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

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19 views

Injectivity and surjectivity on algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an algebraic group defined over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
0
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0answers
25 views

Making sense out of the definition for “morphism of geometric spaces”

I'm trying to read "Introduction to Algebraic Geometry and Algebraic Groups" by Demezure and Gabriel and I'm already stuck on the following definition. A geometric space is defined to be a pair $(X, ...
2
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0answers
57 views

Real form and real structure on a complex Lie group

E.B.Vinberg and A.L.Onishchik in their book give the following two definitions. For a complex Lie group $G$ its real Lie subgroup $H$ is called a real form of $G$, if a) the Lie algebra $L(H)$ of ...
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0answers
3 views

Is any F-stable maximal torus contained in some F-stable maximal Borel subgroup?

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
2
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0answers
27 views

Inclusion of Tori induces surjection of character groups?

Let $k$ be an algebraic closed field. Let $T, T'$ be algebraic Tori in the classical sense, meaning $T \cong \mathbb{A}_k^n \setminus V(X_1 \cdots X_n)$, $T' \cong \mathbb{A}_k^{n'} \setminus V(X_1 ...
0
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1answer
48 views

Order of element in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
2
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0answers
24 views

Order of elements in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
0
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1answer
21 views

Surjectivity of ring homomophism induced by Frobenius endomorphism

Denote by $F_q$ the finite field with $q$ elements, and denote by $\bar{F_q}$ its algebraic closure. Let $V$ be an affine $\bar{F_q}$-variety and $F$ be the Frobenius endomorphism corresponding to an ...
3
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1answer
33 views

Test for a $G$-torsor to be trivial?

I just have a very short question, why is a $G$-torsor trivial precisely when it has a section?
2
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1answer
26 views

Definition of a $k$-structure

I came across the following definition: Let $\Omega$ be algebraically closed, $k \subseteq \Omega$ a subfield, and $V$ an $\Omega$-vector space. A $k$-structure on $V$ is a $k$-vector space $V_k$ ...
1
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4answers
112 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
0
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1answer
31 views

Decomposition of standard Borel subgroup.

I am reading the lecture notes. On page 3, formulas (1.7), (1.8), (1.9), let $P$ be the standard Borel subgroup, we have $P=MN$. Why $M, N$ must be (1.8), (1.9)? I know that $P$ should be a upper ...
0
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0answers
10 views

Etale Fundamental group of an algebraic group

I want to calculate the algebraic fundamental group of a an algebraic group over a riemann surface over $\mathbb C$ (or a smooth algebraic projective curve). Let me state the first case where ...
4
votes
0answers
49 views

homomorphism between smooth algebraic groups of the same dimension

For Lie groups, we have a theorem: Suppose $G$ and $G'$ are Lie groups of the same dimension, $G'$ is connected, and $f : G \to G'$ is a homomorphism of Lie groups with discrete kernel. Then, $f$ ...
3
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1answer
33 views

Does there exist a non-quasi-split torus?

In a homework, I was asked to prove that any torus is isomorphic to a quotient of a finitely many product of Weil restrictions $Res_{L/k}\mathbb{G}_m$. While solving this, I got an impression that ...
0
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1answer
36 views

Definition of an algebraic set being “defined over $k$” in terms of tensor product

Let $\Omega$ be an algebraically closed field, $I$ a radical ideal, and $k$ a subfield of $\Omega$. If $I = \mathcal I(A)$ for some closed set $A \subseteq \Omega^n$, then $I_k := I \cap k[X_1, ... , ...
2
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0answers
17 views

Are linear reductive algebraic groups closed under extensions?

Say we have a ses of algebraic groups $1 \to A \to B \to C \to 1$ where $A,C$ are linear reductive algebraic groups. Does it follow that $B$ is also a linear reductive algebraic group? In other ...
1
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1answer
18 views

nilpotent algebraic groups in terms of extensions

Let $N$ be a nilpotent linear algebraic group over a field $k$. If $k = \mathbb{C}$ and $N$ is connected, one can write $N = U \times T$, where $U$ is a unipotent algebraic group and $T$ is a ...
3
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0answers
55 views

(stability-theoretic) ¨weakly normal groups" are closed under subgroups

Let me first introduce two definitions: For a structure $\mathcal{M}$ in a language $\mathscr{L}$ and a subset $X \subseteq M^n$, the fully induced structure on $X$ is a structure $\mathcal{X}$ with ...
8
votes
1answer
192 views

Reduction modulo p of a linear group over the rational numbers

A paper (http://arxiv.org/pdf/1407.3158v2.pdf) contains the following theorem: Suppose $\mathbb{G}$ is a connected, simply connected, semisimple algebraic group defined over $\mathbb{Q}$, and let ...
0
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0answers
11 views

Prime ideals in Iwahori-Hecke algebras

Results on the ideals (especially the prime, completely prime ones) of Iwahori-Hecke algebras (espcially the ones with finite order) is needed. Thank you very much.
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1answer
43 views

The map $x \mapsto gx$ gives a homeomorphism $G \rightarrow G$ for algebraic groups.

