Split-quaternion - Citizendia

In abstract algebra, the split-quaternions or coquaternions are elements of an associative algebra introduced by James Cockle in 1849 under the latter name. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive Sir James Cockle FRS FRAS FCPS FMS ( 14 January 1819 &ndash 27 January 1895) was an English They are also known as para-quaternions (particularly in recent literature on para-quaternionic geometry) or hyperbolic quaternions, although historically the latter term has a different meaning. In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Unlike the quaternion algebra, the split-quaternions contain zero divisors, nilpotent elements, and nontrivial idempotents. In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation

The set ${1, i, j, k}$ forms a basis. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. The products of these elements are

$i j = k = -j i, \quad j k = -i = -k j, \quad k i = j = -i k,$

$i^2 = -1, \quad j^2 = +1, \quad k^2 = +1,$

and hence ijk = 1. In the para-quaternionic literature, an alternative sign convention, in which k is replaced by -k, is often used: in this convention 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 isomorphic to the dihedral group of a square. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections

A coquaternion

$q~= w + x i + y j + z k$

has a conjugate

$q^* ~= w - x i - y j - z k$

and multiplicative modulus

$qq^* ~= w^2 + x^2 - y^2 - z^2$ .

This quadratic form has split signature, in contrast to the positive definite form on the algebra of quaternions.

When the modulus is non-zero, then q has a multiplicative inverse, namely q*/qq*. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which

$U = \{q : qq^* \ne 0 \}$

is the set of units. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i The set P of all coquaternions forms a ring (P, +, •) with group of units (U, •). In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The coquaternions with modulus qq* = 1 form a non-compact topological group SU(1,1), shown below to be isomorphic to SL(2,R). In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, the Special linear group SL2( R) is the group of all real 2 × 2 matrices with Determinant one

The split-quaternion basis can be identified as the basis elements of either the Clifford algebra Cℓ_1,1(R), with {1, e₁=i, e₂=j, e₁e₂=k}; or the algebra Cℓ_2,0(R), with {1, e₁=j, e₂=k, e₁e₂=i}. In Mathematics, Clifford algebras are a type of Associative algebra. They are also isomorphic to the algebra of real 2 by 2 matrices.

Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra. Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. The tessarines are a mathematical idea introduced by James Cockle in 1848 Linear algebra is the branch of Mathematics concerned with

1 Matrix representations
2 Profile
3 Pan-orthogonality
4 Counter-sphere geometry
5 Application to kinematics
6 Historical notes
7 References
8 See also

Matrix representations

Let

$q = w + x i + y j + z k, ~ ~ u = w + x i, ~ ~ v = y + z i$

where u and v are ordinary complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Then the complex matrix

$\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix}$ ,

with $u^* = w - x i\ ,\ v^* = y - z i$ (complex conjugates of u and v), represents q in the ring of matrices in the sense that multiplication of split-quaternions behaves the same way as the matrix multiplication. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix For example, the determinant of this matrix is

$u u^* - v v^* = q q^* \!$ . In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

The appearance of the minus sign, where there is a plus in H, distinguishes coquaternions from quaternions.

Besides the complex matrix representation, another linear representation associates coquaternions with real matrices (2 x 2). The 2 x 2 real matrices are the Linear mappings of the Cartesian coordinate system into itself by the rule (xy \mapsto (xy\begin{pmatrix}a & c This isomorphism can be made explicit as follows: Note first the product

$\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}$

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 j k = −i in the coquaternion ring. Then for an arbitrary matrix there is the bijection

$\begin{pmatrix} a & c \\ b & d\end{pmatrix} \leftrightarrow q = [(a+d) + (c-b)i + (b+c)j + (a-d)k]/2,$

which is in fact a ring isomorphism. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Furthermore, computing squares of components and gathering terms shows that $q q^* = ad - bc \!$ , which is the determinant of the matrix. Consequently there is a group isomorphism between the unit sphere of coquaternions and SL2(R) = {g ∈ M(2,R) : det g = 1 }, and hence also with SU(1,1): the latter can be seen in the complex representation above. In Mathematics, the Special linear group SL2( R) is the group of all real 2 × 2 matrices with Determinant one

