Two Representations of Quaternions

Representations of the Quaternions

The various possible multiplication tables of octonions, twisted octonions, and sedenions depend directly on a few basic properties of multiplication of quaternions:

there are 4 basis elements:
- e₀ (equivalent to the unit element of the real numbers), and
- 3 other distinct "imaginary" elements e₁, e₂, e₃ such that e_i² = -1
  (historically these imaginary elements were first called i, j, k)

the basis elements multiply according to one of the two following "multiplication tables":

	e₀	e₁	e₂	e₃
e₀	e₀	e₁	e₂	e₃
e₁	e₁	-e₀	-e₃	e₂
e₂	e₂	e₃	-e₀	-e₁
e₃	e₃	-e₂	e₁	-e₀

	e₀	e₁	e₂	e₃
e₀	e₀	e₁	e₂	e₃
e₁	e₁	-e₀	e₃	-e₂
e₂	e₂	-e₃	-e₀	e₁
e₃	e₃	e₂	-e₁	-e₀

(the interrelationship between these two alternatives will be discussed below, and is important in generalizing to higher dimensions)

"Left-handed"

"Right-handed"

i²=j²=k²=kji=-1

i²=j²=k²=ijk=-1

an arbitrary quaternion is of the form r₀e₀ + r₁e₁ + r₂e₂ + r₃e₃, with r_i real numbers
the real coefficients r_i are commutative and associative with any multiplicative combination of the basis elements,
e.g. r_ie_i((r_je_j)(r_ke_k)) = r_ir_jr_k(e_i(e_je_k))
multiplication is distributive over addition

With these properties the multiplication of any 2 quaternions gives a unique quaternion as the product; the exact values depend on which of the two multiplication tables above ("Left-handed" vs. "Right-handed") is chosen.

Restricting attention to just the set {+e_i, -e_i}, 0 ≤ i ≤ 3, gives a finite group (in the case of the quaternions) or loop (nonassociative analog of a group) in higher dimensions; the essence of multiplication in the system being investigated is encapsulated in these finite loops, and most of the following presentation will focus on these finite cases. In particular, symmetry groups are simplified - in the quaternion case, actions of the permutation group on {1,2,3} and reflection of one or more axes will map a multiplication table above onto itself or the multiplication table of opposite "handedness".

The "-handed" labels are a reflection of terminology in 3D geometry and physics - a system of 3D axes is termed "Left-handed" if these axes, in ascending order of index (e.g. x,y,z or i,j,k or e₁, e₂, e₃) have the same orientation as the thumb, index finger, and middle finger of the left hand; similarly for "Right-handed", in mirror-image. Parity and chirality are concepts in physics intimately related to this notion of handedness. Note that a transformation of one table via axis inversion will give the other table. The same convention defining handedness (i.e. a triad of basis elements (e_i, e_j, e_k) i < j < k is "Right-handed" if e_ie_j = +e_k, and "Left-handed" if e_ie_j = -e_k) will be used to define a more complex notion of parity in higher-dimensions, where axis inversion is not equivalent to handedness reversal.

One further observation that will be helpful later - the indexes i, j, k in the above multiplication tables obey

e_ie_j = [+/-] e_i^j (modulo sign),

where '^' denotes bitwise exclusive OR (XOR).