Differential Algebraic Framework for Group Algebras: Explicit Parameterizations and Differential Equations
shifa liu · Zenodo (CERN European Organization for Nuclear Research) · 2025
This paper establishes a comprehensive differential algebraic framework for group algebras, extending methodologies from geometric analysis and algebraic geometry. We construct differential algebraic closures for group algebras and develop explicit, constructive parameterization methods for their elements. This framework provides unified solutions to differential equations defined on group algebras, with significant applications in representation theory, harmonic analysis, and mathematical physics. We establish rigorous existence and uniqueness theorems, develop computational algorithms accompanied by certified error bounds, and provide detailed,verified examples encompassing cyclic groups, symmetric groups, and compact Lie groups. The work bridges differential algebra, group theory, and non-commutative geometry while maintaining the highest standards of mathematical rigor, including formal verification and validated numerics. A key innovation is the categorical for mulation of differential structures and the integration of formal verification methods throughout the theoretical development.