Some model theory of compact Lie groups
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 by Ali Nesin and Anand Pillay PDF
 Trans. Amer. Math. Soc. 326 (1991), 453463 Request permission
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 nonAbelian 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 $\mathbb {R}$.References

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Additional Information
 © Copyright 1991 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 326 (1991), 453463
 MSC: Primary 03C40; Secondary 03C60, 22E15
 DOI: https://doi.org/10.1090/S00029947199110029229
 MathSciNet review: 1002922