Lie-Differential Algebraic Closure: A Unified Framework for Structure and Representation Theory of Lie Algebras

This paper establishes a comprehensive Lie-differential algebraic framework that extends the Hopf-differential closure theory to the realm of Lie algebras. Our main innovation is the construction of the Lie-differential algebraic closure Kg of a Lie algebra g, which provides explicit solutions to structural equations and representation-theoretic problems. We prove that all finite-dimensional representations of g can be explicitly constructed within this closure, with a unified solution formula:n−1 ρ(x)vk = λ(n−1)(x)+m=1 Φm(y(x))1/nωm(k−1)n vk, 0≤k≤n−1,where λ(n−1) is the average eigenvalue, y = (y(0),...,y(n−2)) are Lie critical values, Φm ∈ U(g)[y] are explicit Lie-polynomials with combinatorial correction terms, and ωn is a primitive n-th root of unity. Our framework provides: • Explicit solution formulas for Lie-algebraic equations that transcend classical limitations • Unified treatment of symmetry principles through Lie-Fourier transforms • Deep connections with Tannaka-Krein duality and differential cohomology • Practical algorithms with rigorous complexity bounds (O(n3) for degree-n equations) • Physically relevant applications to gauge theories and integrable systems Supplementary Material File (lie.pdf) - Download - 623.41 KB Information & Authors Information Version history Copyright This work is licensed under a Creative Commons Attribution 4.0 International License

Authors Metrics & Citations Metrics Article Usage 304views 156downloads Citations Download citation Dongqi Liu, shifa liu. Lie-Differential Algebraic Closure: A Unified Framework for Structure and Representation Theory of Lie Algebras. Authorea. 17 October 2025. DOI: https://doi.org/10.22541/au.176072430.03749703/v1 DOI: https://doi.org/10.22541/au.176072430.03749703/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu.