Draft:Hyperquaternions

A hyperquaternion is an extension of a quaternion, formulated within the framework of Clifford algebras in n dimensions. It is defined as a tensor product of quaternion algebras (or subalgebra therof). This approach presents the advantage that the hyperquaternionic product is defined independently of the choice of the generators which facilitates their use in various mathematical and physical applications.

History

In 1878, W. K. Clifford (1845 − 1879) made a synthesis of the extensive 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 $x'=axa^{-1}{\text{ }}(a\in C^{+})$ and thereby rediscovered the (even) Clifford algebras. In 1922, C. L. E. Moore (1876 − 1931) was to call Lipschitz’ algebras ”hyperquaternions”, a term which today appropriately designates the tensor product of quaternion algebras (or subalgebra thereof).

Definition

Let $\mathbb {H}$ be the quaternion algebra and $q=a+bi+cj+dk$, a quaternion 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 $\mathbb {H} ^{\otimes m}$ of $m$ quaternion algebras is defined by

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

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. It is to be noticed that the tensor product is defined intrinsically, independently of the choice of the generators.

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

where $\mathbb {H} _{c}$ is the quaternion conjugation.

The Clifford algebra $C_{n}(p,q)$ has $n=p+q$ generators $e_{1},e_{2},...,e_{n}$ multiplying according to $e_{i}e_{j}+e_{j}e_{i}=0{\text{ }}(i\neq j)$ with $e_{i}^{2}=+1$ ($p$ generators) and $e_{i}^{2}=-1$ ($q$ generators). The algebra contains scalars $(S)$, vectors $(V){\text{ }}e_{i}$, bivectors $(B){\text{ }}e_{i}e_{j}{\text{ }}(i\neq j)$, etc. inducing a multivector structure endowed with an associative exterior product. 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 hyperquaternions $\mathbb {H} ^{\otimes m}$ ($m$ even or odd and the even subalgebras $C^{+}$) yielding the following Clifford algebras $C_{n}(p,q)$ with the parameter $(s=p-q)$

${\begin{aligned}\mathbb {H} ^{\otimes 2m}&\simeq C_{4m}(2m+1,2m-1),{\text{ }}(s=2)\\\mathbb {H} ^{\otimes (2m-1)}&\simeq C_{4m}(2m-2,2m),{\text{ }}(s=-2)\\\end{aligned}}$

and the subalgebras $C^{+}$

${\begin{aligned}\mathbb {C} \otimes \mathbb {H} ^{\otimes (2m-1)}&\simeq C_{4m-1}(2m+1,2m-2),{\text{ }}(s=3)\\\mathbb {C} \otimes \mathbb {H} ^{\otimes (2m-2)}&\simeq C_{4m-3}(2m-2,2m-1),{\text{ }}(s=-1)\\\end{aligned}}$

All hyperquaternions have a definite signature $(p,q)$. Other signatures can be obtained by complexifying the generators, yielding split-hyperquaternions . The table below lists a few hyperquaternion algebras.

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

Due to the isomorphism $\mathbb {H} ^{\otimes 2}\simeq m(4,\mathbb {R} )$ where $m(4,\mathbb {R} )$ denotes the $4\times 4$ real matrices, hyperquaternions yield all real, complex and quaternionic square matrices. Furthermore, since $(\mathbb {H} ^{\otimes 2})^{*}\simeq [m(4,\mathbb {R} ]^{T}$ where $T$ is the matrix transposition, the hyperconjugation generalizes the concepts of matrix transposition, adjoint and transpose quaternion conjugate.

