When discussing with awllower about this question, I begin to think about another one: For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = \overline{[H,H]}$? where $[H,H]$ denote the derived subgroups of $H$. I think, if I define the map $\phi: G \times G \rightarrow G, (x,y) \mapsto xyx^{-1}y^{-1}$, […]

Let us work over an algebraically closed field of characteristic $0$. Let $G$ be a semisimple, or perhaps reductive algebraic group, so we are working with the Zariski topology. Let $\mathfrak{g}$ be the Lie algebra associated to $G$. If possible, could we not appeal to Lie groups and/or exponentiation, unless we explain how this relates […]

There is a well-known classification of finite abelian groups into products of cyclic groups. What about finite abelian group schemes, where we may put in the qualifiers “affine”, “etale”, or “connected” if it helps? There are some easy examples showing that the theory is richer than that of groups: The constant group scheme $\mathbb{Z}/n\mathbb{Z}$, The […]

I am reading some lie groups/lie algebras on my own.. I am using Brian Hall’s Lie Groups, Lie Algebras, and Representations: An Elementary Introduction I was checking for some other references on lie groups and found J. S. Milne’s notes Lie Algebras, Algebraic Groups,and Lie Groups It was written in introductory page of algebraic groups […]

Let us be given a linear algebraic group $G$ over a field $K$ of characterstic zero. This group $G$ is defined as the common zeroes of a finite set of polynomials $\{f_1, \ldots ,f_r\}$ $\in K [T_1,\ldots,T_n]$with some additional properties like having the structure of a group along with 2 morphisms of varieties. Now, the […]

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite dimensional $G$-invariant linear subspace of $A$.) By Nagata’s theorem, $A^G$ is a f.g. $k$-algebra. We have the affine GIT quotiont […]

I was just wondering whether or not it is possible to view $\mathrm{PGL}_2$ as an affine algebraic group.

I am looking for a reference for the above statement that every parabolic subgroup of $\mathrm{Sl}_n(\Bbbk)$ stabilizes some flag in $\Bbbk^n$. I have gone through a large pile of books and can’t seem to find one. Thanks a bunch in advance! Edit: I understand $G=\mathrm{Sl}_n(\Bbbk)$ as a connected algebraic group and define a parabolic subgroup […]

Let $B= \Bigg\{\begin{bmatrix} * & *&* \\ 0 & *&*\\ 0&0&* \end{bmatrix} \Bigg\}< SL(3,\mathbb C)$. What is $SL(3,\mathbb C)/B$? Do we use these facts: Borel fixed point theorem. Algebraic actions of unipotent groups are cells $U_-\cdot x_0=U_-/H$? I actually need a detailed solution because I don’t have enough background in this subject and I have […]

Let $p$ be a prime number, and let $k=\mathbb{F}_p$ be the field of $p$ elements. Let $G=GL_n(k)$. We know that $$|G|=\prod_{i=0}^{n-1}(p^n-p^i)=p^{\binom{n}{2}}\prod_{i=0}^{n-1}(p^{n-i}-1)$$ so that the Sylow $p$-subgroups of $G$ have order $p^{\binom{n}{2}}$. One such subgroup is $U$, the upper-triangular unipotent subgroup consisting of all upper-triangular matrices with $1$’s on the diagonal. Let $A_{ij}=I_n+E_{ij}$ for $j>i$, where […]

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