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.

learn more… | top users | synonyms

0
votes
0answers
10 views

Weil Pairing on Linear Algebraic Groups

I've been studying the Weil pairing on elliptic curves recently and discovered that it has a generalisation to an abelian variety $A$ with its dual $A^{\vee}$, which then becomes a pairing on an ...
2
votes
1answer
27 views

Can $\mathrm{PGL}_2$ be viewed as an affine algebraic group?

I was just wondering whether or not it is possible to view $\mathrm{PGL}_2$ as an affine algebraic group.
4
votes
0answers
31 views

Understanding $G_2$ inside Spin(7)?

This is a rather embarrassing question, so please let me know of any duplication and I will happily remove it. I am seeking to understand the $\mathbb Q$-split form of the algebraic group $G_2$, and ...
0
votes
0answers
39 views

Representations of algebraic group ($S_{\mathfrak{m}}$)

I'm studying Serre's book "Abelian $\ell$-adic Representations and Elliptic Curves" and in chapter II $\S$2.4 we have this proposition: Consider $v$ a finite place of $K$ and $F_v \in Gal(K^{ab}/K)$ ...
2
votes
0answers
50 views

On why $k(X)^{G}$ is a finitely generated field extension

In a book I was reading, from the assumptions that we have a linear algebraic group $G$ acting on an irreducible (affine) variety $X$, the author writes that $k(X)^{G}$ is a finitely generated field ...
1
vote
1answer
34 views

Algebraic groups with no small subgroups

I have read in many textbooks proofs that any Lie group has no small subgroups, that is, there is an open neighborhood of the unity element containing no nontrivial subgroups. In particular, ...
2
votes
0answers
34 views

Different definitions of projective matrix groups, with one giving an algebraic group but not the other

Recently my professor told me that the usual definition of PSL(2,$\mathbb{R}$) = SL(2,$\mathbb{R}$)/{$\pm$I} does not give an algebraic group, but the following definition does: PSL(2,$\mathbb{R}$) ...
1
vote
0answers
13 views

Zariski closure of a group generated by Jordan cell

Suppose we have a subgroup $G \subset \mathrm{GL}(n,\mathbb C)$ generated by a Jordan cell. What is the closure of $G$ in Zariski topology? (That seems for me a rather natural question for which I ...
2
votes
0answers
79 views

Questions about Affine algebraic group scheme over an infinite field K

For an easily comprehension of my questions I write some definitions: An affine algebraic group scheme over $K$ is a representable group-functor from $K$-algebras category, with a finitely generated ...
2
votes
0answers
15 views

Understanding $SL_3(D)$ where D is a central division algebra

Suppose that $K$ is a non-archimedean local field of positive characteristic and $D$ is a four-dimensional central division algebra over $K$. The group $SL_{3}(D)$ can be embedded as a $K$-form of ...
0
votes
0answers
26 views

A question about the pull-back and proper pushforward functors (convolution product of perverse sheaves) on $SL(2,\mathbb C)$

Let $G = SL(2,\mathbb C)$, which is an algebraic group of type $A_1$ over $\mathbb C$. Let $B$ be a Borel subgroup of $G$. Let $X =G/B$. Then $X \cong \mathbb P^1$ and it has a stratification $X = ...
0
votes
0answers
10 views

trying to understand the Bruhat-Tits building for a non-split group

Suppose that $D$ is a central division algebra of dimension four over a p-adic field $\mathbb{Q}_{p}$. What do the apartments of the Bruhat-Tits building for $SL_{3}(D)$ look like? Do they look the ...
3
votes
1answer
27 views

Find normal affine subgroup $N$ such that $G/N $ is an abelian variety ( $G=\mathbb{A}^1 \setminus 0 $)

I'm reading Shafarevich Basic Algebraic Geometry. I read Chevalley Theorem. It asserts that every algebraic group $G$ has a normal subgroup $N$ such that $N$ is affine and $G/N$ is an abelian variety. ...
3
votes
2answers
70 views

Maximal closed subgroups in algebraic groups

Let $G \leq GL(V)$ be an affine algebraic group, over an algebraically closed field. Say that $M$ is a proper subgroup of $G$ which is maximal among the closed proper subgroups of $G$. Does $M$ have ...
0
votes
0answers
16 views

Group action on closed subschemes

Let $G$ be a connected, linear, semi-simple algebraic group and $P \subset G$ the maximal parabolic subgroup. We know that $Z=G/P$ is a projective variety. Then, 1) Does $Z$ contain a line? 2) In ...
0
votes
0answers
13 views

Radical of reductive but not connected linear algebraic groups

Let $G$ be a linear algebraic group over a field $k$ of characteristic zero. A definition of $G$ being reductive is that the radical of $G^0$ (the connected component of the identity of $G$) over ...
3
votes
2answers
104 views

How is the multiplicative group an algebraic variety?

