“Split-quaternion”的意思、由来-开放百科全书

In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. (For example, {{nowrap|{{sfrac|1|2}}(1 + j)}} is an idempotent zero-divisor, and {{nowrap|i − j}} is nilpotent.) As a mathematical structure, they form an algebra over the real numbers, which is isomorphic to the algebra of 2 × 2 real matrices. For other names for split-quaternions see the Synonyms section below.

and hence ijk = 1. It follows from the defining relations that the set {1, i, j, k, −1, −i, −j, −k} is a group under coquaternion multiplication; it is isomorphic to the dihedral group D₄, the symmetry group of a square.

Due to the anti-commutative property of its basis vectors, the product of a coquaternion with its conjugate is given by an isotropic quadratic form:

Given two coquaterions p and q, one has {{nowrap|1=N(p q) = N(p) N(q)}}, showing that N is a quadratic form admitting composition. This algebra is a composition algebra and N is its norm. Any {{nowrap|q ≠ 0}} such that {{nowrap|1=N(q) = 0}} is a null vector, and its presence means that coquaternions form a "split composition algebra", and hence a coquaternion is also called a split quaternion.

When the norm is non-zero, then q has a multiplicative inverse, namely q^∗/N(q). The set

is the set of units. The set P of all coquaternions forms a ring {{nowrap|(P, +, •)}} with group of units {{nowrap|(U, •)}}. The coquaternions with {{nowrap|1=N(q) = 1}} form a non-compact topological group {{nowrap|SU(1, 1)}}, shown below to be isomorphic to {{nowrap|SL(2, R)}}.

Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra.

Matrix representations

and consider u = w + xi, and v = y + zi as ordinary complex numbers with complex conjugates denoted by u^∗ = w − xi, v^∗ = y − zi. Then the complex matrix

represents q in the ring of matrices, i.e. the multiplication of split-quaternions behaves the same way as the matrix multiplication. For example, the determinant of this matrix is

The appearance of the minus sign, where there is a plus in H, distinguishes coquaternions from quaternions. The use of the split-quaternions of norm one (qq* = 1) for hyperbolic motions of the Poincaré disk model of hyperbolic geometry is one of the great utilities of the algebra.

Besides the complex matrix representation, another linear representation associates coquaternions with 2 × 2 real matrices. This isomorphism can be made explicit as follows: Note first the product

and that the square of each factor on the left is the identity matrix, while the square of the right hand side is the negative of the identity matrix. Furthermore, note that these three matrices, together with the identity matrix, form a basis for M(2, R). One can make the matrix product above correspond to {{nowrap|1=jk = −i}} in the coquaternion ring. Then for an arbitrary matrix there is the bijection

which is in fact a ring isomorphism. Furthermore, computing squares of components and gathering terms shows that {{nowrap|1=qq^∗ = ad − bc}}, which is the determinant of the matrix. Consequently, there is a group isomorphism between the unit quasi-sphere of coquaternions and {{nowrap|1=SL(2, R) = {g ∈ M(2, R) : det g = 1},}} and hence also with {{nowrap|SU(1, 1)}}: the latter can be seen in the complex representation above.

For instance, see Karzel and Kist^[1] for the hyperbolic motion group representation with 2 × 2 real matrices.

In both of these linear representations the norm is given by the determinant function. Since the determinant is a multiplicative mapping, the norm of the product of two coquaternions is equal to the product of the two separate norms. Thus coquaternions form a composition algebra. As an algebra over the field of real numbers, it is one of only seven such algebras.

