1
vote
1answer
36 views

Variety of Connected Components

In Milne's text http://www.jmilne.org/math/CourseNotes/iAG.pdf (A71), he introduces the "variety of connected components" of a finite type scheme $X$ over $k$ as the universal example of a zero ...
0
votes
1answer
20 views

maximal subtorus of a connected commutative algebraic linear group [closed]

I'm wondering the following: is the maximal subtorus of a connected commutative algebraic linear group over $k$ a) normal and closed b) defined over $k$ (for $k$ a field of characteristic zero, ...
2
votes
1answer
54 views

When is “being a linear algebraic $k$-group” preserved?

Let $G$ be a linear algebraic group over a field $k$, with Char$(k)=0$. What "group-theoretical operations" preserve the property of "being a $k$-linear algebraic group"? For example When ...
0
votes
0answers
15 views

QR decomposition, borel groups and generalizations

Then every matrix $M$ in $M_{m\times m} (\mathbb{C})$ can be written in the form: $QR=M$, where $Q$ is unitary and $R$ is upper-triangular. My question is simple, does this generalize in the ...
1
vote
1answer
31 views

Relative center of relative group scheme

This might be an easy question. Let $p: X \rightarrow S$ be a relative group scheme. In particular the fibers are group schemes. I want to know if there are constructions like the ``relative ...
2
votes
1answer
50 views

Two equivalent definitions of GIT semistable points

Let $X$ be a projective variety, acted on by a reductive algebraic group $G$. We fix a linearization given by the $G$-equivariant ample line bundle $L\to X$. I am aware of two definitions of the ...
3
votes
2answers
66 views

T//W for adjoint type group PGL3

Let $G$ be a reductive algebraic group and $T$ a maximal torus (over $\mathbb{C}$). It is well known that if $G$ is simply connected type then $T//W = \mathbb{A}^r$. I want to verify that the ...
1
vote
0answers
36 views

$p$-divisible group of tori

I am looking for a reference of the following question which should be well known. Let $k$ be any field and $T$ an algebraic torus over $k$ which is not necessarily split. Let $T(l)$ be the ...
2
votes
1answer
25 views

Maximal tori in $SO(n,\mathbb{C})$

What are maximal tori in $SO(n,\mathbb{C})$? (not $SO(n,\mathbb{R})$) Can a maximal torus in $SO(n,\mathbb{C})$ be written as $T\cap SO(n,\mathbb{C})$ for some maximal torus $T$ in ...
1
vote
1answer
62 views

Algebraic groups of multiplicative type in char 0

For a number field $k$ (so of char 0), are algebraic $k$-groups of multiplicative type always linear?
2
votes
2answers
31 views

The set of one-parameter subgroup of the Multiplicative group $G_m$ is Z

Let $G_m= k^{*}=k-{0}$ be the multiplicative group. We know this is an Algebraic group also. How does one prove any algebraic group morphism $G_m \rightarrow G_m$ is of the form $t \mapsto t^{n}$ for ...
4
votes
1answer
48 views

Closed immersion factors through closed immersion

I'm currently working through the proof of Theorem 1, III.12 in Mumford's "Abelian Varieties". Let $G$ be a finite $k$-group scheme acting on an affine $k$-scheme $X:=\text{Spec}~A$ and let $\phi: ...
3
votes
1answer
40 views

Field of Definition of an Algebraic Group

Linear Algebraic Groups- James E. Humphreys Chapter-XII Let $K$ be an algebraically closed field and $k$ be a arbitrary sub-field of $K.$ A closed set X in $A^n=K\times ...$(n times)$\times K$ is ...
0
votes
0answers
38 views

$SU(n)$ as a variety

Consider the algebraic group $SU(n)$ as an algebraic group scheme over $\mathbb R$. Is it birational to an affine space over $\mathbb R$?
1
vote
0answers
20 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
23 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
104 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
72 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
54 views

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
37 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
59 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
31 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
51 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
87 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
59 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
68 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
47 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
2answers
124 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
84 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
76 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
75 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
40 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
209 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
96 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
57 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
157 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
72 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
51 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
56 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
49 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
85 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
91 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
179 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
124 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
82 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
0answers
83 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
55 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
233 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 ...