Infinite-Dimensional Lie Groups on Nachbin-Weighted Spaces

This paper investigates the construction of infinite-dimensional Lie group structures on weighted function spaces of the form CW (R, k), where k is a topological locally convex Lie algebra.Building on foundational work on Nachbin-weighted spaces of continuous functions, we establish criteria under which CW (R, k) admits a topological locally convex Lie algebra structure.Specifically, for a Nachbin family W of weights on R satisfying W W W , we show that the pointwise Lie bracket induces a well-defined topological Lie algebra structure on CW (R, k).Furthermore, when k is the Lie algebra of a Banach Lie group G, we construct an associated Lie group structure on the subgroup of G R generated by exponentials of g-valued weighted functions.This structure is realized via the Baker-Campbell-Hausdorff formula, with analyticity of group operations established through composition operators and local arguments on weighted spaces.Our results extend classical finite-dimensional Lie theory to a broad class of weighted function spaces, emphasizing the role of admissibility conditions on weights and compatibility with infinite-dimensional Lie-theoretic frameworks..