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Let $k$ be an algebraically closed field. By a finitely generated subfield of $k$ I mean a subfield $k_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ …

In the book Lie groups and algebraic groups by Onishchik and Vinberg, Theorem 3 in Section 5.2.1 on page 240 says: Let $S$ be a real structure on a simply connected complex semisimple Lie group $G$. …

I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved. Let $k$ be a perfect field and $k^s$ its fixed separable closure. Let $X^s$ be a variety …

I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed. …

Question 1. Does Élie Cartan's paper Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355 contain a classification of $\Bbb C$-linear involutions of simple …

I answer Question 1. It is just a calculation. Instead of a real torus, say ${\bf T}$, I consider a pair $(T,\sigma)$, where $T$ is a complex torus and $\sigma\colon T\to T$ is an anti-holomorphic inv …

In addition to Marty's reference, I would recommend to look into §6 "Le groupe fondamental algébrique des groupes algébriques linéaires connexes via les resolutions flasques" of Colliot-Thélène's pape …

I asked this basic question in MSE and got a comment "This belongs to Mathoverflow", so I ask my question here. Let $G$ be a linear algebraic group over a field $k$, and $H\subset G$ be a $k$-sub …

$\newcommand{\Tors}{{\rm Tors}} \newcommand{\tf}{{\rm\, t.f.}} \newcommand{\Gt}{{\Gamma\!,\,\Tors}} \newcommand{\Gtf}{{\Gamma\!,\tf}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\Z}{{\mathbb Z}} \newcomm …

Let ${\frak g}$ be a simple Lie algebra over $\Bbb C$, and let $\theta$ be an inner involution of ${\frak g}$, that is, an inner automorphism of ${\frak g}$ of order dividing 2. Such automorphisms are …

A reference: Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92, Corollary 3.11. In positive characteristic $p$: see loc. cit., Theorem 3.14. It says that if (and only if) $ …

Let $R$ be an irreducible root system with a basis $\Pi$. We obtain the Dynkin diagram $D$ and the extended Dynkin diagram ${\widetilde{D}}$ of $R$ with respect to $\Pi$. Let $Q^\vee\subset P^\vee$ de …

If you are interested in algebraic groups over complex and real numbers only, try Onishchik and Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag 1990. This book contains also representation t …

Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$. We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{ …

This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763. It got upvotes, but no answers or comments, and so I ask it here. Let $G$ …