Generators

The $2^{n}$ generators $e_{\alpha }$ of $\mathbb {H} ^{\otimes n}$ can be chosen in various ways. One choice is

${\begin{aligned}e_{1}&=j\otimes 1\otimes \cdots \otimes 1{\text{ }}(n{\text{ terms}})\\e_{2}&=k\otimes i\otimes \cdots \otimes 1\\e_{3}&=k\otimes j\otimes \cdots \otimes 1\\&\cdots \\e_{2\beta }&=k\otimes \cdots \otimes k\otimes i\otimes 1\cdots \otimes 1\\e_{2\beta +1}&=k\otimes \cdots \otimes k\otimes j\otimes 1\cdots \otimes 1\\&\cdots \\e_{2n}&=k\otimes k\cdots \otimes k\\\end{aligned}}$

where $i,j$ stand at the $(\beta +1)$th place from the left with $0<\beta <n-1$. These generators anticommute among themselves and square to $\pm 1$.

The generators of the algebra $\mathbb {C} \otimes \mathbb {H} ^{\otimes n-1}$ which is the even subalgebra $\mathbb {C} ^{+}$ of $\mathbb {H} ^{\otimes n}$ can be defined as

$f_{1}=e_{1}e_{2},f_{2}=e_{1}e_{3},...f_{2n-1}=e_{1}e_{2n}$

The generators of a few hyperquaternions are given in the following table

Algebra	$C_{n}(p,q)$	Generators
$\mathbb {C}$	$C_{1}(0,1)$	$i$
$\mathbb {H}$	$C_{2}(0,2)$	$j,k$
$\mathbb {C} \otimes \mathbb {H}$	$C_{3}(3,0)$	$iI,iJ,iK$
$\mathbb {H} ^{\otimes 2}$	$C_{4}(3,1)$	$j,kI,kJ,kK$
$\mathbb {C} \otimes \mathbb {H} ^{\otimes 2}$	$C_{5}(2,3)$	$iI,iJ,iKl,iKm,iKn$
$\mathbb {H} ^{\otimes 3}$	$C_{6}(2,4)$	$j,kI,kJ,kKl,kKm,kKn$
$\mathbb {C} \otimes \mathbb {H} ^{\otimes 3}$	$C_{7}(5,2)$	$(iI,iJ,iKl,iKm,iKnL,iKnM,iKnN)$
$\mathbb {H} ^{\otimes 4}$	$C_{8}(5,3)$	$(j,kI,KJ,kKl,kKm,kKnL,kKnM,kKnN)$

The small $i,j,k$ stand for the first quaternionic system, the capital $I,J,K$ for the second one, $l,m,n$ for the third one $l=1\otimes 1\otimes i\otimes 1,etc.$ and the capital $L,M,N$ for the fourth one $(L=1\otimes 1\otimes 1\otimes i,etc.)$ ; all distinct quaternionic systems commuting with each other.

Multivector calculus

Interior and exterior products between two vectors $a,b$ can be defined by

$ab=\lambda a.b+\mu a\wedge b$

where $\lambda ,\mu$ are constant factors [p. 362]. Postulating $a.b=b.a,a\wedge b=b\wedge a$, one obtains

$2a.b=\lambda ^{-1}(ab+ba),2a\wedge b=\mu ^{-1}(ab-ba)$

In the examples below, $\lambda ,\mu \in (\pm 1)$. A full multivector calculus with $\lambda =\mu =1$ is developed in G. Casanova

Example 1: Quaternions $\mathbb {H}$

Having discovered the quaternion group in 1843, W. R. Hamilton was to spend much of his life to develop a 3D calculus. Quaternions were to be replaced by the vector calculus, still in use today. From a modern point of view, the quaternion algebra $\mathbb {H} \simeq C_{2}(0,2)$ being a Clifford algebra having two generators

$e_{1}=i,e_{2}=j,e_{1}e_{2}=k{\text{ }}(e_{1}^{2}=e_{2}^{2}=-1)$.

is appropriate for a 2D modeling. A general element of $\mathbb {H}$ 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 $(B)$. Interior and exterior products can be defined by

