### Abstract

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 F_{4} is constructed in terms of 24 integral and 24 'half integral' quaternions. The root lattice of E_{8} expressed as 240 integral octonions are obtained by pairing two sets of roots of F_{4}. 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), F_{4} and E_{8}. The roots of maximal subgroups SO(16), E_{7}*SU(2), E_{6}*SU(3), SU(9) and SU(5)*SU(5) of E_{8} are identified with a simple method. Subsets of the discrete subgroup of SU(2) leaving maximal subgroups of E_{8} are obtained. Constructions of E_{8} 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.

Original language | English |
---|---|

Article number | 006 |

Pages (from-to) | 1469-1493 |

Number of pages | 25 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 22 |

Issue number | 10 |

DOIs | |

Publication status | Published - 1989 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Journal of Physics A: Mathematical and General*,

*22*(10), 1469-1493. [006]. https://doi.org/10.1088/0305-4470/22/10/006

**Division algebras with integral elements.** / Koca, M.; Ozdes, N.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 22, no. 10, 006, pp. 1469-1493. https://doi.org/10.1088/0305-4470/22/10/006

}

TY - JOUR

T1 - Division algebras with integral elements

AU - Koca, M.

AU - Ozdes, N.

PY - 1989

Y1 - 1989

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0345983836&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0345983836&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/22/10/006

DO - 10.1088/0305-4470/22/10/006

M3 - Article

VL - 22

SP - 1469

EP - 1493

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 10

M1 - 006

ER -