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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1answer
14 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
50 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
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1answer
190 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 ...
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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
41 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 ...
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0answers
5 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 ...
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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 ...
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1answer
44 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
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1answer
51 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
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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
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0answers
96 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 ...
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1answer
30 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
48 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 ...
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0answers
39 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 ...
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0answers
89 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
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1answer
127 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 ...
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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
31 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 ...
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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
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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
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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
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1answer
33 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 ...
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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
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
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0answers
38 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
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1answer
43 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
52 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
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1answer
61 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 ...
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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 ...
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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 ...
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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 ...
0
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1answer
66 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
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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 $U=D_{n}\cap SO(n)$: ...
1
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1answer
29 views

$B$-action on $U$.

Let $G$ be an algebraic, $B$ Borel subgroup, and $U$ unipoent subgroup of $G$. For example, we take $G=GL_n$, $B$ the subgroup of lower triangular matrices, and $U$ unipoent upper triangular matrices. ...
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0answers
42 views

Jacquet-Langlands double coset decomposition

Let $G=\mathrm{GL}(2,F)$ with $F$ a non-archimedean local field. Let $K=\mathrm{GL}(2,\mathcal{O}_F)$ be a maximal compact subgroup. Every element of $G$ lies in one of the double cosets ...
0
votes
1answer
30 views

twists of unipotent algebraic groups

Let $U$ be a unipotent linear algebraic group over some field $k$ with char$k$=0. Let $U'$ be a linear algebraic group over $k$ such that $U'_{\bar{k}} = U_{\bar{k}}$ (ie $U'$ is a $\bar{k}/k$-twist ...
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0answers
35 views

Albert- Algebras and Traceforms

Im new to the topic so this could be basic nonsense to you. Any Albert-Algebra $A$ has a trace map $T:A \rightarrow k$ and thus one can assign a quadratic form $q_A$ of rank $27$ by setting $q_A(x) = ...
1
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3answers
80 views

on the condition “$G$ is defined over $\mathbb{Q}$”

This might be a stupid question, but I cannot understand the "technical condition" when studying some basics of arithmetic groups, that is an algebraic group is defined over $\mathbb{Q}$. ...
2
votes
1answer
72 views

A point in $ PGL(R) $ not in $ GL(R)/R^{\times} $

A bit of notational background first. Let $k$ be a field and define $ PGL_{n} = Spec(k[x_{ij}]_{(det)}) $, where $i,j = 1,...,n$ and where $k[x_{ij}]_{(det)}$ denote the degree $0$ part of the graded ...
1
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1answer
88 views

Which categories of linear representations are semisimple?

Let $k$ be a field of characteristic $0$. For which smooth algebraic groups $G$ over $k$ does the abelian category of linear representations $\mathsf{Rep}_k(G)$ (not assumed to be finite-dimensional) ...
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0answers
19 views

Representation groups over Dedekind domains

I am interested on groups defined over $O_K$ the ring of integers of a number field $K$. Given a linear representation $T:Gl_N(O_K)\rightarrow Gl(W)$ with $W$ a free $O_K$-module, What are the main ...
1
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1answer
34 views

action of $GL_3$ on $P^2$

Find the action of $GL_3(K)$ on $\mathbb P_k^2 $, and compute its orbits and also the isotropy groups for all its orbits. ($K$ is an algebraically closed field) I know that $GL_3$ acts on ...
3
votes
1answer
86 views

Understanding the stack $B\mathbb{Z}$

Here, let $\mathbb{Z}$ be the group scheme whose functor of points is the constant functor which takes a connected affine scheme to the group $\mathbb{Z}$. I'm having a bit of trouble understanding ...
2
votes
1answer
23 views

connected linear algebraic group over the algebraic closure of a field

Let $G$ be a connected linear algebraic group over a field $k$ of characteristic 0. A paper I'm reading seems to imply that $\overline{G}:= G \times_k \overline{k}$ will also be connected, but I don't ...
2
votes
1answer
34 views

A sufficient condition for irreducibility of a $G$-variety

Let $G$ be an algebraic group over a field $k$ and let $V$ be a variety on which $G$ acts. Suppose $U\subset V$ is a closed, irreducible, $G$-stable subset which intersects every $G$-orbit ...
3
votes
1answer
59 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
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1answer
60 views

The identity element of a group

We define the process in Z. Then, is a group. In this group,which is the identity element? The correct answer is the element 10. why ?
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2answers
23 views

Polynomial Ring of Linear Algebraic Group

During lectures, we defined the Linear Algebraic group as the algebraic set $ GL(V):=k^{n^2}-V(Det) $ Where $V(Det)$ are the matrices with $0$ determinant. Then we proceed by identifying the ...