Hyperquaternion Conformal Groups

Abstract

The paper gives a new representation of conformal groups in n dimensions in terms of hyperquaternions defined as tensor products of quaternion algebras (or a subalgebra thereof). Being Clifford algebras, hyperquaternions provide a good representation of pseudo-orthogonal groups such as $O(p+1,q+1)$ isomorphic to the nD conformal group with $n=p+q.$ The representation yields simple expressions of the generators, independently of matrices or operators. The canonical decomposition and the invariants are discussed. As application, the 4D relativistic conformal group is detailed together with a worked example. Finally, the formalism is compared to the operator representation. Potential uses include in particular, conformal geometry, computer graphics and conformal field theory.

Appendices

Appendix A. Lie Algebra of the nD-Conformal Group

Consider an nD space embedded in an $n+2$ hyperquaternion algebra with the generators $e_{0},e_{1},\ldots ,e_{n},e_{n+1}.$ The Lie generators of the rotations, translations, transversions and dilations of the restricted conformal group are respectively

$$\begin{aligned} M_{ij}= & {} \frac{1}{2}e_{i}e_{j} \quad \left( 1\le i,j\le n,i\ne j\right) , \end{aligned}$$

(A.1)

$$\begin{aligned} P_{i}= & {} \left( \frac{e_{0}+e_{n+1}}{2}\right) e_{i},\quad \text { } K_{i}=\left( \frac{e_{n+1}-e_{0}}{2}\right) e_{i}, \end{aligned}$$

(A.2)

$$\begin{aligned} D= & {} \frac{e_{0}e_{n+1}}{2}. \end{aligned}$$

(A.3)

One derives easily the following Lie commutators $\left[ A,B\right] =AB-BA,$ with $\eta _{ij}=\left( e_{i}e_{j}+e_{j}e_{i}\right) /2.$

$$\begin{aligned} \left[ M_{ij},M_{kl}\right]= & {} \eta _{jk}\text { }M_{il}+\eta _{il}\text { } M_{jk}-\eta _{jl}\text { }M_{ik}-\eta _{ik}\text { }M_{jl}, \end{aligned}$$

(A.4)

$$\begin{aligned} \left[ M_{ij},P_{k}\right]= & {} \eta _{jk}P_{i}-\eta _{ik}\text { }P_{j},\quad \left[ M_{ij},K_{k}\right] =\eta _{jk}K_{i}-\eta _{ik}\text { }K_{j}, \end{aligned}$$

(A.5)

$$\begin{aligned} \left[ M_{ij},D\right]= & {} \left[ P_{i},P_{j}\right] =\left[ K_{i},K_{j} \right] =0, \end{aligned}$$

(A.6)

$$\begin{aligned} \left[ K_{i},P_{j}\right]= & {} 2\eta _{ij}D+M_{ij},\quad \left[ D,P_{i}\right] =-P_{i},\text { }\left[ D,K_{i}\right] =K_{i}. \end{aligned}$$

(A.7)

Appendix B. Multivector structure of ${\mathbb {H}}\otimes {\mathbb {H}}\otimes {\mathbb {H}}$

A general hyperquaternion A is a set of 64 terms which can be grouped into a set of 16 quaternions $\left[ q_{i}\right] =a_{i}+b_{i}l+c_{i}m+d_{i}n$ with respect to the sets ijk/IJK/lmn

$$\begin{aligned} A= & {} \left[ q_{1}\right] +I\left[ q_{2}\right] +J\left[ q_{3}\right] +K\left[ q_{4}\right] \nonumber \\&+ i\left[ q_{5}\right] +iI\left[ q_{6}\right] +iJ\left[ q_{7}\right] +iK \left[ q_{8}\right] \nonumber \\&+ j\left[ q_{9}\right] +jI\left[ q_{10}\right] +jJ\left[ q_{11}\right] +jK \left[ q_{12}\right] \nonumber \\&+ k\left[ q_{13}\right] +kI\left[ q_{14}\right] +kJ\left[ q_{15}\right] +kK \left[ q_{16}\right] \end{aligned}$$

(B.1)

yielding a multiplication table which can be implemented (algebraically or numerically) on Mathematica using its quaternion product $\left[ q_{i}\right] **\left[ q_{j}\right] .$ The complete multivector structure is given below, where $e_{0123}=e_{0}e_{1}e_{2}e_{3},$ etc..

$$\begin{aligned} \left[ \begin{array}{llll} 1 &{} l=e_{34} &{} m=e_{42} &{} n=e_{23} \\ I=e_{1234} &{} I l=e_{21} &{} I m=e_{31} &{} I n=e_{41} \\ J=e_{2034} &{} J l=e_{02} &{} J m=e_{03} &{} J n=e_{04} \\ K=e_{10} &{} Kl=e_{1034} &{} Km=e_{0124} &{} Kn=e_{1023} \\ i=e_{012345} &{} il=e_{1025} &{} im=e_{1035} &{} in=e_{1045} \\ iI=e_{50} &{} iI l=e_{3045} &{} iI m=e_{4025} &{} iI n=e_{2035} \\ iJ=e_{51} &{} iJ l=e_{3145} &{} iJ m=e_{4125} &{} iJ n=e_{2135} \\ iK=e_{2345} &{} iKl=e_{52} &{} iKm=e_{53} &{} iKn=e_{54} \\ j=e_{5} &{} jl=e_{345} &{} jm=e_{542} &{} jn=e_{235} \\ jI=e_{12345} &{} jI l=e_{215} &{} jI m=e_{531} &{} jI n=e_{541} \\ jJ=e_{32450} &{} jJ l=e_{025} &{} jJ m=e_{035} &{} jJ n=e_{045} \\ jK=e_{105} &{} jKl=e_{10345} &{} jKm=e_{01245} &{} jKn=e_{10235} \\ k=e_{10234} &{} kl=e_{012} &{} km=e_{013} &{} kn=e_{014} \\ kI=e_{0} &{} kI l=e_{034} &{} kI m=e_{204} &{} kI n=e_{023} \\ kJ=e_{1} &{} kJ l=e_{134} &{} kJ m=e_{214} &{} kJ n=e_{123} \\ kK=e_{324} &{} kKl=e_{2} &{} kKm=e_{3} &{} kKn=e_{4} \end{array} \right] \end{aligned}$$

(B.2)

A Mathematica notebook, concerning ${\mathbb {H}}\otimes {\mathbb {H}}\otimes {\mathbb {C}}$ is provided in [13].