Semialgebraic groups and generalized affine buildings

TL;DR

This paper develops the theory of algebraic groups over real closed fields, constructs a geometric object as an affine $ ext{Lambda}$-building, and generalizes classical Lie group decompositions using model theory and semialgebraic methods.

Contribution

It introduces a novel construction of affine $ ext{Lambda}$-buildings from semialgebraic groups over real closed fields, extending classical Lie theory to a broader algebraic setting.

Findings

Constructed a geometric $ ext{Lambda}$-building from algebraic groups over real closed fields.

Proved generalized decompositions (Iwasawa, Cartan, Bruhat) using model theoretic transfer.

Established semialgebraic versions of key Lie group results like Kostant's convexity.

Abstract

We develop the theory of algebraic groups over real closed fields and apply the results to construct a geometric object $\mathcal{B}$ and to prove that $\mathcal{B}$ is an affine $\Lambda$-building. We use a model theoretic transfer principle to prove generalizations of statements about semisimple Lie groups. In this direction we give proofs for the Iwasawa-decomposition $KAU$, the Cartan-decomposition $KAK$ and the Bruhat-decomposition $BWB$. For unipotent subgroups we prove the Baker-Campbell-Hausdorff formula and use it to analyse root groups. We give a proof of the Jacobson-Morozov Lemma about subgroups whose Lie algebra is isomorphic to $\mathfrak{sl}_2$ and we describe other rank 1 subgroups which are the semisimple parts of Levi-subgroups. We prove a semialgebraic version of Kostant's convexity. Over the reals, semisimple Lie groups are closely related to the symmetry groups of symmetric spaces of non-compact type. These symmetric spaces can be described semialgebraically, which allows us to consider their semialgebraic extension over any real closed field. Starting from these non-standard symmetric spaces we use a valuation (with image some non-discrete ordered abelian group $\Lambda$) on the fields to define a $\Lambda$-pseudometric. Identifying points of distance zero results in a $\Lambda$-metric space $\mathcal{B}$. Assuming that the root system of the associated Lie group is reduced, we prove that $\mathcal{B}$ is an affine $\Lambda$-building. The proof relies on a thorough analysis of stabilizers.