
Journal of Lie Theory 26 (2016), No. 4, 10691077 Copyright Heldermann Verlag 2016 Toroidal Affine Nash Groups Mahir Bilen Can Dept. of Mathematics, Tulane University, 6823 St. Charles Ave, New Orleans, LA 70118, U.S.A. mcan@tulane.edu [Abstractpdf] A toroidal affine Nash group is the affine Nash group analogue of an antiaffine algebraic group. In this note, we prove analogues of Rosenlicht's structure and decomposition theorems: (1) Every affine Nash group $G$ has a smallest normal affine Nash subgroup $H$ such that $G/H$ is an almost linear affine Nash group, and this $H$ is toroidal. (2) If $G$ is a connected affine Nash group, then there exist a largest toroidal affine Nash subgroup $G_{\rm ant}$ and a largest connected, normal, almost linear affine Nash subgroup $G_{\rm aff}$. Moreover, we have $G=G_{\rm ant}G_{\rm aff}$, and $G_{\rm ant}\cap G_{\rm aff}$ contains $(G_{\rm ant})_{\rm aff}$ as an affine Nash subgroup of finite index. Keywords: Real algebraic groups, antiaffine algebraic groups, Rosenlicht's theorem, affine Nash groups, abelian groups. MSC: 22E15, 14L10, 14P20 [ Fulltextpdf (294 KB)] for subscribers only. 