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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 forms a basis. The products of these elements are :ij = 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 is a group under coquaternion multiplication; it is isomorphic to the dihedral group of a square. A coquaternion :''q'' = ''w'' + ''x''i + ''y''j + ''z''k, has a conjugate :''q *'' = ''w'' − ''x''i − ''y''j − ''z''k, and multiplicative modulus *'' = ''w''2 + ''x''2 − ''y''2 − ''z''2. 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 *''. The set :''U'' = 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 ''C''ℓ1,1(R), with ; or the algebra ''C''ℓ2,0(R), with . Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra. ==Matrix representations== Let :''q'' = ''w'' + ''x''i + ''y''j + ''z''k, and consider ''u'' = ''w'' + ''x''i, and ''v'' = ''y'' + ''z''i as ordinary complex numbers with complex conjugates denoted by ''u *'' = ''w'' − ''x''i, ''v *'' = ''y'' − ''z''i. Then the complex matrix :, represents ''q'' in the ring of matrices, i.e. the multiplication of split-quaternions behaves the same way as the matrix multiplication. For example, the determinant of this matrix is :''uu *'' − ''vv *''. The appearance of the minus sign, where there is a plus in H, distinguishes coquaternions from quaternions. The use of the split-quaternions of modulus one (''qq *'' = 1) for hyperbolic motions of the Poincaré disk model of hyperbolic geometry is one of the great utilities of the algebra. Besides the complex matrix representation, another linear representation associates coquaternions with 2 × 2 real matrices. This isomorphism can be made explicit as follows: Note first the product : 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 jk = −i in the coquaternion ring. Then for an arbitrary matrix there is the bijection : which is in fact a ring isomorphism. Furthermore, computing squares of components and gathering terms shows that ''qq *'' = ''ad'' − ''bc'', which is the determinant of the matrix. Consequently there is a group isomorphism between the unit quasi-sphere of coquaternions and SL(2, R) = , and hence also with SU(1, 1): the latter can be seen in the complex representation above. For instance, see Karzel and Kist〔Karzel, Helmut & Günter Kist (1985) "Kinematic Algebras and their Geometries", in ''Rings and Geometry'', R. Kaya, P. Plaumann, and K. Strambach editors, pp 437–509, esp 449,50, D. Reidel ISBN 90-277-2112-2 〕 for the hyperbolic motion group representation with 2 × 2 real matrices. In both of these linear representations the modulus is given by the determinant function. Since the determinant is a multiplicative mapping, the modulus of the product of two coquaternions is equal to the product of the two separate moduli. Thus coquaternions form a composition algebra. As an algebra over the field of real numbers, it is one of only seven such algebras. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Split-quaternion」の詳細全文を読む スポンサード リンク
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