Search Results

$\newcommand{\nN}{{\mathcal N}} \newcommand{\SL}{{\rm SL}} \newcommand{\G}{{\bf G}} \newcommand{\Q}{{\Bbb Q}} \newcommand{\R}{{\Bbb R}} \newcommand{\C}{{\Bbb C}} $The answer is No. Write $\nN$ for the …

$\newcommand{\Hom}{{\rm Hom}} \newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}} \newcommand{\X}{{\sf X}} $ I am looking for a reference for the following lemma (for which I know a proof): Lemma. Let $\var …

$ \newcommand{\g}{{\mathfrak g}} \newcommand{\h}{{\mathfrak h}} \newcommand{\t}{{\mathfrak t}} \newcommand{\C}{{\mathbb C}} \newcommand{\Ad}{{\rm Ad}} \newcommand{\ad}{{\rm ad}} $Theorem. Let $G$ be …

$\DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\ad}{ad} \DeclareMathOperator{\Lie}{Lie} \newcommand{\g}{{\mathfrak g}} \newcommand{\z}{{\mathfrak z}} \newcommand{\s}{{\mathfrak s}} \newcommand{\O …

No. Take $G={\rm SL}(2,{\Bbb R})$, $\ H=\{\,h(\lambda)={\rm diag}(\lambda, \lambda^{-1})\ |\ \lambda\in {\Bbb R}, \lambda>0\,\}$, $$U=\bigg\{ u(a)= \begin{pmatrix} 1 &a\\ 0&1 \end{pmatrix}\ \ \bigg|\ …

The answer is YES. Theorem 1. Let $G$ be a connected semisimple linear algebraic group defined over a number field $k\subset{\mathbb{R}}$. There exists a natural number $d=d(G_{\bar{k}})$ with the fo …

(EDITED taking in account the comment of Yves.) The group denoted in the question by $SU(2,1)(K)$ is the group of $\mathbf{Q}$-points $G(\mathbf{Q})$ for a suitable $\mathbf{Q}$-group $G$, and the gro …

Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$. Let $W=W(R)$ denote its Weyl group. Let $S\subset R$ be a basis of $R$ (a system of simple roots). Let $D=D(R,S)$ denote the c …

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 …

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. …

Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$. The compact group $G_2:={\rm Aut}(O)$ naturally acts on $V$, and clearl …

See below a detailed version of the comment of @LSpice. (Edited taking into account a comment of @YCor.) This is an answer to the question on homomorpisms of real algebraic groups. Proposition. Let $ …

Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$. Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$. Let $\sigma$ be an anti-holomorph …

I prefer to use the language of algebraic groups. All algebraic groups and Lie algebras are defined over $\Bbb C$. 1. Let ${\mathfrak g}$ be a semisimple Lie algebra. Consider the automorphism group $ …

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 …