We develop a comprehensive theory of Lie algebraic meta-operational mathematics, extending the earlier frameworks of meta-operational mathematics, operator-algebraic meta-operations, and Hopf algebraic meta-operations to the setting of Lie algebras and their duals.Twelve axioms (L1–L12) are introduced to characterize the space L = C∞(g,g) of smooth operations on a convenient Lie algebra g. The endomorphism operad Pg is constructed, and it is shown that its unary part carries a natural Lie algebra structure,with primitive elements classified as derivations. A rigorous bornological convergence theory is developed, including Mackey–Cauchy equivalence, completeness, and integral representations of fractional derivations.Exponential and logarithm meta-operations are used to define fractional iterates f◦t for t ∈ C, and their analytic continuation is studied, revealing logarithmic branch points at negative integers and a natural boundary. The dynamics of self-action is analysed, introducing the notion of non-idempotency degree and a criterion for collapse in weighted one-parameter families. Hopf–Lie algebraic structures (coproduct, counit, antipode) are constructed, and a morphism Φg to the Connes–Kreimer Hopf algebra is established, interpreting renormalized path integrals as the counit of this morphism. Applications to noncommutative geometry include Lie-algebraic spectral triples, stability under bornological limits, and an index theorem for the noncommutative torus. Categorification yields a strict 2-category 2LieMetOp and an (∞,1)-operad LieMetOp∞ via the dendroidal nerve. Classical Lie theoretic objects–Weyl reflections, Cartan matrices, and Hasse–Weil zeta functions–are reinterpreted within the meta-operational framework, leading to a continuous version of the zeta function and an equivalent for mulation of the Riemann hypothesis. Numerical algorithms with optimal complexity and rigorous error bounds are provided. All conjectures and open problems from the original research program are proved as theorems; a few related open directions are listed in Chapter 12. This work provides a unified language for Lie theory, quantum field theory, noncommutative geometry, and higher category theory.
Liu S (Wed,) studied this question.