Elementary equivalence and Diophantine problem of Affine Kac-Moody groups

The paper is devoted to model-theoretic properties of Kac-Moody groups with the focus on elementary equivalence of Kac-Moody groups. We show that elementary equivalence of (untwisted) affine Kac-Moody groups implies coincidence of their generalized Cartan matrices and the elementary equivalence of their ground fields. We study also the Diophantine problem in affine Kac-Moody groups. We show that for the loop group the Diophantine problem is polynomial time equivalent (more precisely, Karp equivalent) to the problem in the ground ring. Finally, we show that in affine Kac-Moody groups over finite fields the Diophantine problem is undecidable.