Questions tagged [algebraic-groups]

For questions about groups which have additional structure as 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
7 views

Finiteness of the volume $G(k) \backslash G(\mathbb A)^1$ from that of $G(k) Z(\mathbb A) \backslash G(\mathbb A)$

It follows from the construction of a Siegel domain that $G(k) Z(\mathbb A) \backslash G(\mathbb A)$ has finite volume, where $G$ is a connected, reductive group over a number field $k$, $Z$ is the ...
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0answers
10 views

Unipotent upper triangular matrices with integer entries is Zariski dense

Let $N$ be the group of matrices $\begin{bmatrix} 1 & z \\ 0 & 1 \end{bmatrix}$ for $z \in \mathbb{C}$, let $\Gamma$ be the subgroup of $N$ with $z \in \mathbb{Z}$. I wish to show that $\Gamma$...
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0answers
32 views

Fixed points of the adjoint representation for an algebraic group

Let $G$ be a reductive affine algebraic group over $k$, and consider the adjoint action $G \xrightarrow{\text{Ad}} \text{GL}(\frak{g})$, where $\frak{g}$$=\text{Lie}(G)$ is the Lie algebra of $G$. I ...
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0answers
23 views

Levi subgroups and subsystems of root systems

Let $G$ be a connected reductive algebraic group over a local field $F$ with fixed maximal torus $T$, and denote by $R = R(G,T)$ the set of roots of $T$ in $G$, namely, the set of all nontrivial ...
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0answers
20 views

Are equivariant perverse sheaves constructible with respect to the orbit stratification?

Consider a variety $X$ over a field $k$ (complex numbers is fine) with the action of a group scheme $G$, and a $G$-equivariant perverse sheaf $F$ over $X$. Is it true that there exists a ...
2
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0answers
48 views

Automorphism Aut/Inn/Out of the unitary group $U(N)$

Given a group $G$, we denote the center Z$(G)$, we like to know the automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences: $$...
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2answers
57 views

Meaning of G(k) being dense in G for an algebraic group G/k

Let $G$ be an algebraic group over a field $k$ of characteristic 0. I have read that: if $G$ is connected, then $G(k)$ is dense in $G$ for the Zariski topology. I do not understand what kind of ...
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0answers
33 views

Every character of a diagonalizable algebraic subgroup is a weight of a representation.

Let $G$ be an affine algebraic group containing a diagonalizable closed subgroup $T$. I want to show, that every character of $T$ is a weight of at least one representation of $G$. Can someone help?
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0answers
31 views

Conjugating a root subgroup by a Weyl group element

Fix a field $k$. This is perhaps unnecessary, but assume $\operatorname{char} k = 0$. Let $G$ be a reductive isotropic quasi-split algebraic $k$-group. Let $S \subset G$ be a maximal split torus (of ...
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0answers
10 views

Bounding the norms $\|\rho(g).a\|/\|\rho(g\gamma).a\|$ for $\gamma \in G(k)$ in reduction theory

Let $G$ be a connected, reductive group over a number field $k$, and $P$ a parabolic $k$-subgroup of $G$. From the theory of algebraic group quotients, there is a finite dimensional $k$-vector space $...
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1answer
33 views

Rational representation of the additive group $\mathbb{C}$ as an algebraic group

The additive group $\mathbb{C}$ is a linear algebraic group via the embedding into $SL_2(\mathbb{C})$, $$z \mapsto \begin{bmatrix} 1 & z \\ 0 & 1 \end{bmatrix}.$$ In this way, the Lie algebra ...
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1answer
46 views

'Algebraic' computation of the differential of a Lie algebra homomorphism (using vector fields)

The following is an exercise from section 1.5.4 of Goodman and Wallach's textbook. Let $$ \phi(A) = \begin{bmatrix} \det(A)^{-1} & 0 \\ 0 & A \end{bmatrix}$$ for $A \in GL(n,\mathbb{C})$. Show ...
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0answers
108 views

Etale fundamental groups of algebraic groups

Is the étale fundamental group of an algebraic group always abelian? Is a complete calculation of them available somewhere, at least for reductive groups over fields of char. 0?
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1answer
48 views

Differential of a rational representation of algebraic groups

Let $(\pi,V)$ be a rational (regular) representation of $G \subset GL(n,\mathbb{C})$, $\pi^*$ the dual representation, and set $\rho = \pi \otimes \pi^*$. Let $T : W \otimes W^* \to \text{End}(W)$ ...
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1answer
36 views

Why are finite groups linearly reductive?

