Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields

TL;DR

This paper develops an axiomatic framework for differential calculus, manifolds, and Lie groups over arbitrary infinite fields and rings, extending classical theories to more general algebraic settings.

Contribution

It introduces a unified axiomatic approach to differential calculus and Lie group theory over arbitrary infinite fields and rings, including non-Archimedean fields.

Findings

Established a general theory of differentiable mappings over non-discrete topological fields.

Extended classical differential calculus to non-locally convex spaces.

Unified treatment of manifolds and Lie groups over diverse algebraic structures.

Abstract

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces over non-discrete topological fields, in particular ultrametric fields or the fields of real and complex numbers. In the latter case, a theory of differentiable mappings between general, not necessarily locally convex spaces is obtained, which in the locally convex case is equivalent to Keller's C^k_c-theory.