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