Lie groups over non-discrete topological fields

Abstract

We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test function groups, diffeomorphism groups, and weak direct products of Lie groups. The specific tools of differential calculus required for the Lie group constructions are developed. Notably, we establish differentiability properties of composition and evaluation, as well as exponential laws for function spaces. We also present techniques to deal with the subtle differentiability and continuity properties of non-linear mappings between spaces of test functions. Most of the results are independent of any specific properties of the topological vector spaces involved; in particular, we can deal with real and complex Lie groups modeled on non-locally convex spaces.

128 pp, LaTeX

Cited by in corpus (25)

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• Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds
• Regularity properties of infinite-dimensional Lie groups, and semiregularity
• Finite order differentiability properties, fixed points and implicit functions over valued fields
• The diffeomorphism group of a non-compact orbifold
• Projective unitary representations of infinite dimensional Lie groups
• Implicit functions from topological vector spaces to Fréchet spaces in the presence of metric estimates
• A Geometric Approach to Noncommutative Principal Torus Bundles
• The Lie group of real analytic diffeomorphisms is not real analytic
• Continuity of LF-algebra representations associated to representations of Lie groups
• Free Group Actions from the Viewpoint of Dynamical Systems
• Differentiable mappings between spaces of sections
• Homotopy groups of ascending unions of infinite-dimensional manifolds
• Aspects of p-adic non-linear functional analysis
• Continuity of convolution of test functions on Lie groups
• Manifolds of absolutely continuous curves and the square root velocity framework
• Universal continuous bilinear forms for compactly supported sections of Lie algebra bundles and universal continuous extensions of certain current algebras
• Symmetry Groups of Principal Bundles Over Non-Compact Bases
• Solutions to open problems in Neeb's recent survey on infinite-dimensional Lie groups
• Difference Problems and Differential Problems
• Smoothing operators for vector-valued functions and extension operators
• A Lie group structure on the group of real analytic diffeomorphisms of a compact real analytic manifold with corners
• Ultrametric and non-locally convex analogues of the general curve lemma of convenient differential calculus
• Universal central extensions for groups of sections on non-compact manifolds
• Lie groups of real analytic diffeomorphisms are $L^1$-regular