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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 r0166812hk163851.
• Split-quaternion wikiPageID "1461265".
• Split-quaternion wikiPageLength "16876".
• Split-quaternion wikiPageOutDegree "95".
• Split-quaternion wikiPageRevisionID "659996250".
• Split-quaternion wikiPageWikiLink 2_×_2_real_matrices.
• Split-quaternion wikiPageWikiLink Absolute_value.
• Split-quaternion wikiPageWikiLink Abstract_algebra.
• Split-quaternion wikiPageWikiLink Advances_in_Applied_Clifford_Algebras.
• Split-quaternion wikiPageWikiLink Alexander_Macfarlane.
• Split-quaternion wikiPageWikiLink Algebra_over_a_field.
• Split-quaternion wikiPageWikiLink Angle_of_parallelism.
• Split-quaternion wikiPageWikiLink Anisotropic.
• Split-quaternion wikiPageWikiLink Anisotropy.
• Split-quaternion wikiPageWikiLink Annals_of_Mathematics.
• Split-quaternion wikiPageWikiLink Anticommutativity.
• Split-quaternion wikiPageWikiLink Arthur_Cayley.
• Split-quaternion wikiPageWikiLink Associative_algebra.
• Split-quaternion wikiPageWikiLink Azimuth.
• Split-quaternion wikiPageWikiLink Basis_(linear_algebra).
• Split-quaternion wikiPageWikiLink Bicomplex_number.
• Split-quaternion wikiPageWikiLink Bijection.
• Split-quaternion wikiPageWikiLink Category:Hyperbolic_geometry.
• Split-quaternion wikiPageWikiLink Category:Quaternions.
• Split-quaternion wikiPageWikiLink Category:Special_relativity.
• Split-quaternion wikiPageWikiLink Clifford_algebra.
• Split-quaternion wikiPageWikiLink Compact_space.
• Split-quaternion wikiPageWikiLink Complex_conjugate.
• Split-quaternion wikiPageWikiLink Complex_number.
• Split-quaternion wikiPageWikiLink Complex_plane.
• Split-quaternion wikiPageWikiLink Composition_algebra.
• Split-quaternion wikiPageWikiLink Cross_product.
• Split-quaternion wikiPageWikiLink Cylindrical_coordinate_system.
• Split-quaternion wikiPageWikiLink Determinant.
• Split-quaternion wikiPageWikiLink Differential_geometry.
• Split-quaternion wikiPageWikiLink Dihedral_group.
• Split-quaternion wikiPageWikiLink Dimension.
• Split-quaternion wikiPageWikiLink Dual_number.
• Split-quaternion wikiPageWikiLink Dual_numbers.
• Split-quaternion wikiPageWikiLink Field_(mathematics).
• Split-quaternion wikiPageWikiLink Group_(mathematics).
• Split-quaternion wikiPageWikiLink Hyperbolic-orthogonal.
• Split-quaternion wikiPageWikiLink Hyperbolic_geometry.
• Split-quaternion wikiPageWikiLink Hyperbolic_motion.
• Split-quaternion wikiPageWikiLink Hyperbolic_orthogonality.
• Split-quaternion wikiPageWikiLink Hyperbolic_quaternion.
• Split-quaternion wikiPageWikiLink Hyperboloid.
• Split-quaternion wikiPageWikiLink Hypercomplex_number.
• Split-quaternion wikiPageWikiLink Hypercomplex_numbers.
• Split-quaternion wikiPageWikiLink Idempotent_element.
• Split-quaternion wikiPageWikiLink Inertial_frame_of_reference.
• Split-quaternion wikiPageWikiLink International_Congress_of_Mathematicians.
• Split-quaternion wikiPageWikiLink Isomorphic.
• Split-quaternion wikiPageWikiLink Isomorphism.
• Split-quaternion wikiPageWikiLink James_Cockle.
• Split-quaternion wikiPageWikiLink James_Cockle_(lawyer).
• Split-quaternion wikiPageWikiLink Kinematics.
• Split-quaternion wikiPageWikiLink Line_(geometry).
• Split-quaternion wikiPageWikiLink Linear_algebra.
• Split-quaternion wikiPageWikiLink Linear_representation.
• Split-quaternion wikiPageWikiLink Lorentz_group.
• Split-quaternion wikiPageWikiLink Manifold.
• Split-quaternion wikiPageWikiLink Mathematical_structure.
• Split-quaternion wikiPageWikiLink Matrix_multiplication.
• Split-quaternion wikiPageWikiLink Multiplicative_inverse.
• Split-quaternion wikiPageWikiLink Musean_hypernumber.
• Split-quaternion wikiPageWikiLink Musean_hypernumbers.
• Split-quaternion wikiPageWikiLink Nilpotent.
• Split-quaternion wikiPageWikiLink Perpendicular.
• Split-quaternion wikiPageWikiLink Philosophical_Magazine.
• Split-quaternion wikiPageWikiLink Poincaré_disk_model.
• Split-quaternion wikiPageWikiLink Quadratic_form.
• Split-quaternion wikiPageWikiLink Quaternion.
• Split-quaternion wikiPageWikiLink Quaternion_Society.
• Split-quaternion wikiPageWikiLink Ray_(geometry).
• Split-quaternion wikiPageWikiLink Real_number.
• Split-quaternion wikiPageWikiLink Representation_theory.
• Split-quaternion wikiPageWikiLink Ring_homomorphism.
• Split-quaternion wikiPageWikiLink Ring_isomorphism.
• Split-quaternion wikiPageWikiLink Ring_theory.
• Split-quaternion wikiPageWikiLink Rubin_vase.
• Split-quaternion wikiPageWikiLink SL2(R).
• Split-quaternion wikiPageWikiLink Scalar_(mathematics).
• Split-quaternion wikiPageWikiLink Set_(mathematics).
• Split-quaternion wikiPageWikiLink Spacetime.
• Split-quaternion wikiPageWikiLink Split-biquaternion.
• Split-quaternion wikiPageWikiLink Split-complex_number.
• Split-quaternion wikiPageWikiLink Split-octonion.
• Split-quaternion wikiPageWikiLink String_theory.
• Split-quaternion wikiPageWikiLink Subalgebra.
• Split-quaternion wikiPageWikiLink Subring.
• Split-quaternion wikiPageWikiLink Tangent.
• Split-quaternion wikiPageWikiLink Tessarine.
• Split-quaternion wikiPageWikiLink Topological_group.
• Split-quaternion wikiPageWikiLink Unit_(ring_theory).
• Split-quaternion wikiPageWikiLink Unit_circle.

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