Generation from split-complex numbers

Kevin McCrimmon ^[2] has shown how all composition algebras can be constructed after the manner promulgated by L. E. Dickson and Adrian Albert for the division algebras C, H, and O. Indeed, he presents the multiplication rule

to be used when producing the doubled product in the real-split cases. As before, the doubled conjugate

so that

Profile

The subalgebras of P may be seen by first noting the nature of the subspace {{nowrap|{zi + xj + yk : x, y, z ∈ R}.}} Let

The parameters z and r(θ) are the basis of a cylindrical coordinate system in the subspace. Parameter θ denotes azimuth. Next let a denote any real number and consider the coquaternions

These are the equilateral-hyperboloidal coordinates described by Alexander Macfarlane and Carmody.^[3]

is a subring of P that is isomorphic to the plane of split-complex numbers just as when v is in I then

Note that for every {{Nowrap|r ∈ E}}, {{nowrap|1=(r + i)² = 0 = (r − i)²}} so that {{nowrap|r + i}} and {{nowrap|r − i}} are nilpotents. The plane {{nowrap|1=N = {x + y(r + i): x, y ∈ R}{{null}}}} is a subring of P that is isomorphic to the dual numbers. Since every coquaternion must lie in a D_p, a C_v, or an N plane, these planes profile P. For example, the unit quasi-sphere

consists of the "unit circles" in the constituent planes of P: In D_p it is a unit hyperbola, in N the "unit circle" is a pair of parallel lines, while in C_v it is indeed a circle (though it appears elliptical due to v-stretching).These ellipse/circles found in each C_v are like the illusion of the Rubin vase which "presents the viewer with a mental choice of two interpretations, each of which is valid".

Pan-orthogonality

When coquaternion {{nowrap|1=q = w + xi + yj + zk}}, then the scalar part of q is w.

Definition. For non-zero coquaternions q and t we write {{nowrap|q ⊥ t}} when the scalar part of the product qt^∗ is zero.

Counter-sphere geometry

The quadratic form qq^∗ is positive definite on the planes C_v and N. Consider the counter-sphere {q: qq^∗ = −1}.

Since points on the counter-sphere must line on the conjugate of the unit hyperbola in some plane {{nowrap|D_p ⊂ P}}, m can be written, for some {{nowrap|p ∈ J}}

Let φ be the angle between the hyperbolas from r to p and m. This angle can be viewed, in the plane tangent to the counter-sphere at r, by projection:

as in the expression of angle of parallelism in the hyperbolic plane H² . The parameter θ determining the meridian varies over the S¹. Thus the counter-sphere appears as the manifold S¹ × H².

Application to kinematics

The collection of these mappings bears some relation to the Lorentz group since it is also composed of ordinary and hyperbolic rotations. Among the peculiarities of this approach to relativistic kinematic is the anisotropic profile, say as compared to hyperbolic quaternions.

Reluctance to use coquaternions for kinematic models may stem from the {{nowrap|(2, 2)}} signature when spacetime is presumed to have signature {{nowrap|(1, 3)}} or {{nowrap|(3, 1)}}. Nevertheless, a transparently relativistic kinematics appears when a point of the counter-sphere is used to represent an inertial frame of reference. Indeed, if {{nowrap|1=tt^∗ = −1}}, then there is a {{nowrap|1=p = i sinh(a) + r cosh(a) ∈ J}} such that {{nowrap|t ∈ D_p}}, and a {{nowrap|b ∈ R}} such that {{nowrap|1=t = p exp(bp)}}. Then if {{nowrap|1=u = exp(bp)}}, {{nowrap|1=v = i cosh(a) + r sinh(a)}}, and {{nowrap|1=s = ir}}, the set {{nowrap|{t, u, v, s} }} is a pan-orthogonal basis stemming from t, and the orthogonalities persist through applications of the ordinary or hyperbolic rotations.

Historical notes

The coquaternions were initially introduced (under that name)^[4] in 1849 by James Cockle in the London–Edinburgh–Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 Bibliography^[5] of the Quaternion Society. Alexander Macfarlane called the structure of coquaternion vectors an exspherical system when he was speaking at the International Congress of Mathematicians in Paris in 1900.^[6]

The unit sphere was considered in 1910 by Hans Beck.^[7] For example, the dihedral group appears on page 419. The coquaternion structure has also been mentioned briefly in the Annals of Mathematics.^[8]^[9]

Synonyms