WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 24, 2025
Infinite-Dimensional Lie Groups on Nachbin-Weighted Spaces
Authors: ,
Abstract: This paper investigates the construction of infinite-dimensional Lie group structures on weighted
function spaces of the form CW $$(\mathbb{R}, \mathfrak{t})$$, where $$\mathfrak{t}$$ is a topological locally convex Lie algebra. Building on foundational
work on Nachbin-weighted spaces of continuous functions, we establish criteria under which CW $$(\mathbb{R}, \mathfrak{t})$$ admits a
topological locally convex Lie algebra structure. Specifically, for a Nachbin family W of weights on $$\mathbb{R}$$ satisfying
W WW, we show that the pointwise Lie bracket induces a well-defined topological Lie algebra structure
on CW $$(\mathbb{R}, \mathfrak{t})$$. Furthermore, when $$\mathfrak{t}$$ is the Lie algebra of a Banach Lie group G, we construct an associated Lie
group structure on the subgroup of $$G^{\mathbb{R}}$$ 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.
Search Articles
Keywords: Weighted function spaces, Infinite-dimensional Lie groups, Nachbin families, Locally convex Lie
algebras, Banach Lie groups
Pages: 631-634
DOI: 10.37394/23206.2025.24.62