1
vote
0answers
23 views

Is $U/U(w) = U \cap w U^- w^{-1}$? [closed]

Let $U$ be the maximal upper unipotent subgroup of $GL_n$ and $U^{-}$ maximal lower unipotent subgroup of $GL_n$. Let $U(w) = U \cap wUw^{-1}$. Is $U/U(w) = U \cap w U^- w^{-1}$? Thank you very much.
1
vote
1answer
140 views

Felix Klein's view on algebraic geometry

I think, as a first approach one would say that a geometry on a set $X$ is given by an inner product on $X$. Klein then links geometry to group theory by identifying a geometry on $X$ with a group of ...
4
votes
1answer
97 views

Find a closed subset of an algebraic group, closed under products, which does not contain $e$.

The accepted answer for this question proves the following statement: If $S$ is a closed subset of an algebraic group $G$ which contains $e$ and is closed under taking products in $G$, then $S$ is ...
0
votes
0answers
27 views

definition of semidirect product on two projective spaces

What is the definition of semidirect product on two projective spaces $\mathbb CP^1 \rtimes\mathbb CP^2$ I can not undrestand it.
1
vote
0answers
30 views

Coset variety of an algebraic group

Let $k$ be an algebraically closed field of characteristic $p\geq0$. An affine algebraic group $G$ is an affine algebraic variety (a Zariski closed subset of $k^m$ for some $m$) such that ...
0
votes
1answer
45 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
6
votes
1answer
55 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
4
votes
1answer
46 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
1
vote
1answer
80 views

Question about solvable cocompact subgroups in linear algebraic group over a finite extension of the p-adic numbers

Let $Q_p$ be the p-adic numbers, where p is any prime number. Then $Q_p$ is a locally compact, Hausdorff, totally disconnected (non-discrete) topological field. Let $GL(n,Q_p)$ be the general linear ...
5
votes
1answer
103 views

What is the coordinate ring of $G/U$?

Let $G$ be an algebraic group and $U$ its subgroup consisting all upper triangular matrices. For example, $G=GL_n(k)$ and $U$ the subgroup consisting of all upper triangular unipotent matrices in ...
2
votes
1answer
34 views

Representation-preserving isomorphism

Suppose we have two algebraic groups $G,H$ over a field $K$ (maybe reductive) and we know in advance that $G(A)\cong H(A)$ for some $K$-algebra A. If we attach algebraic representations $\rho_{G(A)}, ...
0
votes
0answers
28 views

parabolic subalgebra

Let $G$ be a semisimple lie group, a parabolic subgroup of $P$ is a connected subgroup that contains a conjugate of $B$, (which $B$ is Borel subgroup of $G$) then I can not see why lie algebra of $P$ ...
2
votes
0answers
152 views

Exceptional isomorphisms of classical algebraic groups

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. An affine algebraic group $G$ is an affine variety over $k$ with a group structure such that multiplication and inversion are ...
1
vote
1answer
83 views

relation between the group O(3) and SU(2)

Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$: a) $O(3)=SO(3)\otimes \{1,-1\}$ b) $SO(3)\simeq SU(2)/Z_2$ Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If ...
3
votes
2answers
175 views

Picard group of a Affine scheme

How do we define a Picard group of an Affine scheme? Is there way to define as for commutative ring? Thanks
3
votes
0answers
97 views

Are simple algebraic groups absolutely simple?

Let $k$ be a field. By a simple algebraic group over $k$ I mean an affine group scheme $G$ of finite type over $k$ such that $G$ is connected, non-commutative and every normal closed subgroup of $G$ ...
3
votes
1answer
133 views

Galois group of $K(X)/K$

Let $K$ be an infinite field, if $K(X)$ is the field of rational function I want to find the Galois group of the extension $K(X)/K$. Lemma 1: If $L$ is a field such that $K\subsetneq L\subseteq ...
5
votes
2answers
148 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 ...
7
votes
1answer
90 views

Finiteness of groups preserving a symmetric positive definite bilinear form

This question arises from reading the note Hodge cycles on abelian varieties by P. Deligne (notes by J.S. Milne). Suppose we are given a group $G$ (for example, either a fundamental group $\pi_1(S, ...
4
votes
2answers
145 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 ...
0
votes
1answer
48 views

how to make factorization by a group action

any algebra and numerical example for Projectivization http://en.wikipedia.org/wiki/Projectivization which book or paper teaching this
8
votes
1answer
414 views

Automorphism group of the elliptic curve $y^2 + y = x^3$

Consider the elliptic curve $E : y^2+y = x^3$ over $\overline{\mathbb{F}_2}$. It has the biggest automorphism group $G$ among all elliptic curves, namely with order $24$. What is the structure of $G$? ...
0
votes
1answer
176 views

Factor group of quaternion group

I have $Q_8=\{I,A,A^2,A^3,B,AB,A^2B,A^3B\}$ where $A=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $B=\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$ Now I take a look at $Q_8/N$ ...
1
vote
0answers
27 views

Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
6
votes
2answers
233 views

For what algebraic curves do rational points form a group?

For what real algebraic curves do rational points form a group ? How does this relate to Jacobian Varieties ?
2
votes
0answers
138 views

Finite automorphism groups of $\mathbb{P}^1$

I would like to know all finite groups of $\operatorname{Aut}(\mathbb{P}^1)$. I am aware of that any automorphism of $\mathbb{P}^1$ is given by Möbius transformation $$ z\mapsto\frac{az+b}{cz+d} $$ ...
2
votes
0answers
53 views

groups acting on curves

Can anybody help me to prove the following statement ? *Let $X$ be a smooth, connected projective curve defined over a number field $k \subset \mathbb{C}$. Let $G$ be a finite group acting on $X$. ...
8
votes
1answer
163 views

Any affine algebraic group is linear.

It is a well-known result that any affine algebraic group is a closed subgroup of some $\mathrm{Gl}_n(\Bbbk)$. However, I would like to see a proof for that, so I looked it up in various books, more ...
1
vote
1answer
150 views

Show that affine transformations are associative

I'm having trouble with the associativity part of showing that $\text{Aff}(\mathbb{R}^2)$ is a group. Recall that if $\left[ A, \overline{r} \right]$ denotes an affine transformation (where $A$ is ...
4
votes
1answer
594 views

Orbit-stabilizer theorem for Lie groups?

Let $G$ be a finite-dimensional lie group, with a transitive action on the points of a smooth finite-dimensional manifold $S$. Let $p$ be some point of $S$ and let $T$ be the stabilizer of $p$ in $G$. ...
1
vote
1answer
243 views

Fencing the Group size,and its implication to Finiteness of Tate-Shafarevich Group

This question is an interesting one,not like my previous one. Can we judge the size of a Quotient Group by seeing the size of its constituents ? To add something ,Suppose consider a group ...
2
votes
0answers
79 views

Is it true that $\forall G\leq Aut(\mathbb{P}^1)=PGL_2(\mathbb{C})$ the map $\mathbb{P}^1\rightarrow \mathbb{P}^1/G$ is defined over $\mathbb{Q}$?

Is it true that for every finite $G\leq Aut(\mathbb{P}^1_{\mathbb{C}})=PGL_2(\mathbb{C})$ the morphism $\mathbb{P}^1_{\mathbb{C}}\rightarrow \mathbb{P}^1_{\mathbb{C}}/G$ descends as a morphism (not ...
12
votes
1answer
212 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 ...