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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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Equivalent definitions of separable extension

Armand Borel in his textbook "Linear Algebraic Groups" (pp. 4) states that $F$ is said to be separable over $\boldsymbol{k}$ if it satisfies the following equivalent conditions ($p$ denotes the ...
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Is this unipotent group, over characteristic 2, connected?

Let $E_{ij}(x)\in Mat_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x)$, $b(y)=1+E_{23}(y)+E_{45}(y)...
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Why is this universal cover isomorphic to $\text{SL}_2(\mathbb{C})$?

In Lurie's proof of the Borel-Weil theorem http://www.math.harvard.edu/~lurie/papers/bwb.pdf, he states that the universal cover of a Levi factor of $S$, where $C\cup U'B=SB$, is isomorphic to $\text{...
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Every Borel contains a Cartan, and conjugacy theorems: A simple proof?

Conjugacy of Borel subalgebras $\newcommand{\ad}{\mathrm{ad}\,}$ Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. A Borel subalgebra $\mathfrak{b} \subseteq \mathfrak{g}$ is a ...
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$\mathbb R$-points of semisimple real algebraic groups, connectivity, and Cartan involutions: some questions

I am reading about Cartan involutions on semisimple real Lie groups and have a point of confusion I am trying to reconcile with linear algebraic groups. Let $\mathbf G$ be a linear algebraic group ...
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Rebuild a linear algebraic group from its orbits

I have the following situation (following the proof that every linear compact group is algebraic from Vinberg, Gorbatsevich and Onishchik "Lie Groups and Lie algebras III" Chapter 4 Theorem 2.1): Let ...
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Generalisation of Hilbert's 90 Theorem

Let $L/K$ be a finite Galois extension of fields with Galois group $G = Gal(L/K)$. According to the famous Hilbert's 90 we know that the first cohomology vanish: $$H^1(G, L^*)=\{1\}$$ My question is ...
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1answer
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Examples of Lie algebras of the $BC$ root system type

What are some examples of Lie algebras of the $BC$ root system type please? I am actually interested in the corresponding groups too. I heard that there were Lie algebras over $\mathbb{R}$ having $BC$ ...
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126 views

What makes the 'special groups' ($\det A = 1$) special?

This is a rather basic, and open-ended question: in several branches of mathematics and physics, we make an effort to classify linear operators $A$, especially orthogonal or unitary operators, by ...
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Rational Invariants of Algebraic Group Action

Suppose $G$ is a connected complex algebraic group acting on a variety $X$. Write $\mathfrak{g}$ for the Lie algebra of $G$. Then both $G$ and $\mathfrak{g}$ act on $\mathbb{C}(X)$, the ring of ...
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2answers
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Multiplication table with identity and order

This seems relatively easy, however I do not understand what is meant by the order of $x$ for each $x\in G\setminus\{3\}$ is equal to $2$. Can someone point me in the right direction? Clearly $G\...
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1answer
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Why is Borel self-normalizing?

Given a reductive algebraic group $G$ over an algebraically closed field, why is its Borel subgroup $B$ self-normalizing? There is an answer on the site already, though only in $G=GL_n$ case.
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1answer
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If $G$ is a smooth scheme over $S$ of characteristic $p$, is the relative Frobenius morphism $F_{G/S}$ faithfully flat?

Let $G$ be a smooth scheme over $S$ of characteristic $p$, do we have that the reltaive Frobenius morphism $F_{G/S}$ is faithfully flat? There is an excersice in Liu's book saying that this is true ...
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1answer
147 views

Lang Steinberg over separably closed field

Let $K=K^{sep}$ be a separably closed field with $K|\mathbb{F}_q$, where $\mathbb{F}_q$ is the field with $q$ elements. Let $\mathbb{G}$ be a connected linear algebraic group over $\mathbb{F}_q$. ...
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Component Groups of Reductive Groups

Suppose $G$ is a reductive group that is not necessarily connected and $Z \subset G$ is a central subgroup. Suppose $G^0$ is the identity component of $G$. Is it true that $G/G^0Z= \pi_0(G/Z)$? I can ...
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639 views

Element in finite number of Borel subgroups

Let G is a linear algebraic group over algebraic closed field, B is an Borel subgroup of G. Does there exist g$\in$G which is only in a finite numbers of conjugates of B (they are also Borel ...
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Basic question about Bruhat-Tits building of an algebraic group over a local field

I have some basic/elementary questions about construction/interpretation/intuitive understanding of buildings of algebraic groups over local fields. I am basically interested in the case where the ...
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Morphism of algebraic groups defined over real points

I am starting to study algebraic groups and I came across the following statement: let $S=\mathrm{Res}_{\mathbb{C}/\mathbb{R}}(\mathbb{G}_{m,\mathbb{C}})$ be the restriction of scalars of the ...
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Is the group generated by Zariski-dense subgroups Zariski-dense?

Given an algebraic group $G$, two algebraic subgroups $H_1, H_2$, and two discrete groups $\Gamma_i \leq H_i$ s.t. $\Gamma_i$ is Zariski-dense in $H_i$, is it true that $\langle \Gamma_1, \Gamma_2 \...
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Surjectivity of Lang map

Could we prove the surjectivity of Lang's map for $GL(N)$ without using algebraic geometry? In other words, given a invertible matrix $M$ in $GL_N(\mathbb{F})$, there exists another invertible ...
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Character group of torus over the real numbers (Theorie de Hodge II)

I am trying to read the article Théorie de Hodge II (which can be found in French here) and in page 24, when Deligne starts discussing Hodge structures, he makes the following claim about the ...
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Algebraic Groups, dual numbers and differentials

I was looking for a method to compute the explicit differential of a regular map between algebraic groups. More precisely if $X$ is a sub-variety in an algebraic group $G$ (say over a finite field $k$)...
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Restriction of scalars $G \mapsto \operatorname{Res}_{\mathbb C/\mathbb R} G$ is injective?

Let $G$ be a linear algebraic group over $\mathbb C$, and let $G_0 = \operatorname{Res}(\mathbb C/\mathbb R, G)$ be the linear algebraic group over $\mathbb R$ obtained by Weil restriction of scalars. ...
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If $\exp(X) \in G$, and $X$ is Hermitian, do we have $X \in \mathfrak g$?

Let $G$ be the $\mathbb R$-points of an algebraic group in $\operatorname{GL}_n(\mathbb C)$ which is defined over $\mathbb R$. Let $\mathfrak g$ be the Lie algebra of $G$. Assume $G$ is ...
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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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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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51 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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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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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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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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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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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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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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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
115 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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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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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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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
73 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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$Spin_n(q)$, $SO_n(q)$, $\Omega_n(q)$ 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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$\mathbb Q$-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
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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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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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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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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
21 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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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
77 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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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? ...