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

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

$x'=rxr^{-1}=(x_{1}cos\theta -x_{2}sin\theta )i+(x_{1}sin\theta +x_{2}cos\theta )j$

with

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

Example 2: Biquaternions $\mathbb {C} \otimes \mathbb {H}$

Hamilton introduced biquaternions as complex quaternions. During the $20^{th}$ century, biquaternions were often used in the special relativistic context . Yet, since $\mathbb {C} \otimes \mathbb {H} \simeq i\otimes (I,J,K)\simeq C_{3}(3,0)$, biquaternions are naturally suited for a 3D modeling. A general element $Q$ can be expressed as a set of two quaternions $Q=q+iq'$ with $q=a+bI+cJ+dK$ (similarly $q'$). The biquaternion product is given by

$Q_{1}Q_{2}=(q_{1}+iq'_{1})(q_{2}+iq'_{2})=(q_{1}q_{2}-q'_{1}q'_{2})+i(q'_{1}q_{2}+q_{1}q'_{2})$

The multivector structure is given by ${\begin{bmatrix}1&I=e_{3}e_{2}&J=e_{1}e_{3}&K=e_{2}e_{1}\\i=e_{1}e_{2}e_{3}&iI=e_{1}&iJ=e_{2}&iK=e_{3}\\\end{bmatrix}}$

and contains scalars $V_{0}$, vectors $V_{1}$, bivectors $V_{2}$ and trivectors $V_{3}$. Interior and exterior products are defined in the following table with the equivalents of the classical vector calculus (with $B=b\wedge c,B_{1}=a\wedge b,B_{2}=c\wedge d,T_{1}=a\wedge b\wedge c,T_{2}=f\wedge g\wedge h$ and $a,b,c,d,e,f,g,h\in V_{1},T\in V_{3}).$

Multivector calculus	Classical vector calculus
$a.b={\frac {ab+ba}{2}}\in V_{0}$	$a.b$
$a\wedge b=-{\frac {ab-ba}{2}}\in V_{2}$	$a\times b$
$a.B={\frac {aB-Ba}{2}}=a.(b\wedge c)\in V_{1}$	$a\times (b\times c)$
$a\wedge B=-{\frac {aB+Ba}{2}}=a\wedge b\wedge c\in V_{3}$	$a.(b\times c)$
$B_{1}.B_{2}=-{\frac {B_{1}B_{2}+B_{2}B_{1}}{2}}=(a\wedge b).(c\wedge d)\in V_{0}$	$(a\times b).(c\times d)$
$[B_{1},B_{2}]={\frac {B_{1}B_{2}-B_{2}B_{1}}{2}}=[a\wedge b,c\wedge d]\in V_{2}$	$(a\times b)\times (c\times d)$
$T_{1}.T_{2}=-{\frac {T_{1}T_{2}+T_{2}T_{1}}{2}}\in V_{0}$	$[a.(b\times c)][f.(g\times h)$
$B.T=-{\frac {BT+TB}{2}}\in V_{1}$
$a.T={\frac {aT+Ta}{2}}\in V_{2}$

The rotation group $SO(3)$ is expressed by $x'=rxr^{-1}$ with

$r=e^{u\theta /2}(u=u_{1}I+u_{2}J+u_{3}K,u^{2}=-1)\in C^{+}{\text{ }}(\simeq \mathbb {H} )$

and $x=x_{1}I+x_{2}J+x_{3}K$ and similarly $x'$.

A matrix representation of the biquaternions is obtained via the Pauli algebra

${\begin{aligned}e_{1}&=iI=\sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}\\e_{2}&=iJ=\sigma _{2}={\begin{bmatrix}0&-i'\\i'&0\end{bmatrix}}\\e_{3}&=iK=\sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}\\\end{aligned}}$

where $i'$ is the usual complex imaginary.

Example 3: Tetraquaternions $\mathbb {H} ^{\otimes 2}$

Since $\mathbb {H} ^{\otimes 2}\simeq C_{4}(3,1)$, this algebra allows a 4D relativistic modeling.

