Split-biquaternion

Modern definition

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}. The idea is to replace the complex numbers used in an ordinary (Hamilton) biquaternion with split-complex numbers.Thus q = w + x i + y j + z k , with w, x, y, z ∈ D is a Clifford biquaternion. Such a number can also be written q = r + s ω , r, s ∈ H, ω2 = +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 –

References

de:Biquaternion#Clifford Biquaternion