# Definability in the group of infinitesimals of a compact Lie group

## TL;DR

This paper establishes bi-interpretability results linking the infinitesimal subgroups of compact Lie groups and definably compact groups in o-minimal structures to real closed valued fields, revealing their algebraic structure.

## Contribution

It proves that the infinitesimal subgroup of a simple compact Lie group is bi-interpretable with a real closed valued field, extending to definably compact groups in o-minimal structures.

## Key findings

- Infinitesimal subgroup of simple compact Lie groups is bi-interpretable with real closed valued fields.
- Infinitesimal subgroup of definably compact groups decomposes into a Q-vector space and finitely many real closed valued fields.
- Every definable field in a real closed convexly valued field is definably isomorphic to the field itself.

## Abstract

We show that for $G$ a simple compact Lie group, the infinitesimal subgroup $G^{00}$ is bi-intepretable with a real closed valued field. We deduce that for $G$ an infinite definably compact group definable in an o-minimal expansion of a field, $G^{00}$ is bi-interpretable with the disjoint union of a (possibly trivial) $\mathbb{Q}$-vector space and finitely many (possibly zero) real closed valued fields. We also describe the isomorphisms between such infinitesimal subgroups, and along the way prove that every {\em definable} field in a real closed convexly valued field $R$ is definably isomorphic to $R$.