A Math Treat: D.E. Littlewood's Elegant Introduction to Non-Commutative Algebras (Pt. 1)

  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