Search Results

The answer is "yes", see below. Dieudonné in his book "La géométrie des groupes classiques" considers the abstract group $SL_n(K)$ for a field $K$, not necessarily commutative, and writes $PSL_n(K)$ …

Let $X$ be a pointed algebraic variety over the field of complex numbers $\mathbb{C}$. Let $\pi_1^{\rm top}(X)$ and $\pi_1^{\mathrm{\acute{e}t}}(X)$ denote the topological and the étale fundamental g …

Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with $p …

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 $ …

The abstract subgroup generated by $H$ and $K$ is closed. We may assume that $G$ is connected. The groups $G$, $H$, $K$ are the groups of real points of real algebraic groups $\mathbf{G}$, $\mathbf{H …

$\newcommand{\sss}{{\rm ss}} \newcommand{\ssc}{{\rm sc}} \newcommand{\tor}{{\rm tor}} \newcommand{\X}{{\sf X}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\qed}{{$\blacksquare$}} $Let $G$ be a (connected …

The answer is YES. It suffices to assume that $H_1$ and $H_2$ are conjugate over $\mathbb{C}$ or, what is the same, that they are conjugate over $\overline{\mathbb{Q}}$. Theorem 1. Let $G$ be a co …

I know all real forms of ${\rm SL}(2,{\Bbb C}$). They are ${\rm SL}(2,{\Bbb R})$ and ${\rm SU}(2)$. Moreover, ${\rm SL}(2,{\Bbb R})$ is isomorphic to ${\rm SU}(1,1)$. Thus I can say that all real fo …

$\newcommand{\ab}{{\rm ab}} \newcommand{\ord}{{\rm ord}} $Let $G$ be a finite or profinite group. Consider the abelianized group $$G^\ab=G/G'$$ where $G'$ is the commutator subgroup of $G$. Let $H\sub …

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. …

Consider the real diagonal $4\times 4$ - matrix $$I_{2,2}={\rm diag}(1,1,-1,-1)$$ and the corresponding special unitary group $$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I_{2,2}\cd …

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 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 $G$ be a simply connected simple algebraic group over the field of complex numbers $\mathbb C$. Let $H$ be a symmetric subgroup of $G$. This means that there exists an automorphism of order 2 $\si …