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
16 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
14 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 ...
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0answers
38 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
34 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
41 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 ...
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1answer
45 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 ...
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1answer
58 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
30 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
60 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
95 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 $U=D_{n}\cap SO(n)$: ...
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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
37 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
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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
32 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) = ...
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3answers
69 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
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1answer
67 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 ...
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1answer
84 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
16 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 ...
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1answer
33 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
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1answer
80 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
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1answer
20 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 ...
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1answer
24 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
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1answer
56 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
56 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
20 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 ...
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1answer
42 views

Definition of unipotent linear algebraic groups over non algebraically closed fields

Suppose we have a field $F$ with $\text{char}\ F=0$ and $F$ is not necessarily algebraically closed. What is the definition of a unipotent linear algebraic group over $F$? I'd really appreciate ...
1
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1answer
47 views

Lemma 5.2.5 in Springer's Linear Algebraic Groups

I'm stuck trying to understand the first paragraph of this proof. Let $X\rightarrow Y$ be a dominant morphism of affine varieties and denote $B=k[X]$,$A=k[Y]$. Assume there exists $b\in B$ such that ...
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0answers
23 views

What do you get when you pull the Bruhat Decomposition back to the Lie algebra via the exponential map?

If $G$ is a connected, reductive, complex group with Borel subgroup $B < G$ and Weyl group $W$, we can write $$G = \bigsqcup_{w \in W} B w B$$ If $\mathfrak{g}$ is the Lie algebra of $G$, we have ...
3
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0answers
47 views

Is a connected unipotent subgroup always contained in a Borel subgroup?

As the question says, is a connected unipotent subgroup $U$ of a linear algebraic group scheme $G$ always contained in a Borel subgroup of $G$? I have an argument for why the answer is yes, and I ...
1
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1answer
41 views

The structure of maximal tori in finite simple groups

Let $\mathbf{G}$ be a linear algebraic group over an algebraic closed field of characteristic $p$ and $F$ a proper frobenius map on it with fixed point group $\mathbf{G}^f=G$ such that ...
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0answers
26 views

Ideas for seminar talk on Algebraic Groups related to Number Theory

In a few weeks I have to give a seminar talk in an algebraic groups seminar, and the topic is number theory (possibly elliptic curves). I am not very knowledgeable in the subject, so I was hoping I ...
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0answers
39 views

diagonalizable group

The following is an exercise from Humphrey's Linear Algebraic Groups (page 108): Let $G$ be an algebraic group, $ \displaystyle H= \cap_{ \chi \in X (G)} \ker ( \chi) $. Prove that: (a) $H$ ...
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0answers
31 views

Computing cohomology of finite groups of Lie type

Let $G_{/\mathbf{Z}}$ be a Chevalley-Demazure group scheme, i.e. a split reductive group scheme over $\mathbf{Z}$. Let $\rho:G\to \operatorname{GL}(V_{/\mathbf{Z}})$ be a representation. If $k$ is a ...
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0answers
72 views

A Question about Zariski topology

Zariski topology which is used in the definition of an algebraic group is only defined for affine and projective varieties, isn't it?
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0answers
171 views

Representation theory of the general linear group over a finite prime field

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire ...
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0answers
40 views

Quotients of linear algebraic groups in different categories?

I have a question on quotients of linear algebraic groups. Let $G$ be a linear algebraic group and $H$ a linear algebraic group acting on $G$ as an algebraic group. I would like to know what the ...
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0answers
26 views

Is a form of a linear algebraic group over $k$ a linear algebraic group over $k$?

As the title says. $k$ : field with char($k$) = $0$ form: if $G$ is a l.a.g. over $k$ a form of $G$ is an algebraic group $G'$ over $k$ such that $G_{\bar{k}} \cong G'_{\bar{k}}$.
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1answer
55 views

Does a projective variety have a torus fixed point?

Let $X$ be a projective variety over $\mathbb{C}$ and let $T=(\mathbb{C^*})^k$ act on it. Is it true that there is a fixed point of this action on every irreducible component of $X$ just because $X$ ...
2
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2answers
48 views

Projective Special Linear Group is an Linear Algebraic Group

So I was wondering why the group $PSL(2,K)$ is a linear algebraic group, in the case that the characteristic of $K$ is not equal to $2$. Actually there is a description of $PSL(2,K)$, namely: ...
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1answer
54 views

Variety of Connected Components

In Milne's text http://www.jmilne.org/math/CourseNotes/iAG.pdf (A71), he introduces the "variety of connected components" of a finite type scheme $X$ over $k$ as the universal example of a zero ...
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1answer
27 views

maximal subtorus of a connected commutative algebraic linear group [closed]

I'm wondering the following: is the maximal subtorus of a connected commutative algebraic linear group over $k$ a) normal and closed b) defined over $k$ (for $k$ a field of characteristic zero, ...
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0answers
60 views

When is “being a linear algebraic $k$-group” preserved?

Let $G$ be a linear algebraic group over a field $k$, with Char$(k)=0$. What "group-theoretical operations" preserve the property of "being a $k$-linear algebraic group"? For example When ...
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0answers
20 views

QR decomposition, borel groups and generalizations

Then every matrix $M$ in $M_{m\times m} (\mathbb{C})$ can be written in the form: $QR=M$, where $Q$ is unitary and $R$ is upper-triangular. My question is simple, does this generalize in the ...
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1answer
40 views

Relative center of relative group scheme

This might be an easy question. Let $p: X \rightarrow S$ be a relative group scheme. In particular the fibers are group schemes. I want to know if there are constructions like the ``relative ...
1
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1answer
28 views

Question on subgroups of reductive groups

A linear algebraic group $G$ over some field $k$, which I assume being of characteristic 0, is reductive if $R_u(G^0_{\overline{k}})$ is trivial, where $R_u$ denotes the unipotent radical, $G^0$ is ...
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0answers
27 views

The simply connected form of a semisimple algebraic group

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$, so that $G$ is an almost-direct product of its minimal closed connected normal subgroups of positive dimension, ...
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2answers
174 views

Is there a group between $SO(2,\mathbb{R})$ and $SL(2,\mathbb{R})$?

Is there a non-trivial subgroup $H \subset SL(2,\mathbb{R})$ such that $H \supset SO(2,\mathbb{R})$ ? My intuition is that, since $\dim SO(2)=1$ and $\dim SL(2)=3$, there should be some group ...
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2answers
20 views

How to show that $GL_n/U$ is birationally isomorphic to $B^-$?

It is said that $GL_n/U$ is birationally isomorphic to $B^-$. Here $U$ acts by right multiplication on $GL_n$. I think that $GL_n/U$ consisting of cosets. Two matrices in the same coset if any two ...