For questions about groups which have additional structure as a algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. Consider using with the (group-theory) tag.

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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 ...
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1answer
83 views

Reduction of algebraic groups

Let $G$ be an algebraic group over $\mathbb{Z}_p$ embedded in $GL_n$. Let's say that $G$ is a family of equation $f \in \mathbb{Z}_p[(X_{i,j})_{i,j}]$ such that the set of invertible matrices ...
4
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1answer
114 views

Galois action on character group

Let $T$ be an algebraic group of multiplicative type over a field $K$. Let $$X^*(T)=\operatorname{Hom}_{\overline{K}}(T_{\overline{K}},(G_m)_{\overline{K}}) = ...
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1answer
126 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 ...
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0answers
53 views

Open Subgroups of Affine algebraic groups

I saw this in a paper that I've been reading and I've been trying to figure out if this is true or not. Let $G(F)$ be a affine, simple, connected, adjoint, algebraic group over a local field endowed ...
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89 views

Are matroids groups?

A mathematician handed me this note about what he said was about matroids and that matroids are groups (like algebraic groups). Is that true? What is this mathematics about? Is this even about ...
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1answer
68 views

Meaning of linearization of an action

What means the following expression: Every action of an affine algebraic group on an affine algebraic variety can be linearized.
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2answers
243 views

Levi decomposition for the parabolic subgroups

This question is for the algebraic groups. I find I cannot understand Levi decomposition for the parabolic subgroups well. Denote the parabolic subgroup is P=LV, L is Levi subgroup. I guess that for ...
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105 views

Is there a classification of finite abelian group schemes?

There is a well-known classification of finite abelian groups into products of cyclic groups. What about finite abelian group schemes, where we may put in the qualifiers "affine", "etale", or ...
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1answer
69 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 ...
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1answer
83 views

Isomorphism between reductive groups implies isomorphic Levi subgroups of prescribed type?

I've come up with the following question. Assume that $G$ and $G'$ are two isomorphic reductive algebraic groups over an algebraically closed field $k$. If $P$ (resp. $P'$) is the standard parabolic ...
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1answer
100 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 ...
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2answers
92 views

A question concerning unipotent matrices and a basis choice

Let $G$ be a subgroup of $\mathbb{GL}_n$, the group of all invertible $n\times n$ matrices over an algebraically closed field $k$, which consists of unipotent matrices. I don't understand a step in ...
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1answer
142 views

The regular semisimple element in the algebraic group

Suppose some maximal torus $T$ of $G$ is $C_G(T)$, then the set of semisimple elements for which $C_G(s)$ is a torus contains a nonempty open set. Such element are called regular semisimple. I want ...
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0answers
32 views

Does a local commutative algebraic group be commutative?

Does a local commutative algebraic group G be commutative ? Here local commutative means for any point g, there is an open set U containing g in the algebraic group G and for any x and y belong to ...
4
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1answer
81 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 ...
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63 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 ...
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1answer
249 views

Notations in Group theory

I will start by apologizing as many will not like this question. I am reading the paper COHOMOLOGY THEORY OF GROUPS WITH A SINGLE DEFINING RELATION and having focused on typology throughout my ...
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0answers
41 views

Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

I'm new to algebaric groups, so please accept my apology in advance if I post crazy questions. Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ ...
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1answer
72 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 ...
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1answer
70 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 ...
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1answer
79 views

A question about Lie algebras corresponding to Lie groups and algebraic groups

Lie groups and algebraic groups both correspond with Lie algebras, which are by definition the left invariant vector field. But the topology of Lie groups and algebraic groups are different. Are their ...
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0answers
129 views

When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
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33 views

Analogs of group schemes over non-commutative rings

For a commutative ring $R$, I can consider $\operatorname{GL}_n(R)$ as a group scheme over $\operatorname{Spec} R$. Are there analogs of this notion when $R$ is non-commutative, say $R = ...
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2answers
172 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 ...
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2answers
268 views

$A[x,y] \not \simeq A[x^2, xy, y^2]$

Is it true that for any (commutative, unital) ring $A$ that $A[s,t], A[x^2, xy, y^2]$ cannot be isomorphic as rings? This is mentioned in passing in Eisenbud-Harris, Geometry of Schemes Exercise ...
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1answer
83 views

Application of Chevalley's Theorem on algebraic groups over DVR

Let $R$ be a DVR with fraction field $K$ and a perfect residue field $k$. Consider a commutative connected group scheme $G$ over $R$, say smooth and of finite type over $R$. So we may use Chevalley's ...
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0answers
50 views

How does the Frobenius map permute the roots

How can a Frobenius map permute the roots of an algebraic group? According to Carter (in Finite groups of Lie type), a root subgroup $X_{\alpha}$ is the 1-dimensional unipotenet subgroup giving rise ...
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0answers
70 views

Example of algebraic groups without a split $BN$-pair.

