Tagged Questions

4
votes
2answers
89 views

Definition of Unipotent in Positive Characteristic

Let $G$ be an affine algebraic group over an algebraically closed field $k$ whose characteristic is $p>0$. Can $\mathcal{U}(G)$, the set of unipotent elements of $G$, be characterized as all ...
0
votes
0answers
34 views

Eigenvectors of algebraic group representation

In paper http://arxiv.org/abs/math/9905053 of Kollar and Szabo there is a lemma (Lemma $1$) in which the following terminology is used: "Every representation $H\rightarrow GL(n,K)$ has an ...
4
votes
2answers
124 views

Definitions in a Theorem of Lang

I'm trying to understand the following theorem due to Serge Lang (Algebraic Groups over Finite Fields, 1956, Theorem 2): Let $p$ be a prime, and let $k$ be a finite field of $q=p^n$ elements. Let ...
4
votes
1answer
54 views

Why is this group action a morphism of varieties?

I am examining a proof that any affine connected algebraic group is a closed subgroup of some $GL_n$, and I am stuck on a fine point. Let $G$ be an affine algebraic group in the sense that it is a ...
6
votes
1answer
87 views

If a group scheme $G$ operates on another scheme $X$, how do you define orbits?

In my specific case, $G=\mathrm{Spec}(k[M])$ is an algebraic torus acting on a toric variety $X_\Sigma$ corresponding to a fan $\Sigma$ when $k$ is not necessarily algebraically closed (or maybe even ...
1
vote
0answers
43 views

Do “multiples” of open dense sets of an algebraic group union to the whole group?

Let $G$ be an algebraic group. From algebraic geometry we know there exists an open dense subset $U$ of $G$ such that $U$ is nonsingular. Since left multiplication of $U$ by elements of $G$ is an ...
4
votes
1answer
75 views

Characters of diagonalizable algebraic groups with no p-torsion

Let $G$ be a diagonalizable algebraic group and $X$ be the character group of $G$. Let $Y$ be a subgroup of $X$. We define $Y^{\perp}$ to be all the $x\in G$ such that $\chi(x)=1$ for all $\chi\in Y$. ...
1
vote
0answers
92 views

Character of affine algebraic groups

Let $G$ be an affine algebraic group. A character of $G$ is a morphism $G\to \mathbb G_m$. Let $X$ be the abelian group of all characters of $G$. Suppose this group is finitely generated, say by ...
1
vote
1answer
90 views

Linear system of divisors on complete variety

I am currently reading Mumford's abelian varieties and Milne's notes on them and I have a problem understanding the proof that they are projective. Both of them use that a complete linear system of ...
2
votes
1answer
46 views

Maximal tori in semi-simple linear algebraic groups

Let $G$ be an algebraic group over an algebraically closed field. Furthermore, let $G$ be semi-simple, i.e. its radical (viz. its maximal closed, connected, solvable normal subgroup) is trivial. One ...
5
votes
1answer
92 views

Which algebraic variety can become a algebraic group?

First, I know the algebraic group must be non-singular and the index of the identity component must be finite. Now given a algebraic variety (especially for a algebraic curve or a algebraic surface ...
3
votes
1answer
58 views

Universal parametrization for orthogonal matrices

Let $k$ be a field whose characteristic is zero and let $n\geq 1$. Say that a matrix $M\in {\cal M}_{n\times n}(k)$ is almost orthogonal if $M^{T}M$ is a nonzero multiple of the identity. Denote the ...
4
votes
0answers
45 views

quotient group scheme

assume I have a group $G$ over a field of char 0 and $H$ a closed subgroup. When is it true that the group $N(H)/H$ is representable? If $G$ has nice properties, like to be reductive or unipotent is ...
5
votes
1answer
66 views

Dimension of the GL-orbit of d-forms in one less variable

Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For ...
5
votes
1answer
60 views

What is the universal deformation of $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$?

