In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. Like the 1843 quaternions of Hamilton, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, coquaternions may be zero divisors or nilpotent. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... William Rowan Hamilton Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... The fundamental concept in linear algebra is that of a vector space or linear space. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero 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 xn = 0. ...

The set { 1, i, j, k } forms a basis. The coquaternion products of these elements are In mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects. ...

ij = k = -ji, j k = - i = - k j, k i = j = - i k

i² = - 1, j² = + 1, k² = + 1 .

With these products the set {1, i, j, k, -1, -i, -j, -k} is isomorphic to the dihedral group of a square. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ...

A coquaternion

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

has conjugate

q* = w - x i - y j - z k and modulus

q q* = w² + x² - y² - z².

When the modulus is non-zero, then q has a multiplicative inverse.

U = {q : q q* ≠ 0 }

is the set of units. The set P of all coquaternions forms a ring (P, +, •) with group of units (U, •). In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... The term group can refer to several concepts: In music, a group is another term for band or other musical ensemble. ...

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. Then the complex matrix The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

where u* = w - x i and v* = y - z i (complex conjugates of u and v), represents q in the ring of matrices in the sense that multiplication of coquaternions behaves the same way as the matrix multiplication. For example, the determinant of this matrix u u* - v v* = q q* ; the appearance of this minus sign where there is a plus in H leads to the alternative name split-quaternion for a coquaternion. Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra. Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ... 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 the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...

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

E = { r ∈ P : r = r(θ), 0 ≤ θ < 2 π }

J = {p(a,r) ∈ P : a ∈ R, r ∈ E } hyperboloid of one sheet

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

Now it is easy to verify that Azimuth is the horizontal component of a direction (compass direction), measured around the horizon from the North point, toward the East, i. ... Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation:  (hyperboloid of one sheet), or  (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ...

{q ∈ P : q² = + 1 } = J ∪ {1,-1}

and that

{q ∈ P : q² = -1 } = I .

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 In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ...

{x + y v : x,y ∈ R } = C_v

is a planar subring of P that is isomorphic to the ordinary complex plane C.

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. 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 A variety of dualities in mathematics are listed at duality (mathematics). ...

SU(1,1) = { q ∈ P : q q* = 1 }

consists of the "unit circles" in the constituent planes of P. 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).

Pan-orthogonality

When coquaternion q = w + xi + yj + zk, then the real part of q is w.
Definition: For non-zero coquaternions q and t we write q ⊥ t when the real part of the product qt* 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.
• For every p ∈ J, if q,t ∈ D_p, then q ⊥ t means these two points are hyperbolic-orthogonal.
• 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: (qu)(tu)* = (uu*)qt* follows from (tu)* = u*t*, a fact based on anti-commutativity of vectors.

A perpendicular line. ... In mathematics, two points in the Cartesian plane are hyperbolically orthogonal if the slopes of their rays from the origin are reciprocal to one another. ...

Counter-sphere geometry

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

m m* = -1 = x² + y² - z² .

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(b p) = sinh b + p cosh b = sinh b + i sinh a cosh b + r cosh a cosh b .

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 φ = x/y = sinh b / (sinh a cosh b) = tanh b / sinh a .

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 metric space S¹ × HP where HP is the hyperbolic plane. In hyperbolic geometry, the angle of parallelism Φ is the angle at one vertex of an right hyperbolic triangle that has two parallel sides. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ...

Historical notes and references

The coquaternions were initially identified and named in the London-Edinburgh-Dublin Philosophical Magazine, series 3, volume 35, pp.434,5 in 1849 by James Cockle under the title "On Systems of Algebra involving more than one Imaginary". At the 1900 Paris meeting of the International Congress of Mathematicians Alexander MacFarlane called the algebra the exspherical quaternion system as he described its profile. MacFarlane examined a differential element of the submanifold {q ∈ P : q q* = - 1 } (the counter-sphere). The sphere itself was considered in German by Hans Beck in 1910 (Transactions of the American Mathematical Society, v.28; e.g. the dihedral group appears on page 419.)In 1942 and 1947 there were two brief mentions of the coquaternion structure in the Annals of Mathematics: The International Congress of Mathematicians (ICM) is the biggest congress in mathematics. ... Alexander MacFarlane (1851 - 1913) was a Scottish-Canadian logician, physicist, and mathematician. ...

• A.A. Albert "Quadratic Forms permitting Composition" 43:161-177
• V. Bargmann "Representations of the Lorentz Group" 48:568-640 .