In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They...

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a...

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton...

spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions. He called its elements...

., xn]. There is a circle of idempotents in the ring of split-quaternions. Split quaternions have the structure of a real algebra, so elements can be...

In mathematics, a versor is a quaternion of norm one (a unit quaternion). Each versor has the form q = exp ⁡ ( a r ) = cos ⁡ a + r sin ⁡ a , r 2 = − 1...

are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion...

In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset {...

just as the quaternion algebra H can be viewed as a union of complex planes, so the hyperbolic quaternion algebra is a pencil of planes of split-complex numbers...

In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus...

so 1 + hi is a null vector. The real subalgebras, split complex numbers, split quaternions, and split-octonions, with their null cones representing the...

representation theory. In the nineteenth century, number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...

⁠ Y 2 + 1 {\displaystyle Y^{2}+1} ⁠, then one obtains the ring of split-quaternions. The anti-commutative property Y X = − X Y {\displaystyle YX=-XY}...

matrix (the first Pauli matrix is an exchange matrix of order two) Split-quaternion This conforms to the convention in mathematics for the matrix exponential...

to be +1 we get the split-octonions. One can also obtain the split-octonions via a Cayley–Dickson doubling of the split-quaternions. Here either choice...

Ellipsoid Paraboloid / Hyperbolic paraboloid Regulus Rotation of axes Split-quaternion § Profile Translation of axes De Sitter space Light cone K. Strubecker:...

original Cayley–Dickson construction to the split-complexes also results in the split-quaternions and then the split-octonions. Albert (1942, p. 171) gave a...

slightly less well-behaved split-quaternions, see Alagös, Oral & Yüce (2012). The rings of quaternions and split-quaternions can both be represented by...

subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford...

spherical linear interpolation, introduced by Ken Shoemake in the context of quaternion interpolation for the purpose of animating 3D rotation. It refers to constant-speed...

*-operations of C*-algebras. One may also define a conjugation for quaternions and split-quaternions: the conjugate of a + b i + c j + d k {\textstyle a+bi+cj+dk}...

coefficients are complex numbers. Split-biquaternions when the coefficients are split-complex numbers. Dual quaternions when the coefficients are dual numbers...

matrix or quaternion notation, calculate the product, and then convert back to Euler axis and angle. The idea behind Euler rotations is to split the complete...

algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions C ⊗ H {\displaystyle \mathbb {C}...

a split algebra: binarions: complex numbers with quadratic form x2 + y2 and split-complex numbers with quadratic form x2 − y2, quaternions and split-quaternions...

F^{\times }} and i j = − j i {\displaystyle ij=-ji} . A quaternion algebra is said to be split over F {\displaystyle F} if it is isomorphic as an F {\displaystyle...

map homomorphism from SU(2) × SU(2) to SO(4) given by quaternion multiplication; see quaternions and spatial rotation. Thus SO(4) is not a simple group...

by James Cockle were published: 1848: On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra, 33:435–9. 1849: On a New Imaginary...

classes of quaternion algebras in the Brauer group of F with the split quaternion algebra as distinguished element and q(a,b) the quaternion algebra (a...

namely the split-complex numbers, the split-quaternions and the split-octonions. If one uses these instead of the complex numbers, quaternions, and octonions...