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Kobi Peterzil will speak about the infinitesimal subgroup of a simple real Lie group.
(w. Martin Bays)
A theorem with Pillay and Starchenko (2000) says that If G is a simple linear Lie group then the pure group (G,.) is either:
(i) unstable, and then bi-interpretable with the real field, e.g when G is compact,
(ii) stable, and then bi-interpretable with the complex field.
Viewing G inside a big real closed field R, we let mu_G be the subgroup of infinitesimals of the identity, with respect to the archimedean valuation ring. Thus, the subgroup mu_G is definable in the associated Real Closed Valued Field (R,v) .
Theorem (w. Bays): For G compact and simple, the structure (mu_G,.) is bi-interpretable with (R,v).
I will discuss the setting and the proof of the above result. In addition, we propose the following generalization,
Conjecture: If G is a simple linear Lie group and (G,.) is unstable then mu_G is bi-interpretable with RCVF and if (G,.) is stable then mu_G is bi-interpretable with ACVF.