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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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8 views

Comparing the relative and absolute Bruhat decompositions for quasi-split reductive groups

Let $G$ be a connected, reductive group over a field $k$. Assume $G$ is quasi-split. Let $A_0$ be a maximal split torus of $G$ with centralizer $T$, and let $B$ be a minimal parabolic (Borel) ...
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43 views

Intuition behind exact sequences as extensions in algebraic geometry

Introduction: There are certain categories of group schemes or group varieties in which one can define exact sequences. For instance, the categories of commutative group schemes of finite type over a ...
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0answers
43 views

Does connected components of a group scheme form a group scheme?

Let $G$ be a group scheme locally finite type and smooth over a base scheme $S$, and assume $S$ is normal and integral. Then does the set of (geometrical) connected components of a group scheme form ...
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37 views

Semisimple elements of a parabolic subgroup are contained in some Levi

Let $P$ be a parabolic $k$-subgroup of a connected, reductive group $G$ over a perfect field $k$. Let $N$ be the unipotent radical of $P$. It is defined over $k$. Let $M$ be a Levi $k$-subgroup of $...
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0answers
30 views

Strong approximation and class number in the adelic setting

$\DeclareMathOperator{\GL}{GL}\DeclareMathOperator{\ord}{ord}\DeclareMathOperator{\SL}{SL}$I have a question about a proposition from Daniel Bump's book, Automorphic Forms and Representations. Here $...
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15 views

Stability under conjugation of a subvector space of Lie algebra of an affine linear algebraic group (in positive characteristic)

Let $G$ a matrix Lie group and $\mathfrak{g}$ its Lie algebra. If a subvector space $V$ of a Lie algebra $\mathfrak{g}$ is stable under conjugation, then it's an ideal. I know this is should be true ...
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50 views

Compute all homomorphisms of $\mathbb G_a$ to $SL_2$ over $\mathbb Z$

How to compute the set of homomorphisms $\mathrm{Hom}\left(\mathbb G_a , SL_2\right)$ between those two group schemes over $\mathbb Z$? Over $\mathbb C$, this can be done using classical algebraic ...
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43 views

Why is split algebraic group quasi-split?

One says a connected linear algebraic group $G$ over a field $k$ is quasi-split over $k$ if there exists a Borel subgroup defined over $k$, and is split if there exists a split maximal torus $T$ over $...
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49 views

Let $\text{T} = \mathbb{Q}(i)^\times/\mathbb{Q}^\times$ compute $\text{T}(\mathbb{A})/\text{T}(\mathbb{Q})$

I have a question about algebraic tori, let $T = \mathbb{Q}(i)^\times/\mathbb{Q}^\times$ and let's try to compute $\text{T}(\mathbb{A})/\text{T}(\mathbb{Q})$. One has a product decomposition for ...
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33 views

Elementary question in reductive group

I am very sorry for asking some elementary question. Let $G$ be a reductive group over a number field $F$ and $N$ a unipotent radical of some parabolic subgroup of $G$. Then I am wondering whether $...
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2answers
95 views

Coordinate ring of a scheme in functorial algebraic geometry

I will preface this by saying that I am new to algebraic geometry, but I am somewhat experienced with category theory. I'm just reading the introduction to Milne's notes "Basic Theory of Affine Group ...
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19 views

Representation of algebraic groups

I have a question related with the notion of representation of algebraic groups. Let $G$ an algebraic group over $k$ and let $V$ a finite dimensional $k$-vector space. We have that $\mathbb{V}=\rm{...
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20 views

Algebraic Groups Connected and Reduced?

Lastly I was a bit surprised about a statement regarding the difference of group schemes to algebraic groups at wiki https://en.wikipedia.org/wiki/Group_scheme Let me quote it: "... Group schemes ...
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0answers
49 views

Can every connected reductive group over $\mathbb F_p^{alg}$ be defined over $\mathbb F_p$?

If I have a connected reductive group $G$ over a field with characteristic $p>0$ (for instance the algebraic closure of $\mathbb F_p$), can it always be defined over $\mathbb F_p$? For groups like $...
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1answer
65 views

Base Change of Algebraic Group

I have some questions about the steps in the proof of COROLLARY 1.35 from Milne's "Algebraic Groups : The theory of group schemes of finite type over a field"(p. 17). Here the excerpt: Let $G$ be a ...
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35 views

$Spin_n(q)$, $SO_n(q)$, $\Omega_n(q)$ and their projective images

I am studying about the structure of orthogonal groups and struggling to understand the relations between groups in the title: From algebraic groups point of view, it is known that $Spin_n(q)$ is ...
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42 views

$\mathbb Q$-simple algebraic groups and restriction of scalars.

