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.
2,143
questions
3
votes
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146
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Extending Tannakian "dictionary" to gerbes
The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories".
Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...
- 323
1
vote
0
answers
44
views
Cartan decomposition over a not-necessarily-discretely-valued field
Let $K$ be a valued field, and let $R$ be the valuation ring of $K$. Let $G$ be a split reductive group over $K$ and $T$ a maximal torus of $G$. On page 107 Berkvoich's book "Spectral theory and ...
- 61
2
votes
0
answers
85
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What is the natural module?
Lemma 2.9 of [1]:
Let $\operatorname{char}(K) \neq 2 $ and let $G$ be $\operatorname{Spin}(m,K)$, $n=\operatorname{rank} G$, and let $V$ be the natural $m$-dimensional module. Suppose $f\in G$ and $f^...
- 165
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votes
0
answers
126
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Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 2,968
1
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0
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153
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Computer computation of the first Galois cohomology of a $p$-adic torus?
Let $T\subset {\rm GL}(N,{\mathbb Q})$ be an $n$-dimensional ${\mathbb Q}$-torus
given by its Lie algebra $\mathfrak{t}\subset \mathfrak{gl}(N,{\mathbb Q})$.
I want to compute, in some sense ...
- 13.8k
6
votes
1
answer
234
views
Extensions of algebraic groups and extensions of fpqc sheaves
There are at least two ways to define $\operatorname{Ext}^p(F,G)$ when $F$ and $G$ are commutative algebraic groups over a field $k$:
Pass to the associated fppf sheaves and use an injective ...
- 7,894
4
votes
0
answers
142
views
Spaces of fixed points
I am reading the paper Space with $\mathbb{G}_{m}$-action, hyperbolic localization and nearby cycles by Timo Richarz and I am having some troubles in understanding the proof of Lemma 1.10.
The setting ...
- 823
8
votes
0
answers
278
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Has the notion of a unipotent group scheme been studied?
The concept of a unipotent algebraic group over a field has been extensively studied and is fundamental in algebraic geometry. However, has the notion of a unipotent group scheme over a general base ...
- 1,139
3
votes
1
answer
78
views
Restriction of scalar commutes with taking maximal subtorus for semisimple group G
I was wondering such a question: for a semisimple complex Lie group $G$, whether it is true that the maximal subtorus of $\mathrm{Res}_{\mathbb{C}/\mathbb{R}}(G)$ is $\mathrm{Res}_{\mathbb{C}/\mathbb{...
- 435
1
vote
0
answers
69
views
Decomposition of $BwBw^{-1}B$
Let $(B,N,W,S)$ be a Tits system with $W$ a finite coxeter group.
Let $w\in W$, consider $BwBw^{-1}B$, then by Bruhat decomposition, it is a disjoint union of some $BxB$, $x\in W$.
My question: Let $X\...
- 551
2
votes
0
answers
66
views
Irreducibility of Białynicki-Birula cells
Let $X\subset \mathbb{P}^n$ be a smooth complex projective variety, and consider a non-trivial action of $\mathbb{C}^*$ on $X$. For any connected fixed component $Y$ of the fixed point locus, we may ...
3
votes
0
answers
169
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 ...
- 611
2
votes
1
answer
262
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Prodiscreteness of rational points of algebraic groups
Let $F$ be a field of characteristic 0 complete for a discrete non-archimedean valuation.
Let $G$ be a commutative smooth algebraic group over $F$.
Let us put on $G(F)$ the topology induced by the ...
- 95
3
votes
0
answers
49
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Algebraicity of the group of equivariant automorphisms of an almost homogeneous variety
The base field is the field of complex numbers. Let $G$ be a connected linear algebraic group. Let $X$ be an almost homogeneous algebraic variety, i.e. $G$ acts on $X$ with a dense open orbit $U \...
- 543
7
votes
2
answers
824
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Hilbert's Satz 90 for real simply-connected groups?
$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K/k$ be a Galois extension. Then one generalisation of Hilbert's Satz 90 states that $H^1(\Gal(K/k),\GL_n(K))=...
- 1,606