Profile

Let

r(θ) = j cos θ + k sin θ (here θ is as fundamental as azimuth)

p(a, r) = i sinh a + r cosh a

v(a, r) = i cosh a + r sinh a

These are the equilateral-hyperboloidal coordinates described by Alexander MacFarlane. Azimuth ( is a mathematical concept defined as the angle usually measured in degrees (° between a reference plane and a point. Alexander Macfarlane may be Alexander Macfarlane (politician (1818-1898 Nova Scotian lawyer and senator Alexander Macfarlane (mathematician

Next, form three foundational sets in the vector-subspace of the ring:

E = { r ∈ P: r = r(θ), 0 ≤ θ < 2 π}

J = {p(a, r) ∈ P: a ∈ R, r ∈ E} catenoid

I = {v(a, r) ∈ P: a ∈ R, r ∈ E} hyperboloid of two sheets

Now it is easy to verify that

{q ∈ P: q² = + 1} = J ∪ {1, -1}

and that

{q ∈ P: q² = -1} = I. A catenoid is a three- Dimensional Shape made by rotating a Catenary Curve around the x axis In Mathematics, a hyperboloid is a Quadric, a type of surface in three Dimensions described by the equation {x^2 \over a^2} +

These set equalities mean that when p ∈ J then the plane

{x + yp: x, y ∈ R} = D_p

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

{x + yv: x, y ∈ R} = C_v

is a planar subring of P that is isomorphic to the ordinary complex plane C. In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis

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

SU(1, 1) = {q ∈ P: qq* = 1}

consists of the "unit circles" in the constituent planes of P. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed In Mathematics, a unit circle is In D_p this is an 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). In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions 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". Rubin's vase (sometimes known as the Rubin face or the Figure-ground vase) is a famous set of cognitive

Pan-orthogonality

When coquaternion $q = w + x i + y j + z k$ , then the real part of q is w. In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z
Definition: For non-zero coquaternions q and t we write q ⊥ t when the real part of the product $q t *$ is zero.

For every v ∈ I, if q, t ∈ C_v, then q ⊥ t means the rays from 0 to q and t are perpendicular. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent
For every p ∈ J, if q, t ∈ D_p, then q ⊥ t means these two points are hyperbolic-orthogonal. In Mathematics, two points in the Cartesian plane are hyperbolically orthogonal if the Slopes of their rays from the origin are Reciprocal to
For every r ∈ E and every a ∈ R, p = p(a, r) and v = v(a, r) satisfy p ⊥ v.
If u is a unit in the coquaternion ring, then q ⊥ t implies qu ⊥ tu.

Proof:

(q u)(t u) * = (u u *) q t *

follows from

(t u) * = u * t *

, which can be established using the anticommutativity property of vector cross products. In mathematics anticommutativity refers to the property of an operation being anticommutative, i In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

Counter-sphere geometry

Take $m = x + y i + z r$ where $r~= j \cos \theta + k \sin \theta$ . Fix theta (θ) and suppose

m m * = - 1 = x 2 + y 2 - z 2

Since points on the counter-sphere must line on a counter-circle in some plane D_p ⊂ P, m can be written, for some p ∈ J

$m~= p \exp{(bp)} = \sinh b + p \cosh b = \sinh b + i \sinh a~\cosh b + r \cosh a~\cosh b$ . In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed

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:

$\tan \phi = \frac{x}{y} = \frac{\sinh b}{\sinh a ~\cosh b} = \frac{\tanh b}{\sinh a}$ . For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation.