Let $G$ be an algebraic group. The product group $G \times G$ (taken as a product of varieties) contains the product topology, and is a product with respect to the canonical projections $G \times G ...
0
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0answers
6 views

Example of nilpotent linear algebaic group, which is not abelian and not unipotent.

I was just searching for an example of a linear algebraic group $G$ (= subgroup of $GL_n$) which is not abelian and not unipotent. By nilpotent I mean that the central series trivializes, hence the ...
1
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0answers
22 views

Morphism from the projective line to an algebraic group

Let $F$ be a field (if require can assume of characteristic $0$) and $\mathbb{P}_F^1$ be the projective line. Suppose $G$ is a connected algebraic group over $F$. We denote the set of $K$-rational ...
1
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1answer
47 views

Are unipotent algebraic groups connected?

Is a unipotent algebraic group over a field of characteristic zero always connected?. As far as I know, every unipotent algebraic group over field of characteristic zero is isomorphic to a closed ...
3
votes
1answer
53 views

How to visualize $\operatorname{Lie}(\operatorname{GL}_n)=\mathfrak{gl}_n$ in positive characteristic?

I'd like an intuitive explanation as to why the Lie algebra of $\operatorname{GL}_n$ is $\mathfrak{gl}_n$ when working over fields of positive characteristic. Below I reproduce how I "see" this fact ...
2
votes
0answers
25 views

PGL_1(A) as a rational group.

Let $A$ be a central simple algebra over a field $F$. How one can see that the group scheme ${\bf\rm{PGL}}_1(A)$ is embedded as an open subgroup of $\mathbb{P}(A)$, the projective space over $A$. ...
4
votes
0answers
98 views

Identifying the cotangent bundle of the flag variety

Suppose $G$ is a Lie group (or I guess a linear algebraic group), $P \subset G$ a Lie subgroup with Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$ respectively. In Chriss and Ginzburg's book ...
0
votes
1answer
31 views

Direct product of algebraic groups

I am attempting to verify that the product variety $G \times G'$ of algebraic groups with the direct product group structure is an algebraic group, though I'm running into trouble. In particular, I'm ...
0
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1answer
50 views

Exercise in Springer: Linear Algebraic Groups (1.4.8 (i) )

This is a pretty bad book to learn algebraic geometry from if you've never seen it before. I'm trying to verify the following assertion. Let $k$ be an algebraically closed field, and $X, Y$ be ...
1
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1answer
38 views

About algebraic groups defined over Q

I'm studying automorphic forms and there's something I don't understand, when we talk about a connected reductive algebraic group $G$ defined over $\mathbb{Q}$, connected means connected as an ...
1
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0answers
43 views

Maximal tori in Lie vs algebraic groups

If $G$ is a Lie group, we define a maximal [Lie] torus in $G$ to be a maximal connected compact abelian Lie subgroup of $G$. These guys correspond to Cartan subalgebras of $\mathfrak{g}=Lie(G)$. If ...
0
votes
0answers
90 views

How to show that $\text{Spec}(\mathbb{Z}[x_1,x_2,\ldots, x_n]) \cong (\mathbb{G}_m)^n$?

How to show that $\text{Spec}(\mathbb{Z}[x_1,x_2,\ldots, x_n]) \cong (\mathbb{G}_m)^n$? Here $\mathbb{G}_m$ is the multiplicative group. Thank you very much. Edit: $\mathbb{G}_m = k - \{0\}$ is ...
1
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1answer
20 views

Poisson actions defined in terms of coactions.

If $(M,\{ \cdot,\cdot \}_{M})$ and $(M',\{ \cdot,\cdot \}_{M'})$ are two Poisson manifolds, then a smooth mapping $\varphi: M \to M'$ is called a Poisson map if it respects the Poisson structures, ...
6
votes
1answer
136 views

Lie algebras of reductive groups

Let $k$ be an algebraically closed field of positive characteristic and let $G$ be a connected split reductive group. We know $G$ is the product of its center $Z(G)$ and derived group $[G, G]$ and ...
0
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0answers
79 views

Proof of Proposition 13.20 in Digne-Michel

I have some problems understanding a part of the proof of Proposition 13.20 in "Representations of Finite Groups of Lie Type" by Digne and Michel. It concerns a bijection between certain varieties ...
1
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1answer
35 views

Describe invariants using coaction.

