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Hamiltonian quaternions∗
mathcam†
2013-03-21 14:00:04
Definition of H
We define a unital associative algebra H over R, of dimension 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 H is of the form
a1 + bi + cj + dk,
a, b, c, d ∈ R
(sometimes denoted by ha, b, c, di or by a + hb, c, di) and the product of two
elements ha, b, c, di and hα, β, γ, δi (order matters) is hw, x, y, zi where
=
aα − bβ − cγ − dδ
x =
aβ + bα + cδ − dγ
y
=
aγ − bδ + cα + dβ
z
=
aδ + bγ − cβ + dα
w
The elements of H are known as Hamiltonian quaternions.
Clearly the subspaces of H generated by {1} and by {1, i} are subalgebras
isomorphic to R and C respectively. R is customarily identified with the corresponding subalgebra of H. (We shall see in a moment that there are other and
less obvious embeddings of C in H.) The real numbers commute with all the
elements of H, and we have
λ · ha, b, c, di = hλa, λb, λc, λdi
for λ ∈ R and ha, b, c, di ∈ H.
∗ hHamiltonianQuaternionsi
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Norm, conjugate, and inverse of a quaternion
Like the complex numbers (C), the quaternions have a natural involution
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 +
2
d )1. (See Euler four-square identity.) This product
√ is used to form a norm k · k
on the algebra (or the ring) H: We define kqk = s where qq = s1.
If v, w ∈ H and λ ∈ R, then
1. kvk ≥ 0 with equality only if v = h0, 0, 0, 0i = 0
2. kλvk = |λ|kvk
3. kv + wk ≤ kvk + kwk
4. kv · wk = kvk · kwk
which means that H qualifies as a normed algebra when we give it the norm
k · k.
Because the norm of any nonzero quaternion q is real and nonzero, we have
qq
qq
=
= h1, 0, 0, 0i
2
kqk
kqk2
which shows that any nonzero quaternion has an inverse:
q −1 =
q
.
kqk2
Other embeddings of C into H
One can use any non-zero q to define an embedding of C into H. If n(z) is
a natural embedding of z ∈ C into H, then the embedding:
z → qn(z)q −1
is also an embedding into H. Because H is an associative algebra, it is obvious
that:
(qn(a)q −1 )(qn(b)q −1 ) = q(n(a)n(b))q −1
and with the distributive laws, it is easy to check that
(qn(a)q −1 ) + (qn(b)q −1 ) = q(n(a) + n(b))q −1
Rotations in 3-space
Let us write
U = {q ∈ H : ||q|| = 1}
With multiplication, U is a group. Let us briefly sketch the relation between U
and the group SO(3) of rotations (about the origin) in 3-space.
An arbitrary element q of U can be expressed cos θ2 + sin θ2 (ai + bj + ck),
for some real numbers θ, a, b, c such that a2 + b2 + c2 = 1. The permutation
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v 7→ qv 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
v 7→ qrv corresponds to the composite rotation F ◦ G. 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
interpretation as well: two unit vectors in opposite directions determine the
same axis of rotation.
On the algebraic side, the quaternions provide an example of a division ring
that is not a field.
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