1
vote
0answers
23 views

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

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
0answers
18 views

What are the elements in $U/U(w)$?

Let $U$ be the maximal upper unipotent subgroup of $GL_n$. Let $U(w) = U \cap wUw^{-1}$. Then $$ U(w) = \{(a_{ij}) \in U: a_{ij} = 0, \text{ if } i<j, w^{-1}(i) < w^{-1}(j) \}. $$ My question ...
0
votes
1answer
22 views

How to compute $U \cap wUw^{-1}$?

Let $U$ be the upper unipotent subgroup of of $GL_n$. It is said that $$ U \cap wUw^{-1} = \{ (a_{ij}) \in U \mid a_{ij}=0, i<j, w^{-1}(i) > w^{-1}(j) \}. $$ How to prove this? I try to compute ...
4
votes
1answer
96 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 ...
3
votes
1answer
66 views

$SU(2)$ as an algebraic group

The $\mathbb R$-valued points of the algebraic group $SU(2)$ can be identified with the real 3-sphere. But how does one define $SU(2)$ over the base field $\mathbb R$ as an algebraic group? What are ...
1
vote
0answers
48 views
+50

Regular elements of a module is open and dense

Let $\mathfrak{g}$ be an algebraic Lie algebra and $V$ be a $\mathfrak{g}$-module, then for each $v\in V$, define $\mathfrak{g}^v = \{x\in\mathfrak{g}:xv = 0\}$. Let $V_{reg}$ be the set of all $v$ ...
0
votes
0answers
18 views

Does taking $G$-invariant sections commute with infinite $\oplus$?

Let $X$ be a projective scheme over an algebraically closed field $k$, and let $G$ be a reductive algebraic group acting on $X$. If $R=\bigoplus_{n\geq 0}H^0(X,L^{\otimes n})$, then $G$ acts on $R$ ...
0
votes
0answers
24 views

The homogeneous coordinate ring attached to a $G$-linearized action

I am reading Huybrechts-Lehn's book on moduli of sheaves, and on page $85$, when they discuss GIT, the setup is as follows: $X$ is a projective scheme over an algebraically closed field $k$, there is ...
0
votes
0answers
26 views

Intuition and examples of weil restriction

It is hard for me to catch the basic ideal of weil restriction, I need some simple examples to understand it, the following exercises are from the Springer's book: Linear Algebraic Groups 11.4.20(3). ...
3
votes
1answer
47 views

Are semisimplicity and regularity closed or open conditions in an algebraic group $G$?

Let $G$ be a connected algebraic group over an algebraically closed field. I'm trying to understand the phrase "the subvariety of semisimple elements in $G$ which are not regular." This tacitly ...
1
vote
0answers
29 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 ...
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 ...
2
votes
2answers
81 views

Dimension of a $G$-variety $X$ that is a finite union of $G$-orbits

Suppose that $G$ is an algebraic group acting on a variety $X$, and $X$ is a finite disjoint union of $G$-orbits $\mathcal{O}_i$, $i=1,\ldots,n$, under this action. Is it true that the dimension of ...
2
votes
1answer
58 views

Confusion about the quotient $G/B$

Let $G$ be an affine, complex, reductive algebraic group and $B$ a Borel of $G$. I have seen and understood the proof that $G/B$ is projective. Now, on the other hand, I have made the following ...
4
votes
1answer
65 views

Linearization of a group action: why the map is equivariant?

I'm using Dolgachev's book on invariant theory to learn linearizations of group actions. Here is a sketch of main construction: let linear algebraic group $G$ act on a quasi-projective variety $X$, ...
1
vote
1answer
39 views

Torus orbit closures and rank-1 subtori

Suppose I have a connected complex torus $K$ acting on a quasi-affine complex variety $X$. Suppose also that I have $p,q\in X$ such that the orbit $Kq$ is closed in $X$ and $q\in ...
5
votes
1answer
101 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 ...
1
vote
1answer
67 views

What is the group of $k$-rational points of an algebraic group?

