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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9
votes
0answers
131 views

a closed subset of an algebraic group with a constant tangent space is a coset

Let $G$ be an algebraic (not necessarily linear) group and let $Z \subset G$ be a Zariski closed irreducible subset. Since tangent bundle of $G$ is trivial, we may identify tangent spaces at all ...
8
votes
0answers
119 views

p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie ...
8
votes
0answers
115 views

Why are parabolic subgroups called “parabolic” subgroups?

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential ...
6
votes
0answers
119 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 ...
6
votes
0answers
157 views

Galois cohomology of Unitary groups

From Hilbert 90 (or, more precisely, a generalisation thereof), we know that the first (Galois) cohomology group of $GL_n$ is trivial, no matter the field of definition. However, for unitary groups, ...
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. ...
5
votes
0answers
113 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 ...
5
votes
0answers
165 views

Spherical building of an exceptional group of Lie type

I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...
4
votes
0answers
80 views

Making the definition of dual root unambiguous

In 5.4 of his book Lectures on Invariant Theory, Igor Dolgachev introduces the dual of a root by requiring that $\check\alpha(t) f_\alpha(x) \check\alpha^{-1}(t)= f_\alpha(x)$ ...
4
votes
0answers
64 views

What is $\rho^{\vee}(-1)$?

What is $\rho^{\vee}(-1)$? By definition, $\rho^{\vee}$ is the sum of all positive coroots. I have some difficulty in computing $\rho^{\vee}(-1)$. For example, in the case of $SL_3$, all positive ...
4
votes
0answers
86 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
4
votes
0answers
84 views

Embedding $\mathbb{G}_a$ into $GL_2$

Let $k$ be an algebraically closed field of characteristic $p$. I'd like to find interesting examples of closed embeddings $\mathbb{G}_a(k)\hookrightarrow GL_2(k)$, where $\mathbb{G}_a(k)$ is ...
4
votes
0answers
64 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 ...
4
votes
0answers
69 views

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 ...
4
votes
0answers
76 views

When is a torus the closure of the cyclic subgroup generated by one of its elements?

$K$ is an algebraically closed field. And $T$ is an algebraic group conatined in $GL(n,K)$. Assume $K$ is not the algebraic closure of a finite field. If $T$ is a torus, show that $T$ is the ...
4
votes
0answers
56 views

Proving $\mathscr L(C_G(H)) \subseteq \mathfrak c_{\mathfrak g}(\mathfrak h) = \{ \mathrm x \in \mathfrak g \mid [\mathrm x, \mathfrak h] = 0\}$

Let $H$ be a closed subgroup of the algebraic group $G$, $C = C_G(H)$. Prove that $\mathfrak{c} = \mathscr{L}(C_G(H)) \subseteq \mathfrak{c}_{\mathfrak{g}}(\mathfrak{h}) = \{ \mathrm{x} \in ...
3
votes
0answers
31 views

Closed subgroups of algebraic group have DCC?

Are there any infinite descending chains of closed subgroups of the general linear group over a field? More specifically, is my argument ok? Can you fill in some of the details? Prop: No. Proof: If ...
3
votes
0answers
36 views

Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
3
votes
0answers
53 views

Computing the fundamental groups of simple algebraic groups of type $A$

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$. Let ...
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 ...
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
0answers
38 views

Subvariety of an Algebraic Group.

Given an algebraic group $G$ over an algebraically closed field $K$, if $H$ is a subvariety of $G$, then is $H$ a subgroup of $G$? This seems rather strong. If it is indeed false, is there a geometric ...
3
votes
0answers
37 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 = ...
3
votes
0answers
146 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 ...
3
votes
0answers
123 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 ...
3
votes
0answers
108 views

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

There is an exercise on page 114 of Humphreys' Linear Algebraic Groups (GTM 21) Prove that $SO(3, \mathbb R)$ (= group of $3 \times 3$ real orthogonal matrices of determinate $1$) is a connected ...
3
votes
0answers
134 views

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 ...
3
votes
0answers
153 views

How to prove the connectedness or irreducibility of a variety?

