Hamiltonian quaternions

Definition of H

We define a unital associative algebra over , of dimensionPlanetmathPlanetmathPlanetmath 4, by the basis {𝟏,𝐢,𝐣,𝐤} and the multiplication table

1 i j k
i -1 k -j
j -k -1 i
k j -i -1

(where the element in row x and column y is xy, not yx). Thus an arbitrary element of is of the form


(sometimes denoted by a,b,c,d or by a+b,c,d) and the productPlanetmathPlanetmath of two elements a,b,c,d and α,β,γ,δ (order matters) is w,x,y,z where

w = aα-bβ-cγ-dδ
x = aβ+bα+cδ-dγ
y = aγ-bδ+cα+dβ
z = aδ+bγ-cβ+dα

The elements of are known as Hamiltonian quaternions.

Clearly the subspacesPlanetmathPlanetmath of generated by {𝟏} and by {𝟏,𝐢} are subalgebrasMathworldPlanetmathPlanetmathPlanetmath isomorphicPlanetmathPlanetmathPlanetmath to and respectively. is customarily identified with the corresponding subalgebra of . (We shall see in a moment that there are other and less obvious embeddingsPlanetmathPlanetmath of in .) The real numbers commute with all the elements of , and we have


for λ and a,b,c,d.

Norm, conjugatePlanetmathPlanetmathPlanetmath, and inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of a quaternion

Like the complex numbers (), the quaternions have a natural involutionPlanetmathPlanetmathPlanetmath called the quaternion conjugate. If q=a𝟏+b𝐢+c𝐣+d𝐤, then the quaternion conjugate of q, denoted q¯, is simply q¯=a𝟏-b𝐢-c𝐣-d𝐤.

One can readily verify that if q=a𝟏+b𝐢+c𝐣+d𝐤, then qq¯=(a2+b2+c2+d2)𝟏. (See Euler four-square identity.) This product is used to form a norm on the algebra (or the ring) : We define q=s where qq¯=s𝟏.

If v,w and λ, then

  1. 1.

    v0 with equality only if v=0,0,0,0=0

  2. 2.


  3. 3.


  4. 4.


which means that qualifies as a normed algebra when we give it the norm .

Because the norm of any nonzero quaternion q is real and nonzero, we have


which shows that any nonzero quaternion has an inverse:


Other embeddings of C into H

One can use any non-zero q to define an embedding of into . If 𝐧(z) is a natural embedding of z into , then the embedding:


is also an embedding into . Because is an associative algebra, it is obvious that:


and with the distributive laws, it is easy to check that


RotationsMathworldPlanetmath in 3-space

Let us write


With multiplication, U is a group. Let us briefly sketch the relationMathworldPlanetmathPlanetmath between U and the group SO(3) of rotations (about the origin) in 3-space.

An arbitrary element q of U can be expressed cosθ2+sinθ2(a𝐢+b𝐣+c𝐤), for some real numbers θ,a,b,c such that a2+b2+c2=1. The permutation vqv of U thus gives rise to a permutation of the real sphere. It turns out that that permutation is a rotation. Its axis is the line through (0,0,0) and (a,b,c), and the angle through which it rotates the sphere is θ. If rotations F and G correspond to quaternions q and r respectively, then clearly the permutation vqrv corresponds to the composite rotation FG. Thus this mapping of U onto SO(3) is a group homomorphism. Its kernel is the subset {1,-1} of U, and thus it comprises a double cover of SO(3). The kernel has a geometric interpretationMathworldPlanetmathPlanetmath as well: two unit vectorsMathworldPlanetmath in opposite directions determine the same axis of rotation.

On the algebraicMathworldPlanetmath side, the quaternions provide an example of a division ring that is not a field.

Title Hamiltonian quaternions
Canonical name HamiltonianQuaternions
Date of creation 2013-03-22 12:35:42
Last modified on 2013-03-22 12:35:42
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 10
Author mathcam (2727)
Entry type Definition
Classification msc 16W99
Synonym quaternion
Related topic EulerFourSquareIdentity
Related topic QuaternionGroup
Related topic HyperkahlerManifold
Related topic MathematicalBiology
Defines quaternion algebraPlanetmathPlanetmath