 = 0
without all the quantities (x 0 , x 1 , x 2 , x 3 ) being zero.
Therefore complex quaternions do not forma a division algebra. It can be shown, however, they are equivalent to the algebra of two rowed matrices, if we set:
Thence we have here one of the most elegant - and simplest- presentations of non-commutative algebras. As well as the basis for what we call 'algebras' in general.
Next: Higher dimensional algebras