Let $k$ be a field and $G$ a linear algebraic group over $k$. What is the group of $k$-rational points of $G$? By definition, $G$ is an algebraic variety. Suppose that $G$ is defined by polynomials ...
2
votes
1answer
73 views

The $\mathbb C((z))$-rational points of a complex semi-simple group $G$

By definition, if $R$ is a $\mathbb C$-algebra and $G$ is a $\mathbb C$-scheme then the set of $R$-valued points on $G$ is $G(R)=\hom_{\text{Sch}_{\mathbb C}}(\operatorname{Spec} R, G)$ In Ginzburg's ...
1
vote
0answers
74 views

P/B is isomorphic to the projective line $\mathbb{P}^1$

Suppose that $P \subset G$ is a parabolic subgroup containing a Borel subgroup $B$. Moreover, let $P$ be a minimal parabolic subgroup properly containing B, i.e., one corresponding to a single root ...
1
vote
0answers
35 views

Orbits and rational points in a $G$-variety

Let $K/k$ be a field extension, let $V_0$ be a variety over $k$, and let $V=V_0\times_k\mathrm{Spec}\;K$, so that we can speak of the $k$-rational points of $V$ as morphisms $\mathrm{Spec }\;k\to ...
7
votes
1answer
174 views

Geometric intuition for linearizing algebraic group action

I am reading an overview of geometric invariant theory and find myself stuck when we begin linearizing the action of an algebraic group on a variety. The definition given in my notes is that given an ...
3
votes
2answers
89 views

“$L/K$ forms of each other”

In section $4$ of these notes, the author says two algebraic groups $G$ and $H$ defined over a field $K$ are "$L/K$ forms of each other" if they are "isomorphic over $L$", where $L$ is a finite field ...
2
votes
1answer
56 views

Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
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
0answers
30 views

Exactness of Hom functor for torus representations?

Given a reductive algebraic group $G$ and a maximal torus $T$. Is it true that the functors $$ Hom_T(-,\lambda) $$ are exact, where $\lambda$ denotes one of the the simple one-dimensional ...
4
votes
1answer
67 views

Is a finite normal subgroup of a reductive algebraic group central?

In a proof I am reading, the author considers the situation where $G$ is a reductive algebraic group (variety) over the complex numbers $\mathbb C$ and $N\trianglelefteq G$ is a closed, normal ...
2
votes
1answer
42 views

Finding the Hopf Algebra Coproduct coming from an Affine Group Scheme

I was wondering if anyone could help with how to, strictly from Yoneda's Lemma, obtain the coproduct map on the Hopf Algebra for an Affine Group Scheme. Particularly for something like $\text{SL}_2$ ...
1
vote
1answer
54 views

Automorphisms of algebraic groups

I am working through Humphreys' Linear algebraic groups, and I am stuck on the following exercise ( ex 4 on pg 57) I need to show that the only automorphisms of $G_m$ (as an algebraic group) is $x ...
3
votes
1answer
48 views

Torsion Subgroups and Periodicity

I am trying to piece together elliptic curves in FLT and would greatly appreciate corrections to my summary (or attempts thereof). Mazur's paper "Number Theory as Gadfly" states, "there is a natural ...
3
votes
0answers
84 views

Formal groups: why the axiom $F(X,Y) \equiv X+Y \pmod {\langle X,Y\rangle^2}$?

Whenever reading about formal groups, this axiom has always appeared to me as a bit artificial, at least compared to the other axioms. To explain what I mean, suppose that $R$ is a ring, and that we ...
0
votes
0answers
86 views

Picard group of a commutative rings

I am trying to read myself about Picard group. This is really interesting for me. A Prüfer domain $R$ is a Bézout domain iff Picard group of $R$ is zero. Is there some good properties of Picard group ...
2
votes
0answers
135 views

Construction of line bundles on the flag variety

Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags... I am a bit confused about the subject in the title. For ...
3
votes
1answer
106 views

How to show that $G_a$ and $G_m$ are connected?