In Finite Groups of Lie Type written by Carter, a $BN$-pair of a group is defined to be two subgroups $B$ and $N$ such that $G$ is generated by $B$ and $N$. $B\cap N$ is normal in $N$. ...
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2answers
449 views

What is a $p$-adic group

I saw the name $p$-adic group on a book I was reading, so I tried to find some related documents. Although I've found something on this topic, there is no definition. Would anyone please explain the ...
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2answers
90 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. ...
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2answers
677 views

References on Linear Algebraic Groups/Lie Theory

I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the ...
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0answers
141 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 ...
4
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1answer
167 views

Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag

I am looking for a reference for the above statement that every parabolic subgroup of $\mathrm{Sl}_n(\Bbbk)$ stabilizes some flag in $\Bbbk^n$. I have gone through a large pile of books and can't seem ...
5
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1answer
215 views

When do all derivations come from automorphisms?

Let $k$ be a field (algebraically closed for simplicity) and let $A$ be an $n$-dimensional algebra over $k$ (not necessarily commutative or even associative). The group $G=\mbox{Aut}(A)$ is an ...
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1answer
76 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 ...
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5answers
50 views

Nature of algebraic structure

I am given $G = \{x + y \sqrt7 \mid x^2 - 7y^2 = 1; x,y \in \mathbb Q\}$ and the task is to determine the nature of $(G, \cdot)$, where $\cdot$ is multiplication. I'm having trouble finding the ...
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1answer
71 views

Geometric difference between two actions of $GL_n(\mathbb{C})$ on $G\times \mathfrak{g}^*$

Let $G=GL_n(\mathbb{C})$. Scenerio 1: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ g.(x,y)=(gx,y). $$ Scenerio 2: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ ...
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1answer
87 views

Dominant weight as a positive combination of simple roots

Let G be a semisimple algebraic group. I can see (geometrically) why every dominant weight has to be a non-negative combination of simple roots (and if it strictly dominant then it has to be a ...
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2answers
81 views

Identifying $SL(2,\mathbb{C})/H$ with $\mathbb{C}^2\setminus \{ 0\}$

Let $G=SL(2,\mathbb{C})$ and let $H$ be the set of unipotent matrices $$ \left\{ \left[ \begin{array}{cc} 1 & b \\ 0 & 1 \\ \end{array} \right] : b\in \mathbb{C}\right\}. $$ I am ...
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0answers
64 views

Why $G\to G/H$ is faithfully flat?

Some questions about algebraic groups. Let $G$ be an affine algebraic group over algebraically closed field $k$. Questions: 1. $G$ is faithfully flat since it is defined over field? 2. Let $H$ be a ...
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2answers
294 views

Connectedness of centralizer exercise

I'm having trouble understanding connectedness from a group theoretic perspective. Let $G$ be the symplectic group of dimension 4 over a field $K$, $$G = \operatorname{Sp}_4(K) = \left\{ A \in ...
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1answer
59 views

Is a codimension one closed subgroup of a connected linear algebraic group is automatically normal?

Let $G$ be a connected linear algebraic group and let $H$ be a closed subgroup of $G$ of codimension one. Is $H$ necessarily normal?
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2answers
115 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 $GL_n(k)$ ?
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1answer
147 views

Highest or positive weights (or roots)

Let $T= (\mathbb{C}^*)^2$ be embedded in $GL_2$ along its diagonal entries, and suppose $T$ acts on $M_2(\mathbb{C})$ via conjugation. Denote $\chi_i(g)=z_i$ where $$ g = \left( \begin{array}{cc} z_1 ...
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2answers
150 views

Diagonalizable linear algebraic group is isomorphic to $(\mathbb{C}^*)^r\times A$, for some finite abelian group $A$

I have three questions about algebraic groups. Let $D$ be a linear algebraic group. Then the following are equivalent: $D$ is diagonalizable. $\mathop{Hom}(D,\mathbb{C}^*)$ is finitely generated ...
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0answers
37 views

The nonexistence of nontrivial solvable series in $M_n(k)$

I am a bit confused about semisimple Lie algebras. For the sake of simplicity, let's take $\mathfrak{g}=M_n(k)$ where $k=\bar{k}$. According to Wiki, $M_n(k)$ is solvable if the radical of $M_n(k)$ ...
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0answers
110 views

Why are $V =K^n$ and its dual isomorphic $SL(2,K)$-modules

In this paper(http://arxiv.org/pdf/1204.6131.pdf), the following statement $V =K^n$ and its dual are isomorphic $SL(2,K)$-modules, seems to be common sense. Here, $K$ is a field of ...
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1answer
77 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. ...
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117 views

Hecke characters and Unitary groups

Let $M/F$ be a quadratic extension of number fields, with Galois group $G=\{1,\tau\}$. Consider the following unitary group $$U_1(R)=\{z\in (R\otimes_FM)^\times :zz^\tau=1\},$$ where $R$ is an ...