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- Split-quaternion abstract "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 zero divisors, nilpotent elements, and nontrivial idempotents. As a mathematical structure, they form an algebra over the real numbers, which is isomorphic to the algebra of 2 × 2 real matrices. The coquaternions came to be called split-quaternions due to the division into positive and negative terms in the modulus function. For other names for split-quaternions see the Synonyms section below.The set {1, i, j, k} forms a basis. The products of these elements areij = k = −ji, jk = −i = −kj, ki = j = −ik,i2 = −1, j2 = +1, k2 = +1,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 of a square.A coquaternionq = w + xi + yj + zk,has a conjugateq* = w − xi − yj − zk,and multiplicative modulusqq* = w2 + x2 − y2 − z2.This quadratic form is split into positive and negative parts, 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*. The setU = {q : qq* ≠ 0}is the set of units. The set P of all coquaternions forms a ring (P, +, •) with group of units (U, •). The coquaternions with modulus qq* = 1 form a non-compact topological group SU(1,1), shown below to be isomorphic to SL(2, R).The split-quaternion basis can be identified as the basis elements of either the Clifford algebra Cℓ1,1(R), with {1, e1 = i, e2 = j, e1e2 = k}; or the algebra Cℓ2,0(R), with {1, e1 = j, e2 = k, e1e2 = i}.Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra.".
- Split-quaternion thumbnail HyperboloidOfOneSheet.PNG?width=300.
- Split-quaternion wikiPageExternalLink 0602171.
- Split-quaternion wikiPageExternalLink 0310415.
- Split-quaternion wikiPageExternalLink 1969129.
- Split-quaternion wikiPageExternalLink r0166812hk163851.
- Split-quaternion wikiPageID "1461265".
- Split-quaternion wikiPageRevisionID "600619777".
- Split-quaternion hasPhotoCollection Split-quaternion.
- Split-quaternion subject Category:Hyperbolic_geometry.
- Split-quaternion subject Category:Quaternions.
- Split-quaternion subject Category:Special_relativity.
- Split-quaternion type Abstraction100002137.
- Split-quaternion type DefiniteQuantity113576101.
- Split-quaternion type Digit113741022.
- Split-quaternion type Four113744304.
- Split-quaternion type Integer113728499.
- Split-quaternion type Measure100033615.
- Split-quaternion type Number113582013.
- Split-quaternion type Quaternions.
- Split-quaternion comment "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 zero divisors, nilpotent elements, and nontrivial idempotents.".
- Split-quaternion label "Coquaternion".
- Split-quaternion label "Kokwaterniony".
- Split-quaternion label "Split-quaternion".
- Split-quaternion sameAs Coquaternion.
- Split-quaternion sameAs Kokwaterniony.
- Split-quaternion sameAs m.053chh.
- Split-quaternion sameAs Q2996919.
- Split-quaternion sameAs Q2996919.
- Split-quaternion sameAs Split-quaternion.
- Split-quaternion wasDerivedFrom Split-quaternion?oldid=600619777.
- Split-quaternion depiction HyperboloidOfOneSheet.PNG.
- Split-quaternion isPrimaryTopicOf Split-quaternion.
