Extensions of definable local homomorphisms in o-minimal structures and semialgebraic groups

• Published in: arXiv.org
• Main Author: Barriga, Eliana
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 25.01.2021
• Subjects: Algebra; Fields (mathematics); Group theory; Homomorphisms; Subgroups; Theorems

We state conditions for which a definable local homomorphism between two locally definable groups \(\mathcal{G}\), \(\mathcal{G^{\prime}}\) can be uniquely extended when \(\mathcal{G}\) 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 \(\widetilde{G}\) of \(G\) is an open locally definable subgroup of \(\widetilde{H\left(R\right)^{0}}\) for some \(R\)-algebraic group \(H\) (Thm. 3.3). Finally, for an abelian definably connected semialgebraic group \(G\) over \(R\), we describe \(\widetilde{G}\) as a locally definable extension of subgroups of the o-minimal universal covering groups of commutative \(R\)-algebraic groups (Theorem 3.4)