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# Questions tagged [algebraic-groups]

Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.

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### Centralizers and algebraic groups

Suppose $G$ is a linear algebraic group - I am also happy to assume $G$ is a simple algebraic group over an algebraically closed field of characteristic zero, but the question won't require this. The ...
1 vote
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### Commuting matrices and cyclic modules

Let $A, B\in M_n(\mathbb{C})$ be matrices that commute. We suppose that there exists a vector $v\in\mathbb{C}^{n}$ such that $(\mathbb{C}[A,B]).v$ generates $\mathbb{C}^{n}$. We call such a pair a ...
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### Functions on products of tori

Let $T$ be an algebraic torus over an algebraically closed field $k$. Let $d\in\mathbb{N}^{*}$. For every $d$-tuple of integers $\underline{n}=(n_1,\dotsc, n_d)$ and a function $f\in k[T]$, we can ...
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### Biquadratic extension of global function fields with cyclic decomposition groups

Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$. Question. What would be an example of a ...
1 vote
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### Equivariant subcategory of Tannakian category

Let $\mathcal C$ be a neutral Tannakian category over a field $K$ of characteristic $0$, with Tannakian fundamental group $U$. Assume there is a pseudo-functor $G \to \operatorname{Aut}_K(\mathcal C)$ ...
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### Rationality of quasi-elementary group actions

I am learning the paper https://arxiv.org/pdf/1604.01005.pdf "Reductive group actions" by Knop and Krotz. They defined that a linear k-algebraic group $H$ with unipotent radical $H_{u}$ is ...
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### Equivariant cohomology with compact support of the generalized Jacobian of a nodal elliptic curve

[Question forwarded from SE for lack of interaction] Given an geometric genus $1$ curve with $1$ node $C$ (hence with arithmetic genus $2$), its normalization is given by the elliptic curve $\tilde{C}$...
1 vote
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### Fields of definition of conjugates

Let $k$ be a field, not necessarily algebraically closed, $G$ an affine group scheme over $k$, $H$ a subgroup of $G$, and $N$ a normal subgroup of $H$, none of them assumed to be smooth. Suppose that ...
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### How are tangent spaces related via geometric quotient?

Let $G$ be a linearly reductive group acting regularly on an irreducible affine variety $X$ (over an algebraically closed field of characteristic zero). Suppose there's a $G$-stable open subvariety $U$...
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### Tits construction of algebraic groups of type D₆ and E₇ via C₃

As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
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### Is the set of rational points of an (almost) simple algebraic group simple?

Let $G$ be an almost simple algebraic group defined over a field $K$. Then we know that, for $H = G/Z(G)$, the set of rational points $H(\overline{K})$ is a simple group (when considered with the ...
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### Is Deodhar decomposition defined over arbitrary field?

Let G be a semisimple algebraic group defined over k. In Deodhar's paper: On some geometric aspects of Bruhat ordering. Ⅰ. A finer decomposition of Bruhat cell. (DOI link) Deodhar give a decomposition ...
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### Derived subgroup of rational points vs. rational points of derived subgroups

Let $G$ be a connected split reductive group over a field $k$. In general, we have an inclusion $$f: [G(k), G(k)] \rightarrow [G,G](k).$$ If $k$ is not algebraically closed, $f$ is not necessarily ...
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### Conditions for a $p$-divisible group to be represented by a formal Lie group

Let $S$ be a scheme where $p$ is locally nilpotent and let $G$ be a $p$-divisible group over $S$. Is connectedness of $G$ equivalent to $G[p] := \ker(p : G \to G) \to S$ radicial (universally ...
$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a reductive algebraic group over field $k = \overline{k}$ and consider the characteristic polynomial $\g \to \g/\!/G := \operatorname{Spec} (k[\g]^G)$ induced ...