Let $G$ be an algebraic group, and $\mu$ is the multiplication in $G$. Define a morphism $G \times G \times G \times G \rightarrow G \times G$ by $\mu \times \mu$, and let $X$ be the inverse image of ...
3
votes
0answers
237 views

question regarding Waterhouse, affine group schemes

Excerpt from Waterhouse, 14.4 Structure of Finite Connected groups. Thm. Let $A$ represent a finite connected group scheme over a perfect field of characteristic $p$. Then $A$ has the form $k[X_1, ...
2
votes
0answers
15 views

Matrix logarithms for various algebraic groups

For any field $k$ of characteristic $p$ the sets $$\left\{g \in GL_n(k) \ \middle| \ g^p = 1\right\} \qquad \text{and} \qquad \left\{x \in \mathbb M_n(k) \ \middle| \ x^p = 0\right\}$$ are in ...
2
votes
0answers
25 views

Simple connected semi-simple group

Here is a question form springer's book Linear Algebraic Groups, 8.4.6(6) Let $G$ be semi-simple and simple-connected, $P$ a parabolic subgroup of $G$ with Levi group $L$, Prove the commutator ...
2
votes
0answers
52 views

finite normal subgroup

$G$ is a subgroup of finite index in $SL(n,Z)$, $n\ge 3$, $N$ is finite normal subgroup of $G$, then I want to know why $N$ is a normal subgroup of $SL(n,Z)$. More generally, $A$ is an arithmetic ...
2
votes
0answers
30 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
2
votes
0answers
49 views

Confused about Borel-Weil theorem

I am trying to understand the Borel-Weil theorem, but I am very confused because of the different conventions used in different sources. I am especially confused about two things: (1) the definition ...
2
votes
0answers
70 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
2
votes
0answers
18 views

Example of algebraic group of type $G_2$

Can anyone point me to a concrete realization of a reductive algebraic group of type $G_2$ over a field of positive characteristic? I have some questions about how the adjoint action permutes certain ...
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 ...
2
votes
0answers
137 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 ...
2
votes
0answers
29 views

Why the finite subgroups of $GL_n$ is closed with respect to Zariski topology?

Let $GL_n$ be the group of all $n$ by $n$ invertible matrices. Why the finite subgroups of $GL_n$ is closed with respect to Zariski topology? Are the zeros defined by some equations? Thank you very ...
2
votes
0answers
71 views

Weyl group of orthogonal group

My question is why a particular element of the Weyl group of $O(8)$ seems to contradict a theorem about root systems. But to tell you the particular element I have to tell you specifically how I'm ...
2
votes
0answers
41 views

Relationship between invariants of a simple algebraic group

Let $G$ be a simple algebraic group over an algebraically closed field $k$. I believe all of the following invariants are well-defined. Besides the coxeter number, I haven't read about the others, ...
2
votes
0answers
50 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 ...
2
votes
0answers
66 views

What is the Weyl group of this group?

Let $G$ be the group $GL_{n_1}(q^{l_1})\times GL_{n_2}(q^{l_2})\times GL_{n_3}(q^{l_3})$. Here $GL_{n_1}(q^{l_1})$ is the rational points of $GL(n,\bar {\mathbf{F}}_q)$. My question iis what is the ...
2
votes
0answers
72 views

Reduction of closed orbit lemma by Galois descent

I am reading a proof of the closed orbit lemma: If $G$ is a smooth $k$-group of finite type acting on a finite-type $k$-scheme $X$, and $x$ is a $k$-point of $X$, then the orbit map of $x$ ...
2
votes
0answers
43 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 $G$ ...
2
votes
0answers
139 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 ...
2
votes
0answers
51 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 ...
2
votes
0answers
75 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 $N$. ...
2
votes
0answers
81 views

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 ...
2
votes
0answers
120 views

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

This is the problem: Exhibit three reductive groups of dimension $4$ and semisimple rank $1$ which are pairwise nonisomorphic (as algebraic groups). I know that for any reductive group $G$ of ...