# 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.

970 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...