Submission date: 17 January 2021

Abstract:

We state conditions for which a definable local homomorphism between two locally definable groups G, G' can be uniquely extended when G is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Thm. 9.1] (see Corollary 2.2). We also prove that Theorem 10.2 in [3] also holds for any definably connected definably compact semialgebraic group G not necessarily abelian over a sufficiently saturated real closed field R; namely, that the o-minimal universal covering group G˜ of G is an open locally definable subgroup of {H(R)^0}˜ for some R-algebraic group H (Thm. 3.3). Finally, for an abelian definably connected semialgebraic group G over R, we describe G˜ as a locally definable extension of subgroups of the o-minimal universal covering groups of commutative R-algebraic groups (Theorem 3.4)

Mathematics Subject Classification: 03C64, 20G20, 22E15, 03C68, 22B99

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Full text arXiv 2101.06782: pdf, ps.