In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

• Biquaternions when the coefficients are complex numbers.
• Split-biquaternions when the coefficients are split-complex numbers.
• Dual quaternions when the coefficients are dual numbers.

This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of the Royal Irish Academy 1844 & 1850 page 388^[1]). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity.

The algebra of biquaternions can be considered as a tensor product ${\displaystyle \mathbb {C} \otimes \mathbb {H} }$ (taken over the reals) where C or ${\displaystyle \mathbb {C} }$ is the field of complex numbers and H or ${\displaystyle \mathbb {H} }$ is the division algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2 × 2 complex matrices M₂(C). They are also isomorphic to several Clifford algebras including H(C) = Cℓ⁰₃(C) = Cℓ₂(C) = Cℓ_1,2(R),^{[2]: 112, 113} the Pauli algebra Cℓ_3,0(R),^{[2]: 112 [3]: 404} and the even part Cℓ⁰_1,3(R) = Cℓ⁰_3,1(R) of the spacetime algebra.^[3]: 386

Definition

Let {1, i, j, k} be the basis for the (real) quaternions H, and let u, v, w, x be complex numbers, then

${\displaystyle q=u\mathbf {1} +v\mathbf {i} +w\mathbf {j} +x\mathbf {k} }$

is a biquaternion.^[4]: 639 To distinguish square roots of minus one in the biquaternions, Hamilton^{[4]: 730 [5]} and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field C by h to avoid confusion with the i in the quaternion group. Commutativity of the scalar field with the quaternion group is assumed:

${\displaystyle h\mathbf {i} =\mathbf {i} h,\ \ h\mathbf {j} =\mathbf {j} h,\ \ h\mathbf {k} =\mathbf {k} h.}$

Hamilton introduced the terms bivector, biconjugate, bitensor, and biversor to extend notions used with real quaternions H.

Hamilton's primary exposition on biquaternions came in 1853 in his Lectures on Quaternions. The editions of Elements of Quaternions, in 1866 by William Edwin Hamilton (son of Rowan), and in 1899, 1901 by Charles Jasper Joly, reduced the biquaternion coverage in favor of the real quaternions.

Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers C. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor. The algebra of biquaternions forms a composition algebra and can be constructed from bicomplex numbers. See § As a composition algebra below.

Place in ring theory

Linear representation

Note the matrix product

${\displaystyle {\begin{pmatrix}h&0\\0&-h\end{pmatrix}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}={\begin{pmatrix}0&h\\h&0\end{pmatrix}}}$.

Because h is the imaginary unit, each of these three arrays has a square equal to the negative of the identity matrix. When this matrix product is interpreted as i j = k, then one obtains a subgroup of matrices that is isomorphic to the quaternion group. Consequently,

${\displaystyle {\begin{pmatrix}u+hv&w+hx\\-w+hx&u-hv\end{pmatrix}}}$

represents biquaternion q = u 1 + v i + w j + x k. Given any 2 × 2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring M(2,C) is isomorphic^[6] to the biquaternion ring.

Subalgebras

Considering the biquaternion algebra over the scalar field of real numbers R, the set

${\displaystyle \{\mathbf {1} ,h,\mathbf {i} ,h\mathbf {i} ,\mathbf {j} ,h\mathbf {j} ,\mathbf {k} ,h\mathbf {k} \}}$

forms a basis so the algebra has eight real dimensions. The squares of the elements hi, hj, and hk are all positive one, for example, (hi)² = h²i² = (−1)(−1) = +1.

The subalgebra given by

${\displaystyle \{x+y(h\mathbf {i} ):x,y\in \mathbb {R} \}}$

is ring isomorphic to the plane of split-complex numbers, which has an algebraic structure built upon the unit hyperbola. The elements hj and hk also determine such subalgebras.

Furthermore,

${\displaystyle \{x+y\mathbf {j} :x,y\in \mathbb {C} \}}$

is a subalgebra isomorphic to the tessarines.

A third subalgebra called coquaternions is generated by hj and hk. It is seen that (hj)(hk) = (−1)i, and that the square of this element is −1. These elements generate the dihedral group of the square. The linear subspace with basis {1, i, hj, hk} thus is closed under multiplication, and forms the coquaternion algebra.