Lubin and Tate show in their paper Formal moduli for one-parameter formal Lie groups that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of ...
2
votes
1answer
68 views

Extension of a short exact sequence of group schemes

Let $S$ be a Dedekind scheme with rational functions field $K$. Consider an exact sequence $$ 0 \to G'_K \to G_K \to G''_K \to 0$$ of smooth $K$-group schemes of finite type. Assume that $G'_K$ and ...
2
votes
2answers
82 views

Understanding Multiplication in a Linear Algebraic Group

I am reading Springer's Linear Algebraic Groups and have a question about how ordinary group multiplication ( in the ordinary group theory sense) translates to that in term so linear algebraic groups. ...
5
votes
0answers
43 views

When is an orbit spherical?

Let's assume we have an affine, reductive, algebraic, complex, generally friendly and cooperative group $G$ acting algebraically on a variety $X$. Let $x\in X$ be some point. Under what conditions on ...
2
votes
0answers
115 views

Fibers of flat morphism are isomorphic as G-modules

Let $f:X\to Y$ be a flat surjective morphism with reduced fibers between affine varieties over an alg. closed field of char. zero, k. Let G be a reductive group acting on X fiberwise. How do I show ...
2
votes
1answer
49 views

Characters of affine algebraic groups and the determinant

Let $G$ be an affine algebraic group (i.e. a $k$-variety which is also a group and the group multiplication and inversion are morphisms of varieties). A character of $G$ is a morphism of algebraic ...
2
votes
2answers
101 views

Embedding elliptic curves into the general linear group

Is it possible to embedd an elliptic curve $E:\;\; y^2=x^3+ax+b$, defined over an algebraically closed field $k$, into some $GL_n(k)$ ?
5
votes
1answer
46 views

When is a subgroup the Weil restriction of another subgroup?

Let $M/F$ be an extension of fields. Let $G$ be an algebraic group over $F$, and consider the $F$-group $H$ defined as the restriction of scalars (a la Weil) of $G$ from $M$ down to $F$, i.e. ...
1
vote
1answer
84 views

$F$-rational points of variety/alg. group in Springer

I am confused about something on page 6 in Springer's Linear Algebraic Groups. Setup: $k$ is an algebraically closed field $X$ is a closed set in $k^n$. $I(X) = \langle f\in k[T_1, \dots, T_n] : ...
4
votes
3answers
105 views

How to define the base extension of a group action on a scheme

Suppose $G/S$ is a group scheme over $S$, $X/S$ is a scheme over $S$. $G$ acts on $X$ by the morphism $ \sigma : G \times_S X \to X$. Let $X'$ be a scheme over $X$. How to deine the group action on ...
2
votes
0answers
51 views

Sufficient condition for surjectivity of a morphism of group schemes

Let $G$ be a group scheme over a field $F$, and let $f:G\to G$ be a homomorphism. Written in my notes, I have the following statement: To check surjectivity (on $F$-rational points), it suffices ...
4
votes
0answers
66 views

Any local algebraic group is birationally equivalent to an algebraic group

In this paper, page $6$ the authors state the following: By Weil’s theorem $[17]$, any local algebraic group is birationally equivalent to an algebraic group. Where $[17]$ A.Weil. On ...
3
votes
1answer
91 views

What's so special about unipotent groups

Why are they so important? I see them appear in Lie theory, algebraic geometry, etc. Can somebody elaborate? For example, can someone explain why they are such natural groups to consider in ...
2
votes
1answer
124 views

Conjugacy classes in GL(n)

Given an element $\gamma$ in $GL(n,F)$, where $F$ is either a global field or a non archimedean local field. Assume $\gamma$ is elliptic, i.e. its characteristic polynomial irreducible. Let $Z(F)$ be ...
2
votes
1answer
77 views

The difference between $k$-closed variety and variety defined over $k$

Let $K$ be an algebraically closed field, and $\mathbb A^n$ the affine-$n$ variety over it. Suppose that $k$ is an arbitrary subfield of $K$. There are definitions on page 217 of Humphreys' Linear ...
3
votes
1answer
145 views

