Draft:Hyperquaternion

• File:Symbol opinion vote.svg Comment: This appears to be generated by AI. It is poorly sourced, confusingly written, and it is also written in a style completely incomprehensible to anyone who is not a mathematician. A quick search of Google Scholar finds many articles with the term hyperquaternion in the title, so presumably the concept is notable - but as the previous reviewer commented, it seems strange that it is not mentioned in the article on Quaternion. If you want to revise this, please read Wikipedia:Manual_of_Style/Mathematics and follow the guidelines there. Also please do NOT use AI-generated text as we are seeing many poor results from this throughout Wikipedia. Lijil (talk) 08:56, 19 September 2025 (UTC)

• File:Symbol opinion vote.svg Comment: Given the asserted notability of this subject, why is it not discussed at all in Quaternion? BD2412 T 00:21, 18 September 2025 (UTC)

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A hyperquaternion number is an element of a Clifford algebra defined as a tensor product of quaternion algebras (or subalgebra thereof).

History

In 1878, W. K. Clifford (1845 − 1879) made a synthesis of the calculus of H. G. Grassmann (1809 - 1877) and the quaternions of W. R. Hamilton (1805 - 1865). He defined his algebras as a tensor product (”compound of algebras”) of quaternion algebras, a concept introduced by B. Peirce (1809 − 1880). In 1880, R. Lipschitz (1832 − 1903) derived the rotation formula of nD Euclidean spaces $x^{\prime }=ax{a}^{-1}(a\in C^{+})$ and thereby rediscovered the (even) Clifford algebras. In 1922, C. L. E. Moore (1876 − 1931) was to call Lipschitz’ algebras ”hyperquaternions”. The term hyperquaternion number designates today the tensor product of quaternion algebras (or subalgebra thereof).

Definition

The quaternion algebra $ℍ$ is composed of quaternions $q=a+bi+cj+dk$, where $i,j,k$ satisfy the relation $i^2=j^2=k^2=ijk=-1$. The quaternion conjugate of $q$ is $q_c=a-bi-cj-dk$. The tensor product ${ℍ}^{\otimes m}$ of $m$ quaternion algebras is defined by

$\begin{array}{cc}{ℍ}^{\otimes m}& =ℍ\otimes ℍ\otimes \cdots \otimes ℍ(m\text{ terms})\\ & =(i,j,k)\otimes (I,J,K)\otimes (l,m,n)\otimes \cdots \\ \end{array}$.

where $(i,j,k)=(i,j,k)\otimes 1\otimes \cdots \otimes 1,(I,J,K)=(I,J,K)\otimes 1\otimes \cdots \otimes 1$, etc. are distinct commuting quaternionic systems. An element of ${ℍ}^{\otimes m}$ is called a hyperquaternion number.

A hyperconjugation is defined by: $({ℍ}^{\otimes m}{)}^{*}=({ℍ}_c^{\otimes m})={ℍ}_c\otimes {ℍ}_c\otimes \cdots \otimes {ℍ}_c$

where ${ℍ}_c$ is the quaternion conjugation.

A Clifford algebra $C{l}_{p,q}(ℝ)$ has $n=p+q$ generators $e_1,e_2,\mathrm{...},e_n$ multiplying according to $e_i{e}_j+e_j{e}_i=0(i\ne j)$ with $e_i^2=+1$ ($p$ generators) and $e_i^2=-1$ ($q$ generators). The algebra contains scalars $(S)$, vectors $(V)e_i$, bivectors $(B)e_i{e}_j(i\ne j)$, etc. inducing a multivector structure. The total number of elements is $2^n$. The even subalgebra $C^{+}$ is generated by the products of an even number of generators.