According to various places, we define an algebraic group as a group that is also an algebraic variety (along with some compatibility conditions). Many places also list some examples, one of which is ...
4
votes
1answer
42 views

Algebraic Peter-Weyl theorem in the case of $G=SL_2$.

The algebraic Peter-Weyl theorem says that for a linear reductive group $G$ we have $\mathbb{C}[G] = \oplus_{V} V \otimes V^* $, where $V$ runs over the set of all non-isomorphic irreducible ...
0
votes
0answers
7 views

definition of an affine F-variety where F is not algebraically closed

In the first chapter of his book "Linear Algebraic Groups", Springer considers a situation where $k$ is an algebraically closed field and $F$ is a subfield, and seeks to define the notion of an ...
0
votes
0answers
9 views

non-uniform arithmetic lattice in a semisimple algebraic group over a local field of positive characteristic

Say I'm considering the group of rational points $G(k)$ where $G$ is the special orthogonal group for the quadratic form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}-x_{5}^{2}$ and $k$ is a ...
0
votes
0answers
27 views

Meaning of “p-adic fields” in Jacques Tits' article on classification of semisimple groups

In Jacques Tits' article "Classification of Algebraic Semisimple Groups", which appears in "Algebraic Groups and Discontinuous Subgroups: Proceedings of Symposia in Pure Mathematics, Volume IX", when ...
0
votes
0answers
9 views

Question on $k$-isomorphic of quasi-split groups which are inner forms of each other

The following appears in Borel's paper, automorphic $L$-function. Let $G$ and $G'$ be two quasi-split connected reductive groups defined over $k$, where $k$ is a field and $\overline k$ is its ...
0
votes
0answers
14 views

Differing conventions regarding “modulus character” of $k$-points of smooth affine $k$-group, $k$ non-Archimedean

Let $\mathbf{G}$ be a smooth connected affine $k$-group, where $k$ is a non-Archimedean local field, $G=\mathbf{G}(k)$ the group of $k$-rational points, a locally $k$-analytic group. Since $G$ is in ...
1
vote
1answer
22 views

Zariski closure of an infinite cyclic group of diagonal matrices

Suppose that $\Gamma=\{$exp $kX\mid k\in\mathbb{Z}\}$ where $X\in\mathfrak{gl}(n,\mathbb{R})$ is a diagonal matrix. How do we prove that the Zariski closure of $\Gamma$ must contain exp $tX$ for all ...
1
vote
1answer
20 views

is “being reductive” extension-closed?

Suppose we have a short exact sequence of linear algebraic groups over a field of characteristic zero $$1 \to N \to G \to G/N \to 1$$ with $N$ and $G/N$ reductive (that is connected with trivial ...
1
vote
0answers
22 views

Injectivity and surjectivity on algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an algebraic group defined over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
0
votes
0answers
29 views

Making sense out of the definition for “morphism of geometric spaces”

I'm trying to read "Introduction to Algebraic Geometry and Algebraic Groups" by Demezure and Gabriel and I'm already stuck on the following definition. A geometric space is defined to be a pair $(X, ...
3
votes
0answers
78 views

Real form and real structure on a complex Lie group

E.B.Vinberg and A.L.Onishchik in their book give the following two definitions. For a complex Lie group $G$ its real Lie subgroup $H$ is called a real form of $G$, if a) the Lie algebra $L(H)$ of ...
0
votes
0answers
4 views

Is any F-stable maximal torus contained in some F-stable maximal Borel subgroup?

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
2
votes
0answers
31 views

Inclusion of Tori induces surjection of character groups?

Let $k$ be an algebraic closed field. Let $T, T'$ be algebraic Tori in the classical sense, meaning $T \cong \mathbb{A}_k^n \setminus V(X_1 \cdots X_n)$, $T' \cong \mathbb{A}_k^{n'} \setminus V(X_1 ...
0
votes
1answer
52 views

Order of element in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
2
votes
0answers
26 views

Order of elements in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
0
votes
1answer
26 views

Surjectivity of ring homomophism induced by Frobenius endomorphism

Denote by $F_q$ the finite field with $q$ elements, and denote by $\bar{F_q}$ its algebraic closure. Let $V$ be an affine $\bar{F_q}$-variety and $F$ be the Frobenius endomorphism corresponding to an ...
3
votes
1answer
45 views

Test for a $G$-torsor to be trivial?