Coherent $G$-sheaf on algebraic varieties

Let $X$ be an algebraic variety (i.e. an integral separated scheme of finite type over an algebraically closed field $k$) and let $G$ be a finite group of automorphisms of $X$. Suppose (as we may in ...
2
votes
0answers
93 views

Examples of reductive groups of dimension $4$ and semisimple rank $1$

This is the problem: Exhibit three reductive groups of dimension $4$ and semisimple rank $1$ which are pairwise nonisomorphic (as algebraic groups). I know that for any reductive group $G$ of ...
1
vote
0answers
183 views

Is the centralizer of a semisimple element in a connected algebraic group always connected

There is an exercise on page 142 of Humphreys' Linear Algebraic Groups: Ex.1 Let $G$ be a connected algebraic group, $x \in G$ is semisimple. Must $C_G(x)$ be connected? When $G$ is solvable, I ...
3
votes
0answers
84 views

The connectedness of $SO(3, \mathbb R)$

There is an exercise on page 114 of Humphreys' Linear Algebraic Groups (GTM 21) Prove that $SO(3, \mathbb R)$ (= group of $3 \times 3$ real orthogonal matrices of determinate $1$) is a connected ...
0
votes
0answers
36 views

An example of nonconnected diagonalizable group, and the case when $X(T)$ and $Y(T)$ are not dual $\mathbb Z$-modules

A diagonalizable (algebraic) group is a group isomorphic to a closed subgroup of the diagonal group $D(n,K)$ for some $n$. For a diagonalizable group $T$, X(T) denotes the character group of ...
1
vote
0answers
87 views

Notation for a canonical quotient of an abelian variety in positive characteristic

This may be a somewhat silly question, but there it goes. Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian variety of dimension $g\geq1$. ...
1
vote
0answers
77 views

Dimension of a quotient

My question is rather "simple" to ask : what is the dimension of the quotient variety $GL_3/U$, where $U$ is the (closed) group of upper triangular unipotent matrices (= upper triangular matrices with ...
3
votes
0answers
96 views

Jordan decomposition/Levi decomposition in GL(n) in positive characteristic

Let $k$ be a non archimedean field of positive characteristic. Lets consider a parabolic subgroup $P \subset GL(n, k)$. I am a little bit confused by the following statement in "Laumon - Cohomology ...
11
votes
0answers
159 views

Does the Zariski closure of a maximal subgroup remain maximal?

Let $k$ be an algebraically closed field and let $G\leq\rm{GL}_n(k)$. Assume that $M<G$ is a maximal subgroup (in the abstract group sense). Denote by $\bar{G}^Z$ the Zariski closure of $G$ in ...
3
votes
0answers
129 views

How to prove the connectedness or irreducibility of a variety?

Let $G$ be an algebraic group, and $\mu$ is the multiplication in $G$. Define a morphism $G \times G \times G \times G \rightarrow G \times G$ by $\mu \times \mu$, and let $X$ be the inverse image of ...
3
votes
0answers
191 views

question regarding Waterhouse, affine group schemes

Excerpt from Waterhouse, 14.4 Structure of Finite Connected groups. Thm. Let $A$ represent a finite connected group scheme over a perfect field of characteristic $p$. Then $A$ has the form $k[X_1, ...
7
votes
2answers
422 views

Is the universal cover of an algebraic group an algebraic group?

Here algebraic group means affine algebraic group in both instances. Also I'm mainly interested in groups over $\mathbb{C}$. In fact I'm taking $\pi_1(G)$ to mean the fundamental group of $G_{an}$, ...
5
votes
1answer
215 views

Checking if Algebraic Groups are simply connected

I have recently been thnking some about algebraic groups and reading parts of Humphreys book on them, and I was wondering if there is a general process to showing they are simply connected. In ...
28
votes
1answer
685 views

Is there an atlas of Algebraic Groups and corresponding Coordinate rings?

I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings. Edit: The previous wording was terrible. Given an algebraic group $G$, with Borel ...