Classification

There are four types of hyperquaternion numbers ${ℍ}^{\otimes m}$ ($m$ even or odd) and the even subalgebras $C^{+}$ yielding the following Clifford algebras $C{l}_{p,q}(ℝ)$ with the parameter $s(s=p-q)$

${ℍ}^{\otimes 2m}\simeq C{l}_{2m+1,2m-1}\mathbb{(}ℝ\mathbb{)},{ℍ}^{\otimes (2m-1)}\simeq C{l}_{2m-2,2m}\mathbb{(}ℝ\mathbb{)}$

and the subalgebras $C^{+}$

$ℂ\otimes {ℍ}^{\otimes (2m-1)}\simeq C{l}_{2m+1,2m-2}\mathbb{(}ℝ\mathbb{)},ℂ\otimes {ℍ}^{\otimes (2m-2)}\simeq C{l}_{2m-2,2m-1}\mathbb{(}ℝ\mathbb{)}$

The table below lists a few hyperquaternion algebras.

Name/Symbol	Dimension $n$	No of elements $2^n$	$C{l}_{p,q}\mathbb{(}ℝ\mathbb{)}$	$s=p-q$
complex number $ℂ$	1	2	$C{l}_{0,1}\mathbb{(}ℝ\mathbb{)}$	-1
quaternions $ℍ$	2	4	$C{l}_{0,2}\mathbb{(}ℝ\mathbb{)}$	-2
biquaternions $ℂ\otimes ℍ$	3	8	$C{l}_{3,0}\mathbb{(}ℝ\mathbb{)}$	3
tetraquaternions ${ℍ}^{\otimes 2}$	4	16	$C{l}_{3,1}\mathbb{(}ℝ\mathbb{)}$	2
$ℂ\otimes {ℍ}^{\otimes 2}$	5	32	$C{l}_{2,3}\mathbb{(}ℝ\mathbb{)}$	-1
${ℍ}^{\otimes 3}$	6	64	$C{l}_{2,4}\mathbb{(}ℝ\mathbb{)}$	-2

The hyperquaternion numbers yield real, complex and quaternionic square matrices due to the isomorphism ${ℍ}^{\otimes 2}\simeq M_{4\times 4}\mathbb{(}ℝ\mathbb{)}$ where $M_{4\times 4}\mathbb{(}ℝ\mathbb{)}$ denotes the $4\times 4$ real matrix.

The hyperconjugation generalizes the concepts of matrix transposition, adjoint and transpose quaternion conjugation since $({ℍ}^{\otimes 2}{)}^{*}=[M_{4\times 4}\mathbb{(}ℝ\mathbb{)}{]}^T$ where $T$ is the matrix transposition.

Example 1: Quaternions $ℍ\simeq C{l}_{0,2}\mathbb{(}ℝ\mathbb{)}$

The quaternion algebra $ℍ\simeq C{l}_{0,2}\mathbb{(}ℝ\mathbb{)}$ is a Clifford algebra with two generators

$e_1=i,e_2=j,e_1{e}_2=k(e_1^2=e_2^2=-1)$.

A general element of $ℍ$ is expressed by $q=s+x_{1}i+x_{2}j+bk$ where $s$ is a scalar, $x=x_{1}i+x_{2}j$ a vector $(V)$ and $bk$ a bivector $(B)$. The subalgebra $C^{+}\text{ is }(s+bk)$.

Interior and exterior products can be defined by

$\begin{array}{ccccccc}x.y& =-(xy+yx)/2& & =x_1{y}_1+x_2{y}_2& & & \in S\\ x\wedge y& =(xy-yx)/2& & =(x_1{y}_2-x_2{y}_1)k& & & \in B\\ x.B& =-(xB-Bx)/2& & =b(-x_{2}i+x_{1}j)& & & \in V\\ \end{array}$.

The rotation group $SO(2)$ is expressed by

$x^{\prime }=rx{r}^{-1}=(x_{1}cos\theta -x_{2}sin\theta )i+(x_{1}sin\theta +x_{2}cos\theta )j$

with

$r=e^{k\theta /2}=(cos\theta /2+ksin\theta /2)\in C^{+}(\simeq ℂ)$.

