A 2X2 Matrix Approach to Hopf Fibrations: Quaternionic Reduction of Octonionic Bi-unitary Matrices and an RLLR/LRRL Bracketing Rule
Abstract
We develop a common two-by-two matrix framework for the four classical Hopf fibrations associated with the real, complex, quaternionic and octonionic numbers. In all four cases the Hopf map is the image of the north pole of a sphere of trace-free Hermitian matrices under conjugation by a bi-unitary matrix, and it depends only on the first column of that matrix, without any bracketing convention, even for the non-associative octonions. Our main results concern general octonionic bi-unitary matrices, beyond the compatible matrices, with entries in a single complex subalgebra, on which earlier work on octonionic Lorentz transformations relies. We prove that every octonionic bi-unitary matrix is in fact quaternionic: its entries lie in a quaternion subalgebra depending on the matrix. We show that requiring Hermitian output leaves exactly four inequivalent ways of bracketing the octonionic triple products, among them an interlaced rule called RLLR/LRRL, and that each of them splits into an ordinary five-dimensional rotation and a four-dimensional scaling governed by a non-commutative determinant. This determines exactly when the evaluation preserves the geometry of the sphere, shows that it may contract or even collapse four dimensions, and explains why it cannot be compatible with matrix multiplication. We also recover in compact form, through the Clifford map, the known fact that suitable compatible matrices, which we interpret as transvections of the eight-sphere, generate the spin group Spin(9) by nested evaluations. Non-associativity thus does not prevent a two-by-two octonionic description of the full Spin(9) symmetry; it prevents that description from being realised through unrestricted multiplication of octonionic matrices.
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BibTeX
@misc{gazeau2026matrix,
title = {{A 2X2 Matrix Approach to Hopf Fibrations: Quaternionic Reduction of Octonionic Bi-unitary Matrices and an RLLR/LRRL Bracketing Rule}},
author = {Jean-Pierre Gazeau and Aneta Sliżewska},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26975},
eprint = {2610.26975},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05842477v1},
}
RIS
TY - UNPB
TI - A 2X2 Matrix Approach to Hopf Fibrations: Quaternionic Reduction of Octonionic Bi-unitary Matrices and an RLLR/LRRL Bracketing Rule
AU - Jean-Pierre Gazeau
AU - Aneta Sliżewska
PY - 2026
DA - 2026/10/18
PB - HAL
DO - 10.48550/arxiv.2610.26975
UR - https://hal.science/hal-05842477v1
AB - We develop a common two-by-two matrix framework for the four classical Hopf fibrations associated with the real, complex, quaternionic and octonionic numbers. In all four cases the Hopf map is the image of the north pole of a sphere of trace-free Hermitian matrices under conjugation by a bi-unitary matrix, and it depends only on the first column of that matrix, without any bracketing convention, even for the non-associative octonions. Our main results concern general octonionic bi-unitary matrices, beyond the compatible matrices, with entries in a single complex subalgebra, on which earlier work on octonionic Lorentz transformations relies. We prove that every octonionic bi-unitary matrix is in fact quaternionic: its entries lie in a quaternion subalgebra depending on the matrix. We show that requiring Hermitian output leaves exactly four inequivalent ways of bracketing the octonionic triple products, among them an interlaced rule called RLLR/LRRL, and that each of them splits into an ordinary five-dimensional rotation and a four-dimensional scaling governed by a non-commutative determinant. This determines exactly when the evaluation preserves the geometry of the sphere, shows that it may contract or even collapse four dimensions, and explains why it cannot be compatible with matrix multiplication. We also recover in compact form, through the Clifford map, the known fact that suitable compatible matrices, which we interpret as transvections of the eight-sphere, generate the spin group Spin(9) by nested evaluations. Non-associativity thus does not prevent a two-by-two octonionic description of the full Spin(9) symmetry; it prevents that description from being realised through unrestricted multiplication of octonionic matrices.
ER -