Differential Algebraic Methods in Composition Algebras: A Constructive Framework for Explicit Parameterizations and Geometric Analysis
shifa liu · Zenodo (CERN European Organization for Nuclear Research) · 2025
This paper develops a comprehensive differential algebraic framework for composition algebras,extending the constructive methods previously established for algebraic geometry and differential geometry. We define the differential composition algebraic closure KCompAlg through a recursive adjunction process that systematically incorporates norm-preserving maps, Cayley-Dickson doubling constructions, solutions to geometric partial differential equations on composition algebraic manifolds, and insights from recent advances in symmetric tensor powers of composition algebras [11] and functional equations in composition algebra [12]. Our main results include explicit parameterization theorems for the norm-one groups of composition algebras, complete with combinatorial correction terms derived from the algebraic structure and multiplication tables through a novel integration of tree enumeration and recursive trace formulas. We provide a complete classification of composition algebras within our enhanced framework, detailed constructions for quaternions and octonions with certified error bounds using interval arithmetic, and computational algorithms that preserve algebraic structure through careful management of non-associativity. The work establishes deep connections between composition algebra theory, differential algebra, geometric analysis, and modern algebraic geometry through the incorporation of octonion algebras over schemes [13]. Our combinatorial correction framework provides the first systematic treatment of non-associativity in parameterization problems with applications extending beyond composition algebras to other non-associative algebraic structures, while maintaining mathematical rigor through strengthened set-theoretic and functorial foundations.