arXiv subjects

G. P. Wilmot

Publications and source records attributed to G. P. Wilmot.

4 recordsLinked to original sources

arXiv2026

Clifford Algebra Calibration Post-Quantum Cryptography

The double lock scenario allows a secret to be transferred from one person to another without having previously exchanged secrets such as private/public keys. This can be realised by rotations of generalised calibrations in Clifford algebra. Calibrations map to Cayley-Dickson algebras and generalising this provides the mathematical structure enabling the enormous code space needed for a cryptosystem. Commuting rotations in large dimensional spaces are used but the complexity comes from the message being embedded in generalised calibrations. This system uses ideals of Clifford algebras and quasi-algebras that are sometimes embedded in Cayley-Dickson algebras making it a candidate for post-quantum cryptography. Understanding whether the new mathematics of generalised calibrations can produce a secure cryptosystem is the challenge presented in this manuscript.

math.RA↗

arXiv2026

Automorphisms of Sedenions

This paper extends the octonion calibration in Clifford algebra to two related sedenion calibrations. Associative calibrations map quaternion rings in Clifford algebra to those in Cayley-Dickson algebras, with octonions consisting of seven rings and sedenions having 35 rings. A new non-associative calibration is found that is related to the sedenion associative calibration but is invertible, providing a classification of the ideals of the even sub-algebra of Clifford algebra. This leads to subalgebras of certain Clifford algebras and a thorough analysis of the possible automorphisms of sedenions is applied using the Spin and Pin groups. While the calibrations and automorphsms of octonions have minimal divergence, it is found sedenions introduce new algebras and this work clarifies the discrepancy in the results of Schafer and Brown on the automorphisms of sedenions. The non-associative calibrations are found to be an infinite series that matches the finite geometry PG(N,2) series, discovered by Gino Fano, but since this series can be derived from simplices the name Fano hyper-volumes is suggested. The visual representation of sedenions as the 3-D Fano volume, representing the 15 Fano planes of sedenions, allows the power-associative rings to be visually distinguished from the octonion non-associative rings. These are more than loops because the power-associative rings of sedenions contain zero divisors. The visual representation of sedenions may be important for particle physics and Clifford algebra provides analysis tools for the PG(N,2) series that uncovers the hidden structure of Cayley-Dickson algebras within an associative algebra.

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arXiv2026

Structure of the Cayley-Dickson algebras

Viewing the Cayley-Dickson process as a graded construction provides a rigorous definition of associativity consisting of three classes and the non-associative parts dividing into four types. These simplify the Moufang loop identities and Mal'cev's identity, which identifies the non-associative Lie algebra structure. Analysing the non-associativity structure uncovers 3-cycles that distinguish between the Moufang identities and are used to identify three power-associative subalgebras of sedenions and higher level Cayley-Dickson algebras. Power-associativity introduces zero divisors into Cayley-Dickson algebras in a systematic way and it is convenient to replace the terminology hypercomplex numbers with {\it ultracomplex numbers} for the power-associative algebras. The non-associative types show that zero divisors in these algebras occur in multiples of 84 and cycles and modes are uncovered that reduce these down to factors of seven primary zero divisor pairs. It is shown that this is due to the power-associative subalgebras being embedded into ultracomplex numbers in multiples of seven. The graded notation allows the eight octonion and seven power-associative subalgebras of sedenions to be uniquely derived, up to representation. The zero divisors for split sedenion algebras are analysed and mappings between three of these are provided. These split algebras are shown to involve the same power-associative subalgebras as sedenions.

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arXiv2025

Construction of exceptional Lie algebra G2 and non-associative algebras using Clifford algebra

This article uses Clifford algebra of definite signature to derive octonions and the Lie exceptional algebra G2 from calibrations using Pin(7). This is simpler than the usual exterior algebra derivation and uncovers a subalgebra of Spin(7) that enables G2 and an invertible element used to classify six other algebras which are found to be related to the symmetries of G2 in a way that breaks the symmetry of octonions. The 4-form calibration terms of Spin(7) are related to an ideal with three idempotents and provides a direct construction of G2 for each of the 480 representations of the octonions. Clifford algebra thus provides a new construction of G2 without using the Lie bracket. This result is extended to 15 dimensions generating another 100 algebras as well as the sedenions.

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