All questions with tag [math: algebraic-groups]


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 to show that the induced morphism $G_K\to G_K$ is surjective, for some finite extension $K/F$. I would like a reference for this fact. Remarks As suc...
2022-07-25 20:38:41

connected algebraic groups

Let $G$ be a linear algebraic group over $\mathbb C$. Let $\psi$ be a finite dimensional regular representation of $G$ into $GL(V)$. Suppose $G$ is connected. I would like to show for $v$ in $V$ the following are equivalent: 1) $\psi(g)(v)=v$ for all $g$ in $G$. 2) $d\psi(X)(v)=0$ for all $X$ in the Lie algebra of $G$. For 1) implies 2), can I ...
2022-07-20 01:36:24

bilinear forms and linear algebraic groups

Let $G$ be a linear algebraic group and $\phi$ a finite dimensional regular representation of $G$ into $GL(V)$ I would like to know about bilinear forms on $V$ and when they are $G$-invariant. Specifically, I want to show that a bilinear form $Z$ on $V$ is $G$-invariant if and only if $Z(d\phi(X)(v),w)=-Z(v,d\phi(X)(w)) \\ \forall v, w \in V, a...
2022-07-17 17:28:04

Galois cohomology of Unitary groups

From Hilbert 90 (or, extra specifically, a generalisation thereof), we understand that the first (Galois) cohomology team of $GL_n$ is unimportant, despite the area of definition. Nonetheless, for unitary teams, which are external kinds of the basic straight team, the tale is various. As I have actually listened to time and again, the first coh...
2022-07-17 15:12:54

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 algebraic groups of transformations. Amer. J. Math. 77, (1955), 355-391. I would like to know if I can find that Theorem in some book about Algebraic Groups. ...
2022-07-14 05:09:39

What's so special about unipotent groups

Why are they so vital? I see them show up in Lie concept, algebraic geometry, etc. Can someone specify? As an example, can a person clarify why they are such all-natural teams to take into consideration in algebraic geometry?
2022-07-14 04:47:28

How to represent the naive PGL functor?

Let $k$ be a field. Consider the functor $F : \mathrm{Alg}(k) \to \mathrm{Set}, ~ R \mapsto \mathrm{GL}_n(R) / R^*$ [You might call this $\mathrm{PGL}_n^{\text{naive}}$, since it does not coincide on the correct $\mathrm{PGL}_n = \mathrm{GL}_n / \mathbb{G}_m$ - only on local $k$-algebras. See also Milne's script on algebraic groups, Example I.9....
2022-07-14 01:51:13

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 the center of $GL(n,F)$. Is $\gamma$ conjugated to an element in $Z(F)GL(n, o)$, where $o$ is the ring of integers of $F$? Heuristic: The central...
2022-07-12 14:49:02

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 the case of quasi-projective varieties) that for any $x$ in $X$, the orbit $G_x$ of $x$ is contained in an affine open subset of $X$. A classical re...
2022-07-11 23:40:07

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 Algebraic Groups: A subvariety $X$ of $\mathbb A^n$ is $k$-closed if $X$ is the set of zeros of some collection of polynomials having coefficients in...
2022-07-11 08:44:25

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

This is the problem: Exhibit 3 reductive teams of measurement $4$ and also semisimple ranking $1$ which are pairwise nonisomorphic (as algebraic groups). I recognize that for any kind of reductive team $G$ of semisimple ranking $1$, $G = (G,G)Z$, where $(G, G)$, the acquired subgroup of $G$ is semisimple, of measurement $3$, and also $Z$ is ...
2022-07-08 06:51:03

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 think of another fact whose correctedness is proved on the book: Let $H$ be a subgroup (not necessarily closed) of a connected solvable group $G$, $H$...
2022-07-08 06:48:27

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

There is a workout on web page 114 of Humphreys' Linear Algebraic Groups (GTM 21) Prove that $SO(3, \mathbb R)$ (= team of $3 \times 3$ actual orthogonal matrices of determinate $1$) is a linked subgroup of $SL(3, \mathbb C)$ containing semisimple components, yet not commutative. The semisimplicity of the components and also the noncommutat...
2022-07-07 02:37:55

Unitary (algebraic) groups

I am seeking referrals on unitary teams in the algebraic setup: that is, offered a square expansion $E/F$, the unitary teams (if I recognize appropriately) are subgroups of the Weil constraint of scalars of $GL(n)$ from $E$ to $F$, which are dealt with by some involution. Extra especially, I am seeking some type of category, which ones are seemi...
2022-07-07 02:16:26

Iwasawa Decomposition for $SL(n,\mathbb{R})$.

Can a person please straight me to an excellent reference for Iwasawa disintegration of this Lie team?, I read that I require to make use of below orthogonalization procedure of Schmidt from Linear algebra, yet I am not exactly sure just how to do this. Many thanks.
2022-07-06 01:00:10

What are the characters of SL_2 and PSL_2?

How to compute $X^*(SL_2) = \operatorname{Hom}(SL_2,\mathbb{G_m})$ and also $X^*(PSL_2) = \operatorname{Hom}(PSL_2,\mathbb{G_m})$? ($SL_2$ and also $PSL_2$ are considered as algebraic groups over an area $K$) For $SL_2$, I attempted to do it with Hopf algebra, which brings about compute $\operatorname{Hom}(K[X,X^{-1}],K[A,B,C,D]/(AD-BC-1))$, ye...
2022-07-06 00:14:14

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 $g\geq1$. One knows that the $p$-torsion of $A$ is a product: $$A[p]=\hat A[p]\times T_p(A)\otimes(\Bbb Z_p/p\Bbb Z_p).$$ Here $\hat A[p]$ is the maxima...
2022-07-04 21:12:36

Dimension of a quotient

My inquiry is instead "simple" to ask: what is the measurement of the ratio selection $GL_3/U$, where $U$ is the (shut) team of upper triangular unipotent matrices (= upper triangular matrices with 1 gets on the angled). Prior to working with that ratio selection, I've worked with $SL_2/U$ and also I could establish the measurement ma...
2022-07-04 17:38:11

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 of Drinfeld Modular ... ": I have an issue with the following two assertions $P = MN$ has a Levi decomposition over $k$ (pg.123) and $\...
2022-07-03 02:14:19

Why are parabolic subgroups called "parabolic" subgroups?

I made use of to assume that points called "parabolic" have to have something to do with parabolas or their specifying square formulas. Actually, terms like allegorical coordinate, allegorical partial differential formula and more, are without a doubt connected to parabolas and also their formulas. Yet, why are allegorical teams in al...
2022-07-01 19:33:29