I'm trying to understand the following statement: Any simply connected $\mathbb{Q}$-simple (algebraic) group has the form $\mathbf{R}_{F/\mathbb{Q}}(G)$ where $F$ is some field containing $\mathbb{Q}$...
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1answer
63 views

Tannakian duality for $\mathrm{SL}_{2}(\mathbb{R})$

Tannakian duality claims that we can recover any compact group from its finite-dimensional representations. More generally, we can recover affine group scheme from its finite-dimensional ...
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33 views

Permutations on vertices of cubes and hence finding volume enclosed by the vertices

Denote $C$ to be the cube $C=\{(x_1,x_2,x_3)|0 \leq x_1,x_2,x_3 \leq 1\}$ and let $V=\{ (x_1,x_2,x_3)|x_1,x_2,x_3 \in \{0,1 \} \}$ be the set of vertices of the cube. Let $A=$convex$((0,0,0) , (1,0,...
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3answers
55 views

Proof $x\in R^\times \wedge b\in R^\times \Rightarrow ab\in R^\times$ [closed]

Let $R$ be a communitative ring. Prove $a\in R^\times \wedge b\in R^\times >\Rightarrow ab\in R^\times$ with $R^\times := \{x\in R\ |\ x\ \text{ >invertible}\}.$ Do you have any ideas and tips ...
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39 views

Zariski Closure in a Torus and the dimension of the quotient

I'm reading the second chapter of Serre's Abelian l-Adic Representations and Elliptic Curves, and need help with the first exercise. Let $K$ be a number field and let $T$ be the Weil Restriction of ...
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1answer
62 views

Why are all proper ring varieties trivial?

$ \newcommand{\A}{\Bbb A} \newcommand{\Set}{\mathsf{Set}} \newcommand{\Sch}{\mathsf{Sch}} $ Define a ring variety to be a variety$^{[1]}$ $X$ over a field $k$, such that the functor of points $$\...
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1answer
16 views

Union of Bruhat cells is open in $G$

Let $G$ be (the points of) a connected, reductive group over a local field $k$ with maximal split torus $S$, minimal parabolic $P_0$, and Weyl group $W = N_G(S)/Z_G(S)$. Let $\Delta$ be the ...
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0answers
25 views

List all squared groups

$G$ group is called "squared" iff for every non-trivial subgroup $H$ we have that the index of $H$ is $2$, list all such groups ? I managed to prove that if such $G$ exist then there is only one ...
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1answer
70 views

Brauer-Severi varieties as quotients of forms of $\text{GL}_2$

Let $L/F$ be a finite galois extension of fields, with galois group $\Gamma$. Let $X$ be a variety over $F$ such that $X_L \cong \mathbb{P}^1_L$ over $L$, corresponding to a cohomology class $\alpha \...
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0answers
24 views

Definition of simple linear algebraic group

Why is it that many sources define simple (or almost-simple) linear algebraic group to be a connected, semisimple linear algebraic group such that every proper connected normal subgroup is trivial? ...
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0answers
11 views

Adjacent parabolic subgroups and proportionality of $H_P(n')$ to $\alpha^{\vee}$

Let $P = MN$ be a parabolic subgroup of a $p$-adic reductive group $G$ with split component $A_M$. There is bijection from the set of parabolic subgroups of $G$ with Levi $M$ and the chambers of $\...
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1answer
30 views

Prove : $\langle\gcd(a,n)\rangle\leq\langle a\rangle$

Reading a book about Group Theory I came across the following statement and its proof: Given $(\mathbb{Z}_n,+)$ (meaning the group of integers modulo $n$ with binary operation of addition) prove ...
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1answer
109 views

Algebraic 1-cocycles and Galois gerbs

We have the following set up: $K/F$ is Galois, $D$ is an algebraic group of mult. type and $E$ is an extension of groups: $$1\to D(K)\to E\to Gal(K/F)\to 1$$ Now take a linear algebraic group G over F....
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0answers
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Automorphisms of the linear algebraic group PGL_2

Every $\mathbb{R}$-algebra automorphism $\varphi \colon M_2(\mathbb{R}) \to M_2(\mathbb{R})$ gives rise to an automorphism of the linear algebraic $\mathbb{R}$-group $PGL_2$. That is, $\varphi$ ...
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1answer
64 views

$PGL_2(\Bbb R)$ as a scheme

How is $PGL_2(\Bbb R)$ a scheme? Here is my thought process $GL_2(\Bbb R)=Spec(\Bbb{R}[w,x,y,z,q]/((wz-xy)q-1))$ We want $PGL_2(\Bbb R)=GL_2(\Bbb R)/\Bbb{G}_m(\Bbb R)$ somehow. We can find $PGL_2(\...
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0answers
65 views

Quotient of linear algebraic groups: dimension of a faithful representation

Let $k$ be an algebraically closed field (I am mostly interested in $k=\mathbb{C}$ if that matters). Let $G$ be an affine algebraic subgroup of $GL_n(k)$ (ie $G$ is in fact linear). Let $H$ be a ...
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1answer
51 views

Is the functor that assigns to an algebra its algebraic group of units fully faithful?