As b gets large, tanh b nears one. Then tan φ = 1/sinh a . This appearance of the angle of parallelism in a meridian θ inclines one to expect to see the counter- sphere unfold as the manifold S¹ × H² where H² is the hyperbolic plane. In Hyperbolic geometry, the angle of parallelism Φ is the Angle at one vertex of a right Hyperbolic triangle that has two asymptotic parallel sides A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In

Application to kinematics

By using the foundations given above, one can show that the mapping

q → u⁻¹qu

is an ordinary or hyperbolic rotation according as

u = exp(av), v ∈ I or u = exp(ap), p ∈ J.

These mappings are projectivities in the inversive ring geometry of coquaternions. In Mathematics, inversive ring geometry is the extension to the context of Associative rings of the concepts of Projective line, Homogeneous The collection of these mappings bears some relation to the Lorentz group since it is also composed of ordinary and hyperbolic rotations. In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting Among the peculiarities of this approach to relativistic kinematic is the anisotropic profile, say as compared to Hyperbolic quaternions. Anisotropy (pronounced with stress on the third syllable ˌænaɪˈsɒtrəpi is the property of being directionally dependent as opposed to Isotropy, which means homogeneity In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association

Reticence to use coquaternions for kinematic models may stem from the (2, 2) signature when spacetime is presumed to have signature (1, 3) or (3, 1). In Mathematics, signature can refer to The signature of a Permutation is ±1 according to whether it is an even/odd permutation SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Nevertheless, a transparently relativistic kinematics appears when a point of the counter-sphere is used to represent an inertial frame of reference. Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest Indeed, if $t t * = - 1$ , then there is a p ∈ J such that t ∈ D_p, and an a ∈ R such that t = p exp(ap). Then if u = exp(ap) and s = ir, the set {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) in 1849 by James Cockle in the London–Edinburgh–Dublin Philosophical Magazine (Cockle 1849). The Philosophical Magazine is arguably the world’s oldest commercially published Scientific journal. The introductory papers by Cockle were recalled in the 1904 Bibliography of the Quaternion Society (1899 - 1913). A Scientific society, the Quaternion Society was an “International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics” Alexander MacFarlane called the structure of coquaternion vectors an exspherical system when he was speaking in Paris in 1900. Alexander Macfarlane ( April 21 1851 – August 28, 1913) was a Scottish - Canadian Logician Physicist

The unit sphere was considered in 1910 by Hans Beck (Beck 1910: e. g. , the dihedral group appears on page 419). The coquaternion structure has also been mentioned briefly in the Annals of Mathematics (Albert 1942, Bargmann 1947). The Annals of Mathematics (ISSN 0003-486X abbreviated as Ann of Math

Manifolds with para-quaternionic structures are studied in differential geometry and string theory (Ivanov and Zamkovoy 2005, Mohaupt 2006). Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

References

Albert, A. A. (1942), "Quadratic Forms permitting Composition", Annals of Mathematics 43, pp. The Annals of Mathematics (ISSN 0003-486X abbreviated as Ann of Math 161–177.
Bargmann, V. (1947), "Representations of the Lorentz Group", Annals of Mathematics 48, pp. The Annals of Mathematics (ISSN 0003-486X abbreviated as Ann of Math 568–640.
Beck, Hans (1910), in Transactions of the American Mathematical Society 28. Transactions of the American Mathematical Society is a monthly mathematics journal published by the American Mathematical Society.
Cockle, James (1849), "On Systems of Algebra involving more than one Imaginary", Philosophical Magazine (series 3) 35, pp. The Philosophical Magazine is arguably the world’s oldest commercially published Scientific journal. 434–435.
Ivanov, Stefan; Zamkovoy, Simeon (2005), "Parahermitian and paraquaternionic manifolds", Differential Geometry and its Applications 23, pp. 205–234, math.DG/0310415 MR 2006d:53025. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many
MacFarlane, Alexander (1900) Proceedings of the International Congress of Mathematicians, Paris, page 306. The International Congress of Mathematicians (ICM is the largest congress in the Mathematics community
MacFarlane, Alexander (1904) Bibliography of Quaternions and Allied Systems of Mathematics,entries for James Cockle, pp. 17-18.
Mohaupt, Thomas (2006), "New developments in special geometry", hep-th/0602171.

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