I just have a very short question, why is a $G$-torsor trivial precisely when it has a section?
2
votes
1answer
26 views

Definition of a $k$-structure

I came across the following definition: Let $\Omega$ be algebraically closed, $k \subseteq \Omega$ a subfield, and $V$ an $\Omega$-vector space. A $k$-structure on $V$ is a $k$-vector space $V_k$ ...
1
vote
4answers
119 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
0
votes
1answer
35 views

Decomposition of standard Borel subgroup.

I am reading the lecture notes. On page 3, formulas (1.7), (1.8), (1.9), let $P$ be the standard Borel subgroup, we have $P=MN$. Why $M, N$ must be (1.8), (1.9)? I know that $P$ should be a upper ...
0
votes
0answers
12 views

Etale Fundamental group of an algebraic group

I want to calculate the algebraic fundamental group of a an algebraic group over a riemann surface over $\mathbb C$ (or a smooth algebraic projective curve). Let me state the first case where ...
4
votes
0answers
49 views

homomorphism between smooth algebraic groups of the same dimension

For Lie groups, we have a theorem: Suppose $G$ and $G'$ are Lie groups of the same dimension, $G'$ is connected, and $f : G \to G'$ is a homomorphism of Lie groups with discrete kernel. Then, $f$ ...
3
votes
1answer
37 views

Does there exist a non-quasi-split torus?

In a homework, I was asked to prove that any torus is isomorphic to a quotient of a finitely many product of Weil restrictions $Res_{L/k}\mathbb{G}_m$. While solving this, I got an impression that ...
0
votes
1answer
36 views

Definition of an algebraic set being “defined over $k$” in terms of tensor product

Let $\Omega$ be an algebraically closed field, $I$ a radical ideal, and $k$ a subfield of $\Omega$. If $I = \mathcal I(A)$ for some closed set $A \subseteq \Omega^n$, then $I_k := I \cap k[X_1, ... , ...
2
votes
0answers
20 views

Are linear reductive algebraic groups closed under extensions?

Say we have a ses of algebraic groups $1 \to A \to B \to C \to 1$ where $A,C$ are linear reductive algebraic groups. Does it follow that $B$ is also a linear reductive algebraic group? In other ...
1
vote
1answer
18 views

nilpotent algebraic groups in terms of extensions

Let $N$ be a nilpotent linear algebraic group over a field $k$. If $k = \mathbb{C}$ and $N$ is connected, one can write $N = U \times T$, where $U$ is a unipotent algebraic group and $T$ is a ...
3
votes
0answers
62 views

(stability-theoretic) ¨weakly normal groups" are closed under subgroups

Let me first introduce two definitions: For a structure $\mathcal{M}$ in a language $\mathscr{L}$ and a subset $X \subseteq M^n$, the fully induced structure on $X$ is a structure $\mathcal{X}$ with ...
8
votes
1answer
196 views

Reduction modulo p of a linear group over the rational numbers

A paper (http://arxiv.org/pdf/1407.3158v2.pdf) contains the following theorem: Suppose $\mathbb{G}$ is a connected, simply connected, semisimple algebraic group defined over $\mathbb{Q}$, and let ...
0
votes
0answers
12 views

Prime ideals in Iwahori-Hecke algebras

Results on the ideals (especially the prime, completely prime ones) of Iwahori-Hecke algebras (espcially the ones with finite order) is needed. Thank you very much.
-1
votes
1answer
46 views

The map $x \mapsto gx$ gives a homeomorphism $G \rightarrow G$ for algebraic groups.

Let $G$ be an algebraic group. The product group $G \times G$ (taken as a product of varieties) contains the product topology, and is a product with respect to the canonical projections $G \times G ...
0
votes
0answers
9 views

Example of nilpotent linear algebaic group, which is not abelian and not unipotent.

I was just searching for an example of a linear algebraic group $G$ (= subgroup of $GL_n$) which is not abelian and not unipotent. By nilpotent I mean that the central series trivializes, hence the ...
1
vote
0answers
22 views

Morphism from the projective line to an algebraic group

Let $F$ be a field (if require can assume of characteristic $0$) and $\mathbb{P}_F^1$ be the projective line. Suppose $G$ is a connected algebraic group over $F$. We denote the set of $K$-rational ...
1
vote
1answer
59 views

Are unipotent algebraic groups connected?

Is a unipotent algebraic group over a field of characteristic zero always connected?. As far as I know, every unipotent algebraic group over field of characteristic zero is isomorphic to a closed ...