The Baker-Campbell-Hausdorff formula and the Zassenhaus formula in synthetic differential geometry

TL;DR

This paper extends the Baker-Campbell-Hausdorff and Zassenhaus formulas within the framework of synthetic differential geometry, connecting classical Lie group theory with modern geometric approaches.

Contribution

It introduces the formulations of these formulas in synthetic differential geometry, providing a new perspective on Lie group structures.

Findings

Formulation of BCH formula in synthetic differential geometry

Derivation of Zassenhaus formula in the same framework

Bridges classical Lie theory with synthetic differential geometry

Abstract

After the torch of Anders Kock [Taylor series calculus for ring objects of line type, Journal of Pure and Applied Algebra, 12 (1978), 271-293], we will establish the Baker-Campbell-Hausdorff formula as well as the Zassenhaus formula in the theory of Lie groups.