A general element $A$, called tetraquaternion is simply a set of four quaternions $A=q_{0}+iq_{1}+jq_{2}+kq_{3}={\begin{bmatrix}q_{0},\\q_{1},\\q_{2},\\q_{3}\end{bmatrix}}{\text{ }}(q_{i}\in \mathbb {H} )$ and similarly $A'$ with $q_{i}=\alpha _{i}+\beta _{i}I+\gamma _{i}J+\delta _{i}K$ (real coefficients). The product $AA'$ yields a set of four quaternions

$AA'={\begin{bmatrix}q_{0}q'_{0}-q_{1}q'_{1}-q_{2}q'_{2}-q_{3}q'_{3},\\q_{0}q'_{1}+q_{1}q'_{0}+q_{2}q'_{3}-q_{3}q'_{2},\\q_{0}q'_{2}+q_{2}q'_{0}+q_{3}q'_{1}-q_{1}q'_{3},\\q_{0}q'_{3}+q_{3}q'_{0}+q_{1}q'_{2}-q_{2}q'_{1}\end{bmatrix}}$.

The four generators are $e_{0}=j,e_{1}=kI,e_{2}=kJ,e_{3}=kK$. The multivector structure is

${\begin{aligned}{\begin{bmatrix}1&I=e_{3}e_{2}&J=e_{1}e_{3}&K=e_{2}e_{1}\\i=e_{0}e_{1}e_{2}e_{3}&iI=e_{0}e_{1}&iJ=e_{0}e_{2}&iK=e_{0}e_{3}\\j=e_{0}&jI=e_{0}e_{3}e_{2}&jJ=e_{0}e_{1}e_{3}&jK=e_{0}e_{2}e_{1}\\k=e_{1}e_{2}e_{3}&kI=e_{1}&kJ=e_{2}&kK=e_{3}\\\end{bmatrix}}\\\end{aligned}}$

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

If $A_{p}=a_{1}\wedge a_{2}\wedge ...\wedge a_{p}{\text{ }}(p\geq 1)$ denotes a multivector (where $a_{i}$ are vectors) and $x$ is a vector, the interior and exterior products are given by

${\begin{aligned}2x.A_{p}&=(-1)^{p}[xA_{p}-(-1)^{p}A_{p}x],\\2x\wedge A_{p}&=(-1)^{p}[xA_{p}+(-1)^{p}A_{p}x],\\\end{aligned}}$

${\begin{aligned}2B.B'&=S(BB'+B'B)\in S,2B\wedge B'=P(BB'+B'B)\in P,\\2B.P&=(BP+PB)\in B,2T.P=(TP-PT)\in V,\\2T.T'&=-(TT'+T'T)\in S,2P.P'=(TP-PT)\in S,\\\end{aligned}}$

where $S(),P()$ are the scalar and pseudoscalar part. An orthochronous proper Lorentz transformation is given by

$x'=axa^{-1},a\in C^{+}{\text{ }}(\simeq \mathbb {C} \otimes \mathbb {H} )$

with $x=cte_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}$ (similarly $x'$). A matrix representation is obtained via

${\begin{aligned}e_{0}&=j\otimes 1=j={\begin{bmatrix}0&0&-1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\\\end{bmatrix}},e_{1}=k\otimes i=kI={\begin{bmatrix}0&0&1&0\\0&0&0&-1\\1&0&0&0\\0&-1&0&0\\\end{bmatrix}},\\e_{2}&=k\otimes j=kJ={\begin{bmatrix}-1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}},e_{3}=k\otimes k=kK={\begin{bmatrix}0&0&0&-1\\0&0&-1&0\\0&-1&0&0\\-1&0&0&0\\\end{bmatrix}}.\\\end{aligned}}$

See also

• Classification of Clifford algebras
• Clifford algebra
• Hypercomplex number
• Quaternion