The inverse of the hyperquaternion, M 4 (H)
Abstract
The hyperquaternion H ⊗ H ⊗ H, of dimension sixty-four, is the algebra M 4 (H) of quaternionic 4×4 matrices and the Clifford algebra Cl(5, 1) = Cl(6, 0) = Cl(2, 4) = Cl(1, 5); a companion paper gave the inverse of the tetraquaternion T = H⊗H in closed form, sixteen cubic components over a quartic. We ask whether the same can be done in sixty-four dimensions. It can, in three structural forms, and it cannot as a printed list of components. The element is a 4 × 4 matrix whose sixteen entries are signed sums of four of its sixteen quaternions, and the inverse is given by the quasideterminants of Gelfand and Retakh, sixteen expressions of 82 terms with quaternion inverses, the same recursion as in the companion paper, now with sixteen distinct entries; the denominator of everything is the Study determinant Det, which is real, never negative, of degree eight, the product of the norms of the four pivots of the elimination, and the square of the Dieudonné determinant. The inverse is also the polynomial
)/ Det of degree seven whose coefficients are polynomials in the eight scalar parts ⟨R k ⟩ 0 , written out; and, for the element read as a quaternion whose four components are tetraquaternions, a rational function of tetraquaternion inverses, each of them the closed form of the companion paper. What cannot be done is the expansion: Det has 1,726,464 monomials in the sixty-four entry variables and one adjugate entry 413,952, against 196 and 31 in sixteen dimensions. The singular set has codimension four, GL(4, H) is connected, and the conjugate is the inverse exactly on the compact Sp(4). The question of the companion paper, whether the element of a body can be a zero divisor, is then asked again. The tetraquaternion sits inside with Det = r 2 , so its cone r = 0 becomes a double zero without sign, and near the cone Det is a sum of four squares, the change of r on the weak singlet and the same linear form on the weak triplet; the scalar quaternion, the Higgs doublet of the unified theory, has Det = |q| 8 and is singular only in the symmetric phase; the singular sheets of a body are Krein collisions of two eigenvalues, and a vacuum component on the transverse isospin axis lifts them at second order, Det = b 2 |v T ηv| 2 (σ 1 σ 2 σ 3 ) 2 , the light cone of the origin included, while the massless ray through the origin is lifted only at eighth order, Det = b 8 . The vacuum does not remove the singular states of a body: it moves them off the cone to a set of codimension three in the eleven-dimensional state space, on which the body alone is invertible and which a worldline does not meet in general.
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BibTeX
@misc{reynaud2026inverse,
title = {{The inverse of the hyperquaternion, M 4 (H)}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749771v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The inverse of the hyperquaternion, M 4 (H)
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749771v2
AB - The hyperquaternion H ⊗ H ⊗ H, of dimension sixty-four, is the algebra M 4 (H) of quaternionic 4×4 matrices and the Clifford algebra Cl(5, 1) = Cl(6, 0) = Cl(2, 4) = Cl(1, 5); a companion paper gave the inverse of the tetraquaternion T = H⊗H in closed form, sixteen cubic components over a quartic. We ask whether the same can be done in sixty-four dimensions. It can, in three structural forms, and it cannot as a printed list of components. The element is a 4 × 4 matrix whose sixteen entries are signed sums of four of its sixteen quaternions, and the inverse is given by the quasideterminants of Gelfand and Retakh, sixteen expressions of 82 terms with quaternion inverses, the same recursion as in the companion paper, now with sixteen distinct entries; the denominator of everything is the Study determinant Det, which is real, never negative, of degree eight, the product of the norms of the four pivots of the elimination, and the square of the Dieudonné determinant. The inverse is also the polynomial )/ Det of degree seven whose coefficients are polynomials in the eight scalar parts ⟨R k ⟩ 0 , written out; and, for the element read as a quaternion whose four components are tetraquaternions, a rational function of tetraquaternion inverses, each of them the closed form of the companion paper. What cannot be done is the expansion: Det has 1,726,464 monomials in the sixty-four entry variables and one adjugate entry 413,952, against 196 and 31 in sixteen dimensions. The singular set has codimension four, GL(4, H) is connected, and the conjugate is the inverse exactly on the compact Sp(4). The question of the companion paper, whether the element of a body can be a zero divisor, is then asked again. The tetraquaternion sits inside with Det = r 2 , so its cone r = 0 becomes a double zero without sign, and near the cone Det is a sum of four squares, the change of r on the weak singlet and the same linear form on the weak triplet; the scalar quaternion, the Higgs doublet of the unified theory, has Det = |q| 8 and is singular only in the symmetric phase; the singular sheets of a body are Krein collisions of two eigenvalues, and a vacuum component on the transverse isospin axis lifts them at second order, Det = b 2 |v T ηv| 2 (σ 1 σ 2 σ 3 ) 2 , the light cone of the origin included, while the massless ray through the origin is lifted only at eighth order, Det = b 8 . The vacuum does not remove the singular states of a body: it moves them off the cone to a set of codimension three in the eleven-dimensional state space, on which the body alone is invertible and which a worldline does not meet in general.
ER -