Let $K$ be a field and $A^1$ the affine line. Let $G_a$ be the affine line $A^1$ with group law $\mu(x, y)=x+y$. Let $G_m$ be the affine open subset $K^* \subset A^1$ with group law $\mu(x,y)=xy$. How ...
1
vote
0answers
79 views

Questions about the book linear algebraic groups by Springer.

I am reading the book linear algebraic groups by Springer. I have a question on Page 53, on line 3, it is said that $d-h \geq p$ implies that ${d-h \choose p} \not\equiv 0 \pmod p$ by Lemma 3.4.2. But ...
3
votes
1answer
73 views

Splitting field of a torus

Let $T$ be a torus over some field $k$ (not necessarily perfect). Is there a smallest extension $k'$ of $k$ such that $T \times_{\operatorname{Spec}k} \operatorname{Spec}k'$ is a split torus over ...
1
vote
1answer
54 views

Questions about the proof of the isomorphism $k[X] \to \mathcal{O}(X)$ in the book Linear algebraic groups by Springer.

I am reading the book linear algebraic group by Springer. I have some questions on page 8. Theorem 1.4.5 is: $\phi: k[X] \to \mathcal{O}(X)$ is an isomorphism. (1) Line 8-9 of the proof of Theorem ...
3
votes
1answer
47 views

$GL(-)$ as a k-group functor

My question is essentially may lye simply in a notational obstruction. For a k-algebra M, Jantzen J. defines the k-group functor $GL(M)$ as: $GL(M)(A):=(End_A(M\otimes_{\mathbb{k}} A)^*$. My ...
4
votes
2answers
207 views

Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group?

By a complex reductive algebraic group I mean the group of complex points of a (possibly disconnected) affine algebraic group defined over $\mathbb{C}$ whose unipotent radical (maximal connected ...
5
votes
0answers
92 views

Motivation?: Lie algebra and algebraic group Cohomology

This is just an a-priori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
3
votes
1answer
62 views

The identity component of an algebraic group is always parabolic

Essentially I was wondering if the quotient of an algebraic group $G$ by its identity component $G^0$ is necessarily always parabolic. My argument: This seems right since $G^0$ is a closed ...
3
votes
1answer
74 views

Why is the map: $GL_n(K)\times GL_n(K) \to GL_n(K)$ regular?

Let $K$ be a field and $GL_n(K)$ the set of all invertible $n$ by $n$ matrices over $K$. Let $m: GL_n(K)\times GL_n(K) \to GL_n(K)$ be the usual multiplication of matrices. Why the map $m$ is regular? ...
4
votes
2answers
158 views

Automorphism Groups of Projective Algebraic Surfaces

When does a projective algebraic surface have an infinite automorphism group? Is there a simple criterion, or at least a sufficient condition?
0
votes
0answers
34 views

Calculating this quotient?

What is the quotient of the algebraic Groups $GL_n$ by $Sp_n$ equal to? I was conisering a different example and would use the universal property to establish it, but I wasn't certain what it should ...
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 $G$ ...
1
vote
0answers
50 views

The symplectic transvection group is isomorphic to what $A^1$, $A^2$?

The transvections generate the symplectic group, but to demonstrate its connectedness I need to establish their connectedness. I was thinking of doing this through an isomorphism of alg. groups with ...
7
votes
1answer
135 views

Is taking inverse automatically well-defined?

By the usual definition, Lie group is a manifold $G$ with a group structure on it such that the multiplication $m\colon G\times G\to G$ and taking inverse $i\colon G\to G$ are both smooth maps. But it ...
2
votes
0answers
49 views

The set of regular points in an algebraic group

I saw the following fact in a paper I was reading and I was wondering if someone could provide a reference. Let $K$ be a non-archimedean local field (say of characteristic $0$), and let $G$ be a ...
5
votes
1answer
103 views

The regular representation for affine group schemes

I want to understand the regular representation of an affine algebraic group. An affine algebraic group as I know it, is a functor from the category of $k $ -algebras to groups that is representable ...
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 ...