Let $A$ be a $d$-dimensional algebra over a field $K$. One can naturally assign to $A$ the linear algebraic $K$-group $\mathbf{GL_1}(A)$ that represents the functor $B \mapsto (A \otimes_K B)^\times$ ...
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12 views

Deduction concerning Weyl chambers

In Humphreys' book on linear algebraic groups, he includes the following argument. Here $G$ is a connected algebraic group and $\mathfrak{B}$ denotes the collection of Borel subgroups of $G$. For ...
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45 views

Showing that an ideal of $\mathbb{C}[x,y]$ is prime.

Let $\mathbb{C}[x,y]$ be a ring of complex polynomials in two variables. Let $A = V(y^2-x, y^2-x^2) := \{(s,t) \in \mathbb{C}^2 |t^2 - s = 0 = t^2 - s^2\}$ (i.e $V(p(x,y))$ is the set of all zeros ...
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1answer
46 views

Classification of $\mathbb{R}/\mathbb{Q}$-forms of an algebraic group

The classification of forms of algebraic groups is generally only done for Galois extensions $L/K$. One gets a description of all $L/K$-forms of an algebraic $K$-group in terms of Galois cohomology. ...
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30 views

Global sections of pullback of $G$-equivariant $D_Z$-modules

Let $G$ be a semi-simple complex algebraic group with lie algebra $\mathfrak{g}$. For a fix Borel subgroup $B$ let $X=G/B$ be the flag variety. Let $i_l,i_r:X \to X \times X=Z$ denote the inclusion of ...
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0answers
36 views

If $\alpha\colon T\to\mathbb{G}_m$ is a root, does $\alpha(t)$ range over $\mathbb{G}_m$ as $t$ ranges over $T$?

Suppose $G$ is a reductive group with maximal torus $T$, and $U$ is a $T$-stable subgroup. For $\alpha$ a root relative to $T$, fix an isomorphism $u_\alpha\colon\mathbb{G}_a\to U_\alpha$ from the ...
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0answers
54 views

Flat affine group scheme $G$ over $\mathbb{Z}$ arises from embedding generic fiber $G_{\mathbb{Q}}$ into $GL_{n,\mathbb{Q}}$.

If we have an connected reductive group (reductive probably doesn't matter, affine group scheme is what matters) $G_{\mathbb{Q}}$ over $\mathbb{Q}$, we may construct a flat affine $\mathbb{Z}$- group, ...
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1answer
157 views

Tannakian theory for Lie algebras

Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
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1answer
49 views

Projective linear group PGLn()

Good morning, Let F a field with valuation and $O_F$ its valuation ring. I have to consider $PGLn(F)/PGLn(O_F)$. I understood that PGLn(F) is the quotient of GLn(F) by the scalar diagonal matrix. Is ...
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1answer
38 views

quotient of algebraic groups in Springer's book

In Springer's 'Linear Algebraic Groups', he proves that if $G$ is a linear (=affine) algebraic group and $H$ is a closed normal subgroup of $G$, then $G/H$ has a linear algebraic group structure with ...
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0answers
47 views

Interesting examples of commutative linear algebraic groups; uniqueness of composition series

We define an algebraic group to be solvable if it has a filtration where the successive quotients are abelian. The Lie-Klochin theorem says that every smooth solvable connected group admits an ...
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1answer
39 views

Conjugacy class that is not closed

I'm reading Springer's Linear algebraic group and I stuck with one exercise. Let $k$ be a field of characteristic 2 and let $G = \mathrm{SL}_{2}$. Then it says that conjugacy class of the matrix $$ ...
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0answers
29 views

$\mathbb{F}_q$-rational elements in unipotent classes of simple algebraic group in positive characteristic

Sorry in advance if this question is trivial or trivially false. I haven't managed to find a satisfactory proof (or reference of one), or a counterexample for it. Let $k$ be the algebraic closure of ...
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0answers
23 views

Sweeping a G-orbit by a different group H

Suppose H and K are algebraic subgroups of a linear algebraic group G. Suppose that G acts on a smooth algebraic variety X. Given $x\in X$, we can consider the map \begin{equation} H\times K \...
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1answer
105 views

How to prove the adjoint representation of $\text{SL}_2$ is irreducible when $\text{char}(k) \ne 2$?

Consider the adjoint representation of $\text{SL}_2$, where it acts by conjugation on its Lie algebra of matrices of the form $\begin{pmatrix} a & b \\ c & -a \\ \end{pmatrix}$, and suppose we ...
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1answer
38 views

Are adjoint quotients of connected reductive groups of adjoint type?

Suppose we have a connected reductive group $G$, and consider the adjoint quotient $G^{ad}=G/Z(G)$. Is $G^{ad}$ a group of adjoint type? I'm having difficulty coming up with a counterexample.
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29 views

Are the Following Endomorphism and General Linear Group Functors Representable?

Let $k$ be a unital commutative ring. Let $k$-Alg denote the category of commutative and unital $k$-algebras. Let Set denote the category of Sets Fix an arbitrary $k$-module $V.$ Consider the ...
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10 views

Real unipotent subgroups and their action

I am confused about the definition and the action of real unipotent algebraic groups. So let $G$ be an abelian complex linear unipotent algebraic group, such that it acts algebraically and on a ...