Example 2: Tetraquaternions ${ℍ}^{\otimes 2}\simeq C{l}_{3,1}\mathbb{(}ℝ\mathbb{)}$

A general element $A$ of ${ℍ}^{\otimes 2}$ is a set of four quaternions called tetraquaternion

$A=q_0+i{q}_1+j{q}_2+k{q}_3=\left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right](q_i\in ℍ)$

and similarly $A^{\prime }$ with $q_i={\alpha }_i+{\beta }_{i}I+{\gamma }_{i}J+{\delta }_{i}K$ (${\alpha }_i,{\beta }_i,{\gamma }_i,{\delta }_i$, real coefficients).

The product $A{A}^{\prime }$ yields a set of four quaternions

$A{A}^{\prime }=\left[\begin{array}{c}{q}_{0}q{\text{'}}_0-q_{1}q{\text{'}}_1-q_{2}q{\text{'}}_2-q_{3}q{\text{'}}_3\\ q_{0}q{\text{'}}_1+q_{1}q{\text{'}}_0+q_{2}q{\text{'}}_3-q_{3}q{\text{'}}_2\\ q_{0}q{\text{'}}_2+q_{2}q{\text{'}}_0+q_{3}q{\text{'}}_1-q_{1}q{\text{'}}_3\\ q_{0}q{\text{'}}_3+q_{3}q{\text{'}}_0+q_{1}q{\text{'}}_2-q_{2}q{\text{'}}_1\end{array}\right]$.

where the order of the elements has to be respected. The four generators of $C{l}_{3,1}\mathbb{(}ℝ\mathbb{)}$ are $e_0=j,e_1=kI,e_2=kJ,e_3=kK$. Tetraquaternions lead to applications in special relativity.

The multivector structure is

$\begin{array}{c}\left[\begin{array}{cccc}1& I=e_3{e}_2& J=e_1{e}_3& K=e_2{e}_1\\ i=e_0{e}_1{e}_2{e}_3& iI=e_0{e}_1& iJ=e_0{e}_2& iK=e_0{e}_3\\ j=e_0& jI=e_0{e}_3{e}_2& jJ=e_0{e}_1{e}_3& jK=e_0{e}_2{e}_1\\ k=e_1{e}_2{e}_3& kI=e_1& kJ=e_2& kK=e_3\end{array}\right]\\ \end{array}$

The multivector structure contains scalars $(S)$, vectors $(V)$, bivectors $(B)$ , trivectors $(T)$ and pseudo-scalars $(P)$ .

If $A_p=a_1\wedge a_2\wedge \mathrm{...}\wedge a_p(p\ge 1)$ denotes a multivector (where $a_i$ are vectors) and $x$ is a vector, the interior and exterior products can be defined

$\begin{array}{cc}2x.A_p& =(-1{)}^p[x{A}_p-(-1{)}^p{A}_{p}x],\\ 2x\wedge A_p& =(-1{)}^p[x{A}_p+(-1{)}^p{A}_{p}x],\\ \end{array}$.

An orthochronous proper Lorentz transformation is given by

$x^{\prime }=ax{a}^{-1},a\in C^{+}(\simeq ℂ\otimes ℍ)$

with $x=ct{e}_0+x_1{e}_1+x_2{e}_2+x_3{e}_3$ (similarly $x^{\prime }$).

A matrix representation is obtained via

$\begin{array}{cc}{e}_0& =j\otimes 1=j=\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right],e_1=k\otimes i=kI=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right],\\ e_2& =k\otimes j=kJ=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],e_3=k\otimes k=kK=\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& -1& 0\\ 0& -1& 0& 0\\ -1& 0& 0& 0\end{array}\right].\\ \end{array}$

See also

• Biquaternion
• Classification of Clifford algebras
• Clifford algebra
• Hypercomplex number
• Quaternion
• Sympletic group