Some model theory of compact Lie groups

Abstract:

We consider questions of first order definability in a compact Lie group $G$. Our main result is that if such $G$ is simple (and centerless) then the Lie group structure of $G$ is first order definable from the abstract group structure. Along the way we also show (i) if $G$ is non-Abelian and connected then a copy of the field $\mathbb {R}$ is interpretable. in $(G, \cdot )$, and (ii) any "$1$-dimensional" field interpretable in $(\mathbb {R}, +, \cdot )$ is definably (i.e., semialgebraically) isomorphic to the ground field $\mathbb {R}$.