Pairing two elements of a given division algebra furnished with a multiplication rule leads to an algebra of higher dimension restricted by 8. This fact is used to obtain the roots of SO(4) and SP(2) from the roots -or+1 of SU(2) and the weights -or+1/2 of its spinor representation. The root lattice of SO(8) described by 24 integral quaternions are obtained by pairing two sets of roots of SP(2). The root system of F4 is constructed in terms of 24 integral and 24 'half integral' quaternions. The root lattice of E8 expressed as 240 integral octonions are obtained by pairing two sets of roots of F4. Twenty four integral quaternions of SO(8) forming a discrete subgroup of SU(2) are shown to be the automorphism group of the root lattices of SO(8), F4 and E8. The roots of maximal subgroups SO(16), E7*SU(2), E6*SU(3), SU(9) and SU(5)*SU(5) of E8 are identified with a simple method. Subsets of the discrete subgroup of SU(2) leaving maximal subgroups of E8 are obtained. Constructions of E8 root lattice with integral octonions in seven distinct ways are made. Magic squares of integral lattices of Goddard, Nahm, Olive, Ruegg and Schwimmer are derived. Possible physical applications are suggested.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics