Split-biquaternion

A split-biquaternion is ring isomorphic to the Clifford algebra Cl0,3(R). This is the geometric algebra generated by three orthogonal imaginary unit basis directions, under the combination rule : e_i e_j = \begin{cases} -1 & i=j, \\ - e_j e_i & i \neq j \end{cases} giving an algebra spanned by the 8 basis elements , with (e1e2)2 = (e2e3)2 = (e3e1)2 = −1 and ω2 = (e1e2e3)2 = +1. The sub-algebra spanned by the 4 elements is the division ring of Hamilton's quaternions, . One can therefore see that : \mathrm{Cl}_{0,3}(\mathbf{R}) \cong \mathbf{H} \otimes \mathbf{D} where is the algebra spanned by , the algebra of the split-complex numbers. Equivalently, : \mathrm{Cl}_{0,3}(\mathbf{R}) \cong \mathbf{H} \oplus \mathbf{H}. 2 = +1 , H the division ring of Hamilton's quaternions. --> == Split-biquaternion group ==

The split-biquaternions form an associative ring as is clear from considering multiplications in its basis . When hyperbolic unit ω is adjoined to the quaternion group one obtains a 16 element group : ( {1, i, j, k, −1, −i, −j, −k, ω, ωi, ωj, ωk, −ω, −ωi, −ωj, −ωk}, × ), which is the internal direct product of the quaternion group and the cyclic group {1, ω} of order 2. == Module ==

Since elements of the quaternion group can be taken as a basis of the space of split-biquaternions, it may be compared to a vector space. But split-complex numbers form a ring, not a field, so vector space is not appropriate. Rather the space of split-biquaternions forms a free module. This standard term of ring theory expresses a similarity to a vector space, and this structure by Clifford in 1873 is an instance. Split-biquaternions form an algebra over a ring, but not a group ring. == Direct sum of two quaternion rings ==

The direct sum of the division ring of quaternions with itself is denoted \mathbf{H} \oplus \mathbf{H}. The product of two elements (a \oplus b) and (c \oplus d) is a c \oplus b d in this direct sum algebra. Proposition: The algebra of split-biquaternions is isomorphic to \mathbf{H} \oplus \mathbf{H}. proof: Every split-biquaternion has an expression q = w + z ω where w and z are quaternions and ω2 = +1. Now if p = u + v ω is another split-biquaternion, their product is : pq = uw + vz + (uz + vw) \omega . The isomorphism mapping from split-biquaternions to \mathbf{H} \oplus \mathbf{H} is given by : p \mapsto (u + v) \oplus (u - v) , \quad q \mapsto (w + z) \oplus (w - z). In \mathbf{H} \oplus \mathbf{H}, the product of these images, according to the algebra-product of \mathbf{H} \oplus \mathbf{H} indicated above, is : (u + v)(w + z) \oplus (u - v)(w - z). This element is also the image of pq under the mapping into \mathbf{H} \oplus \mathbf{H}. Thus the products agree, the mapping is a homomorphism; and since it is bijective, it is an isomorphism. Though split-biquaternions form an eight-dimensional space like Hamilton's biquaternions, on the basis of the Proposition it is apparent that this algebra splits into the direct sum of two copies of the real quaternions. == Hamilton biquaternion ==

The split-biquaternions should not be confused with the (ordinary) biquaternions previously introduced by William Rowan Hamilton. Hamilton's biquaternions are elements of the algebra : \mathrm{Cl}_{2}(\mathbf{C}) = \mathbf{H} \otimes \mathbf{C}. : \mathrm{Cl}_{3,0}(\mathbf{R}) = \mathbf{H} \otimes \mathbf{C}. == Synonyms ==

The following terms and compounds refer to the split-biquaternion algebra: • elliptic biquaternions – , • Clifford biquaternion – , • dyquaternions – • \mathbf{D} \otimes \mathbf{H} where D = split-complex numbers – , • \mathbf{H} \oplus \mathbf{H}, the direct sum of two quaternion algebras – == See also ==