Let $G$ be a linear algebraic group contained in $GL(n)$. $G$ is linearly reductive iff every regular representation is completely reducible. Among the examples of linearly reductive groups, there are ...
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0answers
34 views

Isomorphism of algebraic tori implies equal rank

Let $D_n$ be the algebraic torus of rank $n$ (so $D_n = (\mathbb{C}^\times)^n$). I want to show that if $D_k \cong D_n$ as linear algebraic groups, then $k = n$. Here's what I've got. We assume WLOG ...
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2answers
50 views

$\mathbb{C}$ as the Lie algebra of $\mathbb{C}$

It is known that $\mathbb{C}$ is a linear algebraic group as the set of matrices $\begin{pmatrix} 0 & a \\ 0 & 0 \end{pmatrix}$ with $a \in \mathbb{C}$. I read in some notes that $Lie(\mathbb{...
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1answer
49 views

Coincidence of the coordinate ring and the set of regular functions

If $X$ is an affine variety, and if we denote by k[X] its coordinate ring and by $O(X)$ the set of regular functions on $X$, then it is well-known that $k[X]\cong O(X)$. My question is : Do we have ...
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0answers
43 views

Is $\mathbb{C}^{n}$ an irreducible representation of $O(n, \mathbb{C})$?

I am trying to prove that $\mathbb{C}^{n}$ is an irreducible representation of $O(n, \mathbb{C}) = \left\{ A \in GL(n) | A^{t}A = Id \right\}$. I presume the action is defined in the standard way, i.e....
2
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1answer
51 views

A different group associated with a Lie algebra

Let $\mathfrak g$ be a Lie algebra over a field $k$. Everybody loves to study its automorphism group, $$\mathrm{Aut}(\mathfrak g) := \{\alpha \in Aut_k(\mathfrak g): [\alpha(x),\alpha(y)] = \alpha([x,...
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0answers
50 views

Why is the product map $GL_1(k)\times GL_1(k)\rightarrow GL_1(k)$ not continuous? [duplicate]

I am reading Springer's Invariant Theory. I already have some experience with linear algebraic groups and invariant theory, yet one of the first exercises of the book has already confused me. In the ...
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0answers
49 views

Zariski closure of an algebraic linear group

Let $G$ and $H$ algebraic linear groups and $\phi : G \to H$ a regular group homomorphism. I wonder if $\overline{\phi(G)}$ (the Zariski closure of $\phi(G)$) is again a subgroup and how this could be ...
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1answer
74 views

$SL(2)/U(2)$ is isomorphic to $\mathbb{C}^2 \setminus \left\{ 0 \right\}$?

Let $G= SL(2)$ and $H=U(2)$ the set of $2x2$ matrices of type $\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}$. By Chevalley Theorem $G/H$ can be immersed in $\mathbb{P}(\mathbb{C}^2 \oplus \...
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0answers
35 views

When an Action on open dense subvariety by an Algebraic Group extends to Variety

A toric variety $X$ over $k$ is a variety which contains an algebraic torus ($T= \mathbb{G}_k^s$) as a dense open subset such that the action of the torus on itself extends to the whole of $X$. Slogan:...
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1answer
133 views

Mysterious morphism

Until I get a clearer understanding of what follows, and it is the aim of this post, I am sorry I cannot make the question title more explicit. In the setting of algebraic groups I found the following ...
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1answer
42 views

Automorphism of linear algebraic group $G$ preserves finite dimensional subspace of $k[G]$

I'm struggling to understand a claim in Springers "linear algebraic groups". Suppose $G$ is a linear algebraic group and $\sigma$ an automorphism of $G$ as an algebraic group. How can I show ...
2
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0answers
29 views

Orbits and fibres in categorical quotients

I am reading these notes about linear algebraic groups and when reading Theorem 1.24, I came up with a question, maybe a trivial one: in point $(v)$ it is only proven the uniqueness of the closed $G$-...
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1answer
52 views

Picard group of a Parabolic subgroup subgroups of a simple or reductive algebraic group. [closed]

Let $G$ be a reductive algebraic group for eg. $G=GL(n,\mathbb{C}).$ Let $P$ be a Parabolic subgroup of $G.$ What is the Picard group of a Parabolic subgroup? Is it the same as that of its Levi ...
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1answer
62 views

Questions about intersection of algebraic subgroups

I have a question related to Proposition 1.49 of the book Algebraic Group of Milne. Here is the excerpt: My question is the last line. I don't really understand the reasoning in that line. From what ...
2
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1answer
51 views

What is the canonical isogeny?

Given a (split connected reductive) root datum $\Psi$, there is another associated root datum $\Psi'$ and a morphism of root data $\Psi \to \Psi'$ called the canonical isogeny. My question is, is the ...
5
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1answer
108 views

Adjoint group of a reductive group

Let $G$ be a (not necessarily quasi-split) reductive group over $\mathbb{Q}$. In order to use results from semi-simple Lie groups for the reductive group $G(\mathbb{R})$, it seems common to implicitly ...
3
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1answer
49 views

Reference for representation theory of classical groups over arbitrary fields

I am looking for a reference for the representation theory of (classical) algebraic groups, that would ideally be something like the second part of Jantzen's "Representations of Algebraic Groups&...
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2answers
23 views

Dimension of normalizer of closed connected subgroup

Let $G$ be unipotent and let $H$ be proper closed, connected subgroup. Show that $\operatorname{dim}(N_G(H)) > \operatorname{dim} H$. We know that $H \triangleleft Z_G(H) \triangleleft N_G(H)$, ...
4
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1answer
85 views

Application of Lang' theorem about finite groups

Let $G$ be a connected algebraic group defined over the finite field $\mathbb{F}_q$, and let $F: G \to G$ be the Frobenius morphism. Show that $G^F = \{g \in G\mid g = F(g)\}$ is a finite group; For $...
1
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1answer
54 views

Prove the only normal unipotent subgroup $G$ is trivial

Let $G$ be an algebraic subgroup of $GL_n(k)$ which acts irreducibly on $V = k^n$, via the natural morphism $G \hookrightarrow GL_n(k)$, where $k= \overline{k}$. How to prove that the only normal ...
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0answers
50 views

Is this representation isomorphic to $V\otimes W$?

