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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 created: h2013-03-21i by: hmathcami version: h32846i Privacy setting: h1i hDefinitioni h16W99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 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 2 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. 3