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I need a construction in linear algebraic groups which uses taking quotient by a central finite group subscheme. My question is, whether it goes through in ``bad'' characteristics, when this group sub …

I am looking for a reference for the following well-known fact: For any subdiagram $\Delta_0$ of of the Dynkin diagram $\Delta=D(G)$ of an adjoint simple group $G$ over an algebraically closed field $ …

I am looking for a reference for the following fact: The orthogonal group $O_{2n}$ over an algebraically closed field of characteristic 2 has exactly two connected components. To be more precise, let …

Let $G$ be a connected linear algebraic group over an algebraically closed field $k$ of characteristic 0. Let $D\subset G$ be a closed diagonalizable subgroup of $G$ (a subgroup of multiplicative type …

This is a continuation of my previous question. Let $G$ be a connected reductive group over an algebraically closed field $k$ of characteristic 0. We assume that $\mathrm{Pic}\ G=0$. This is the same …

Let $m>0$ be a natural number. Consider the following semisimple algebraic groups over ${\mathbb{R}}$: $$ G={\mathrm{SU}}(2m,4m),\ \ H={\mathrm{SU}}(2m,2m)\times{\mathrm{SU}}(2m). $$ We embed $H$ into …

Yes, the toral component of a connected Lie group is equal to the toral component of its solvable radical. Let $G$ be a Lie group, $S$ its solvable radical, and $\mathrm{TC}(G)$ denote the toral co …

Over $\mathbb{C}$, the étale fundamental group is the profinite completion of the topological fundamental group. You certainly assume that your affine algebraic group $G$ is connected. The question re …

Concerning rational parametrizations of algebraic groups similar to the Cayley transform see this answer, which deals with Cayley maps, i.e., equivariant birational isomorphisms between an algebraic g …

A Cayley map is a $G$-equivariant birational isomorphism $\lambda: G\to \mathfrak{g}$ (which does not have to exist). A connected linear algebraic group $G$ over $\mathbb{C}$ is called a Cayley grou …

Consider the subgroup $N:=\langle k\rangle\subset \mathbb{Z}^n$. There exists a basis $f_1,\dots,f_n$ of $\mathbb{Z}^n$ such that $uf_1$ is a basis of $N$, where $u\in \mathbb{Z}$, $u>0$, see Vinberg, …

EDIT: I roll back to the previous proof, in characteristic 0 only. My last proof including characteristic $p$ was false (thanks to Will Sawin for noticing this). Let $G={\rm GL}_{n,k}$\,, where $k$ i …

There is NO such example. Note that any semisimple algebraic ${\mathbb{R}}$-group $H$ of Hermitian type has a compact (anisotropic) maximal torus. Indeed, by a definition of a group of Hermitian t …

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 …

What are the automorphisms of $SL_n$ as an algebraic variety? In other words, let $k$ be an algebraically closed field of characteristic 0 (e.g., $k=\mathbb{C}$). Let $\tau$ be an automorphism of $SL …