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Journal of Lie Theory 26 (2016), No. 4, 1069--1077
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.


A toroidal affine Nash group is the affine Nash group analogue of an anti-affine 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, anti-affine algebraic groups, Rosenlicht's theorem, affine Nash groups, abelian groups.

MSC: 22E15, 14L10, 14P20

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