In the context of quantum mechanics and spinor algebra, the biquaternions hi, hj, and hk (or their negatives), viewed in the M₂(C) representation, are called Pauli matrices.

Algebraic properties

The biquaternions have two conjugations:

• the biconjugate or biscalar minus bivector is ${\displaystyle q^{*}=w-x\mathbf {i} -y\mathbf {j} -z\mathbf {k} \!\ ,}$ and
• the complex conjugation of biquaternion coefficients ${\displaystyle q^{\star }=w^{\star }+x^{\star }\mathbf {i} +y^{\star }\mathbf {j} +z^{\star }\mathbf {k} }$

where ${\displaystyle z^{\star }=a-bh}$ when ${\displaystyle z=a+bh,\quad a,b\in \mathbb {R} ,\quad h^{2}=-\mathbf {1} .}$

Note that ${\displaystyle (pq)^{*}=q^{*}p^{*},\quad (pq)^{\star }=p^{\star }q^{\star },\quad (q^{*})^{\star }=(q^{\star })^{*}.}$

Clearly, if ${\displaystyle qq^{*}=0}$ then q is a zero divisor. Otherwise ${\displaystyle \lbrace qq^{*}\rbrace ^{-\mathbf {1} }}$ is defined over the complex numbers. Further, ${\displaystyle qq^{*}=q^{*}q}$ is easily verified. This allows an inverse to be defined by

• ${\displaystyle q^{-\mathbf {1} }=q^{*}\lbrace qq^{*}\rbrace ^{-\mathbf {1} }}$, if ${\displaystyle qq^{*}\neq 0.}$

Relation to Lorentz transformations

Consider now the linear subspace^[7]

${\displaystyle M=\lbrace q\colon q^{*}=q^{\star }\rbrace =\lbrace t+x(h\mathbf {i} )+y(h\mathbf {j} )+z(h\mathbf {k} )\colon t,x,y,z\in \mathbb {R} \rbrace .}$

M is not a subalgebra since it is not closed under products; for example ${\displaystyle (h\mathbf {i} )(h\mathbf {j} )=h^{2}\mathbf {ij} =-\mathbf {k} \notin M.}$. Indeed, M cannot form an algebra if it is not even a magma.

Proposition: If q is in M, then ${\displaystyle qq^{*}=t^{2}-x^{2}-y^{2}-z^{2}.}$

Proof: From the definitions,

${\displaystyle {\begin{aligned}qq^{*}&=(t+xh\mathbf {i} +yh\mathbf {j} +zh\mathbf {k} )(t-xh\mathbf {i} -yh\mathbf {j} -zh\mathbf {k} )\\&=t^{2}-x^{2}(h\mathbf {i} )^{2}-y^{2}(h\mathbf {j} )^{2}-z^{2}(h\mathbf {k} )^{2}\\&=t^{2}-x^{2}-y^{2}-z^{2}.\end{aligned}}}$

Definition: Let biquaternion g satisfy ${\displaystyle gg^{*}=\mathbf {1} .}$ Then the Lorentz transformation associated with g is given by

${\displaystyle T(q)=g^{*}qg^{\star }.}$

Proposition: If q is in M, then T(q) is also in M.

Proof: ${\displaystyle (g^{*}qg^{\star })^{*}=(g^{\star })^{*}q^{*}g=(g^{*})^{\star }q^{\star }g=(g^{*}qg^{\star })^{\star }.}$

Proposition: ${\displaystyle \quad T(q)(T(q))^{*}=qq^{*}}$

Proof: Note first that gg* = 1 means that the sum of the squares of its four complex components is one. Then the sum of the squares of the complex conjugates of these components is also one. Therefore, ${\displaystyle g^{\star }(g^{\star })^{*}=\mathbf {1} .}$ Now

${\displaystyle (g^{\ast <!--\; \ast \; -->}q{g}^{\star <!--\; \star \; -->})(g^{\ast <!--\; \ast \; -->}q{g}^{\star <!--\; \star \; -->}{)}^{\ast <!--\; \ast \; -->}=g^{\ast <!--\; \ast \; -->}q{g}^{}}$Zdroj:https://en.wikipedia.org?pojem=Biquaternion
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