Let $X$ be an algebraic variety and $G$ be an algebraic group which acts on $X$. We know that an invariant function $f$ in the coordinate ring $\mathbb{C}[X]$ is a function such that $g(f) = f$ for ...
0
votes
0answers
37 views

positive roots remain positive

Let $W$ be the Weyl group of $SL_{n+1}$ and $w \in W$. Let $R^+$ denote the set of positive roots with respect to the Borel subgroup of upper triangular matrices. Define $R^+(w)=\{\alpha \in R^+: ...
4
votes
1answer
52 views

$\mathbb{G}_a$ or $\mathbb{G}_m$ as subgroups of Affine Algebraic Groups

Is it true that every connected Affine Algebraic Group has a subgroup isomorphic to $\mathbb{G}_a$ or $\mathbb{G}_m$? If so- why?
2
votes
1answer
47 views

Is $\mathbb{C}[T \times_T T] = (\mathbb{C}[T] \otimes \mathbb{C}[T])^T$?

Let $T$ be an algebraic group. There is a left $T$ action on $T$ given by left multiplication and a right action on $T$ given by right multiplication. Let $T$ acts in the middle of $T \times T$ by the ...
2
votes
1answer
34 views

Does $G\to X$ have dense image iff $T_eG \to T_{\theta(e)}X$ is surjective?

Let $G$ be a connected algebraic group, $X$ a variety and and $\theta : G\to X$ be a morphism of varieties. (In particular it could be the orbit of some action of $G$.) Consider the corresponding ...
1
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1answer
25 views

Is $\mathbb{C}[T \times_T T] \cong \mathbb{C}[T]$?

Let $T$ be an algebraic group. Do we have the following result: The algebra $\mathbb{C}[T \times_T T]$ is equal to the image of the co-multiplication $\mathbb{C}[T] \to \mathbb{C}[T] \otimes ...
1
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0answers
43 views

line bundle descents?

Let the permutation group $S_4$ act on $\mathbb C^4$ by permuting the coordinates. Consider the categorical quotient $\mathbb P(\mathbb C^4)/S_4$. It is a projective variety by a theorem of Mumford. ...
3
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0answers
41 views

structure of commutative algebraic groups

I've read that there is a structure theorem for commutative algebraic groups over an algebraically closed field $K$ namely they are the direct product of a semisimple group and a unipotent group. ...
3
votes
1answer
45 views

Fixed point subspaces $V^B$ and $V^G$ for a Borel subgroup $B\subset G$ coincide

Assume that $G$ is a linear algebraic group, and let $B \subset G$ a Borel subgroup of it. Let $(V,\rho)$ a rational $G$-module. Define $$V^G := \{ v \in V \mid g \cdot v = v \quad \forall g \in G ...
1
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1answer
53 views

Frobenius map in scheme theory

Let $\mathbb{K}$ be a field of characteristic $p$ and let $f: \mathbb{A}_{\mathbb{K}}^{1} \mapsto \mathbb{A}_{\mathbb{K}}^{1}$ be the morphism of the form $x \mapsto x^p$. We consider ...
4
votes
1answer
62 views

Is “being solvable” a geometric property for linear algebraic groups?

Say $G$ is a solvable linear algebraic group over some field $k$ of characteristic 0. This means that its derived series eventually terminates with a 1. My question is: Is "being solvable" a ...
2
votes
0answers
38 views

Does the adjoint action induce a trivial action on Lie algebra cohomology?

Let $k$ be an algebraically closed field and $\Gamma$ a finite group. $\Gamma$ acts on itself via conjugation, and it is true that the induced action on the cohomology algebra $H^{*}(\Gamma,k)$ is ...
0
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0answers
24 views

$S/R_u(S)$ for solvable groups

Consider a solvable linear algebraic group $S$ over a field $F$ with $char(F)=0$. Let $R_u(S)$ be the unipotent radical of $S$. If $S$ were connected, then $S/R_u(S)$ would be a torus. In ...
1
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0answers
24 views

Solvable algebraic groups and base-change

Let $G$ be an algebraic group over a field $k$ of characteristic zero. My understanding is that solvability is not a geometric property (is this correct though?). This motivates my questions: Let ...