Assume that $G,H$ are linearly reductive groups over $\mathbb{C}$ and let $U$ be a finite dimensional representation of $G\times H$. Assume that we have $U_{\mid G}\cong V^{\oplus \dim W}$ and $U_{\...
0
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1answer
39 views

$\mathbb{C}(x)$ as a representation of $\mathbb{C}$

Let $G$ an algebraic linear group. A representation $V$ of $G$ is a vector space such that the application $\phi : G \to GL(V)$ is defined. Definition Let $G$ a linear algebraic group. A ...
2
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1answer
84 views

Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$ mod $I \otimes I $ for all $x \in I$.

I got stuck on this problem, so if anyone can give me a hint on this, I really appreciate. Let $I$ be the augmentation ideal in Hopf algebra $A$. Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$ mod $...
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0answers
37 views

Why 1 isn’t $a*b=max(a,b)$ identity element?

I had an exam this morning and the first question was in a test form ,the question goes: Imagine there is a binary operation $a*b=max(a,b)$ on N . Which option is correct ? a)$(N,*)$ is not abelian. ...
2
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2answers
83 views

Explicit isomorphism between ${\rm SL}(2,{\Bbb R})$ and ${\rm SU}(1,1)$

I have heard that there exists an isomorphism of real algebraic groups as in the title. I am asking for an explicit isomorphism. Motivation: I need such an isomorphism for a calculation of Galois ...
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1answer
65 views

Tensor product of regular modules is regular

Let $G$ and $H$ two algebraic linear groups, $V$ a regular $G$- module and $W$ a regular $H$ - module. I want to prove that $V \otimes W$ is a regular $G \times H$-module. If $V$ and $W$ are regular ...
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1answer
26 views

bad primes and torsion primes

Let $Φ$ be a root system of type $G_2$, with base ${α, β}$ where $α$ is short. Then $Ψ_1 :=$ ±{$α, 3α + 2β$} is a closed subsystem of type $A_1A_1$, and clearly $|\mathbb{Z}Φ/\mathbb{Z}Ψ_1| = 2$; ...
2
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1answer
52 views

maximal torus and semisimple elements

I am being stupid here. $G$ is a connected algebraic group and $s$ is a semisimple element. Let $T$ be a maximal torus. Then if $T$ is contained in $C_G(s)$, then $s$ is in $T$. I got stuck here. I ...
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0answers
45 views

Similar skew symmetric matrices are orthogonally similar

Let $X$, $Y$ be any two skew-symmetric matrices, such that $gXg^{-1}=Y,g\in GL_n(\mathbb R) $ then I need to show that there exists an orthogonal matrix h, such that $hXh^{-1}=Y,h\in O_n(\mathbb R)$. ...
2
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1answer
25 views

Why does elements in the parabolic subgroups of $SL(n,k)$ take the form of upper block triangular matrices?

Let $G$ be either $SL(n,k)$ (or I guess any linear subgroup of $GL(n,k)$) for a field $k$. And $P$ be a parabolic subgroup of $G$, I have seen the fact that any $A\in P$ looks like the block diagonal ...
3
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1answer
140 views

Galois cohomology of projective linear group

I am currently trying to compute Galois cohomology $H^{1}(\overline{k}/k, PGL_2(\overline{k}))$. As far as I know these cocycles correspond to isomorphism classes of smooth genus-$0$ curves over $k$. ...
3
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1answer
89 views

Is $\text{Hom}_k(A, B)$ a group? What is its group operator?

I want ask a question related to Hom operator. I'm studying affine group schemes, and in the definition, it says that an affine group scheme is just a functor F from k-algebras to groups. If F is ...
7
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0answers
160 views

Are the Deligne-Lusztig varieties for $\mathrm{GL}$ and $\mathrm{SL}$ the same?

Let me first give some definitions for reference. Let $\mathbf G$ be a connected reductive group over an algebraic closure $\overline{\mathbb F}$ of a finite field $\mathbb F_p$, where $p$ is a prime ...
1
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1answer
19 views

Identity component and rational point over algebraically closed field

For an algebraic group $G$ over $\mathbb{C}$ (in general, an algebraically closed field with character $0$), let the identity component of G be denoted by $G_{0}$. I wonder whether the rational point ...
4
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0answers
68 views

Functorial Algebraic Geometry - Schemes as Gluing Construction

Coming from a category theory background and having no experience with smooth manifolds the definition of a scheme as certain functors $\mathsf{CRing} \longrightarrow \mathsf{Set}$ feels much more ...

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