A Clifford biquaternion is a concept from geometric algebra. 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. The collection of all Clifford biquaternions forms a Clifford algebra of dimension 8 over the real line R. A geometric algebra is a multilinear algebra with a geometric interpretation. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, a biquaternion (or complex quaternion) is an element of the (unique) quaternion algebra over the complex numbers. ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Clifford algebras are a type of associative algebra in mathematics. ...

Clifford biquaternion group

The Clifford biquaternions form an associative ring as is clear from considering multiplications in its basis. When 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},•). In mathematics, associativity is a property that a binary operation can have. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ...

See also

• split-octonions

In mathematics, the split-octonions are a nonassociative extension of the quaternions (or the split-quaternions). ...

References

• William Kingdon Clifford (1873), "Preliminary Sketch of Biquaternions", Paper XX, Mathematical Papers, p.181.
• Alexander MacAuley (1898) Octonions: A Development of Clifford's Biquaternions, Cambridge University Press.
• P.R. Girard (1984), "The quaternion group and modern physics", European Journal of Physics, 5:25-32.