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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2
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0answers
49 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 ...
0
votes
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
votes
0answers
26 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
votes
1answer
46 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
votes
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
votes
1answer
20 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
votes
1answer
30 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
votes
1answer
25 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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
0answers
54 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
votes
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.
-1
votes
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
votes
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
vote
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
vote
1answer
46 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
votes
1answer
49 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
vote
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
vote
0answers
42 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
vote
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
133 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
votes
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
vote
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
vote
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
vote
0answers
42 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
votes
0answers
40 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
vote
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
37 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
votes
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
vote
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 ...
0
votes
1answer
67 views

Clarifications about parabolic subgroups of $GL_4$

I'm asked to find all the parabolic groups $P$ which contains $T_4$, the subgroup (borel) of upper triangular invertible matrices. This is my definition of parabolic subgroup Let $P$ be a subgroup ...
2
votes
0answers
104 views

Maximal solvable subgroup not Borel

Given $G$ connected linear algebraic group I want to find a $U \subset G$ maximal solvable subgroup which is not connected. My attempt is to take $G=SO(n)$ for $n \geq 3$ and $U=D_{n}\cap SO(n)$: ...