# Questions tagged [algebraic-groups]

For questions about groups which have additional structure as algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. These include both linear algebraic groups as well as abelian varieties. Consider using with the (group-theory) tag.

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### How to obtain the coroots of $\operatorname{SL}_2$?

Let $G = \operatorname{SL}_2(\mathbb{C})$ and fix the maximal torus $T$ being the diagonal matrices in $G$. Let $\mathfrak{g} = \operatorname{Lie}(G)$, $\mathfrak{h} = \operatorname{Lie}(T)$ as usual, ...
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### Is the product of reductive algebraic groups reductive?

I know some cases in which the product of reductive algebraic groups is reductive. For example, $\text{GL}_n\times\text{GL}_m$ is known to be reductive. A product of algebraic tori $T\times T'$ must ...
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### Is the quotient of a linear complex group by a normal finite subgroup linear? [closed]

Lt $N$ be a finite normal subgroup of a complex linear group $G$. Is the quotient $G/H$ also a linear group?
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### Cohomology class of automorphism group of Galois form

Let $\Gamma$ be the Galois group of a finite Galois extension $K/k$ of fields of characteristic zero. Let $G$ be an algebraic group defined over $k$. Let $G'$ be another algebraic group over $k$. We ...
90 views

### A property of an irreducible root system

Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
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### Is there a scheme-theoretic definition for the center of an algebraic group (or group scheme)?

Since this is the case for other usual group-theoretic concepts like normal subgroups or derived subgroup (i.e. commutator), I was wondering if there exists a similar construction for an object ...
54 views

### Products of algebraic groups and sums of Lie algebras

Let $G$ be a connected solvable algebraic group. Then we know $G = T \ltimes G_u$ is a semi-direct product, where $T$ is a maximal tori, $G_u$ is the unipotent part of $G$. Main Question. Why does ...
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### Theorem of Highest Weight for Reductive vs Semisimple complex algebraic group

Let $G$ be a connected, reductive, complex algebraic group. It is well known that if $G$ is moreover semisimple, i.e. its Lie algebra $\mathfrak{g}$ is semisimple (or equivalently, its center $Z(G)$ ...
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1 vote
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### Cohomology Theory for Flag Varieties

I'm reading these notes about central results of cohomology theory of flag varieties, ie those having the form $G/B$ for $G$ semisimple, simply-connected, complex algebraic group, and $B$ it's (up to ...
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### Hasse principle for quadratic forms vs algebraic groups

Can the Hasse-Minkowski theorem for quadratic forms over a number field be recovered from the Hasse principle for algebraic groups? More specifically, by the former I mean the theorem that two ...
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### Decomposition of a general (not necessarily solvable) real algebraic group $G=(S\times T)\ltimes U$

Let $G$ be a connected real algebraic group. I have seen the following two facts: Based on this notes 1.If $G$ is nilpotent, then $G_s$ is a closed, connected subgroup (hence a torus). $G_u$ is the ...
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### Decomposing central simple algebras

Assume $A$ is a central simple algebra of degree $2^nl$ for some odd number $l$ and of index $2^m$ with $m$ dividing $n$. Question: Can we write $A$ as $B \otimes C$, with $C$ being a central simple ...
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### Homogeneous spaces for unipotent groups are isomorphic to $\mathbb{A}^n$

I am looking for a proof of the following fact: Let $U$ be a unipotent algebraic group over $\mathbb{C}$. Then any $U$-homogeneous space is isomorphic to $\mathbb{A}^n$. Context: Let $G$ be a ...
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### Question about a specific argument in the proof of Geometric Satake

Suppose $\tilde{G}_{\mathbb{Z}_p}$ is a flat affine group scheme over $\text{Spec}(\mathbb{Z}_p)$ (the p-adic integers) such that the fiber over the generic point $\tilde{G}_{\mathbb{Q}_p}$ is known ...
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### (Algebraic) monodromy group of $V \otimes V^*$

The algebraic monodromy group of a continuous semisimple (or irreducible further) representation $$\rho: G \to GL(V)=GL_n(\mathbb C)$$ is, by definition, the Zariski closure of the image of $\rho$ in ...
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### Lie-Kolchin theorem, unipotent matrices and Ad-unipotent elements

Let $G\le \text{SL}(n,\mathbb R)$ be a connected real matrix Lie group, not necessarily solvable The two basic questions I have, and I have been told by people that they "follow from" Lie-...
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### Is an extension of an algebraic group by a multiplicative group a semidirect product?

This is probably a very simple question with a negative answer, but I somehow cannot find a counterexample. Let $X$ be a smooth algebraic variety over an algebraically closed field $k$. Assume that $X$...
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1 vote
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### Affine group schemes

I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups. My question Let $k$ be an ...
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1 vote
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### How is the centralizer in $\mathrm{GL}_n(k)$ a torus

I have the following definition: Suppose that $g \in \mathrm{GL}_n(k)$ is regular and semisimple. Define $T_g := \mathrm{Cent}_{\mathrm{GL}_n(k)}(g)$ to be the centralizer of $g$ in $\mathrm{GL}_n(k)$....
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### Zariski closure of an algebraic subgroup of finite index [closed]

Let $G$ be an algebraic group and $H \subset G$ a subgroup of $G$. Let $H_0$ be a subgroup of $H$ of finite index. Then I guess the Zariski closure of $H_0$ is exactly the Zariski closure of $H$ in $G$...
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### Fibre bundle over Borel subgroup with fibre the complete flag

Let $G = \mathrm{GL}_n$, let $B$ be a Borel subgroup of $G$. The set $\mathcal{B}$ of $G$-conjugates of $B$ can be given an algebraic variety structure and is known as the variety of Borel subgroups ...
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### Connected components of subgroup of torus

Consider a finite field $\mathbb{F}_q$ of characteristic $p>0$. Let $A=(a_{ij})$ be an integer matrix with $k$ columns and a finite number of rows. Consider the algebraic subgroup $\pmb{H}_A$ of ...
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### Classifying representations of GL(n): what's a good reference?

What's a good reference for the classification of finite-dimensional algebraic representations of $\mathrm{GL}(n,k)$ when $k$ is an arbitrary field of characteristic zero? For $k = \mathbb{C}$, ...
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### Center of Levi subgroups in SLn

Fix a partition $\lambda$ of $n$ of length $l$. Consider a Levi subgroup $L_{\lambda}$ in $GL_{n}(\mathbb{C})$. There is the decomposition $L_{\lambda} = \prod_{i} GL_{\lambda_{i}}$. Now the ...
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1 vote
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### Surjective maps between algebraic groups induce surjective maps between connected components

I am reading the paper of F. Beukers, A refined version of the Siegel-Shidlovskii theorem (here is the link). The author mentions the following result in Algebraic group theory without proof. Lemma 2....
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### Algebraic Torus is a group scheme

I am taking a course on toric varieties this semester, and I am a little confused by how the algebraic torus is a group scheme, as we didn't really define what a group scheme is. I was given the ...
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