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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12
votes
0answers
198 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 ...
11
votes
0answers
139 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 ...
10
votes
0answers
152 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
202 views

Representation theory of the general linear group over a finite prime field

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire ...
7
votes
0answers
94 views

Weil Pairing on Linear Algebraic Groups

I've been studying the Weil pairing on elliptic curves recently and discovered that it has a generalisation to an abelian variety $A$ with its dual $A^{\vee}$, which then becomes a pairing on an ...
6
votes
0answers
175 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 ...
6
votes
0answers
133 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
181 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
119 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
196 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
50 views

homomorphism between smooth algebraic groups of the same dimension

For Lie groups, we have a theorem: Suppose $G$ and $G'$ are Lie groups of the same dimension, $G'$ is connected, and $f : G \to G'$ is a homomorphism of Lie groups with discrete kernel. Then, $f$ ...
4
votes
0answers
104 views

Identifying the cotangent bundle of the flag variety

Suppose $G$ is a Lie group (or I guess a linear algebraic group), $P \subset G$ a Lie subgroup with Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$ respectively. In Chriss and Ginzburg's book ...
4
votes
0answers
42 views

Is there anywhere some explicit Bruhat decompositions are written down?

Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given. Also, I calculated the following regarding the Bruhat decomposition of ...
4
votes
0answers
85 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
67 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
115 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
89 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
77 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
194 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 ...
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
86 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
59 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
84 views

Questions about Affine algebraic group scheme over an infinite field K

For an easily comprehension of my questions I write some definitions: An affine algebraic group scheme over $K$ is a representable group-functor from $K$-algebras category, with a finitely generated ...
3
votes
0answers
84 views

Real form and real structure on a complex Lie group

E.B.Vinberg and A.L.Onishchik in their book give the following two definitions. For a complex Lie group $G$ its real Lie subgroup $H$ is called a real form of $G$, if a) the Lie algebra $L(H)$ of ...
3
votes
0answers
62 views

(stability-theoretic) ¨weakly normal groups" are closed under subgroups

Let me first introduce two definitions: For a structure $\mathcal{M}$ in a language $\mathscr{L}$ and a subset $X \subseteq M^n$, the fully induced structure on $X$ is a structure $\mathcal{X}$ with ...
3
votes
0answers
43 views

structure of commutative algebraic groups

I've read that there is a structure theorem for commutative algebraic groups over an algebraically closed field $K$ namely they are the direct product of a semisimple group and a unipotent group. ...
3
votes
0answers
77 views

Is a connected unipotent subgroup always contained in a Borel subgroup?

As the question says, is a connected unipotent subgroup $U$ of a linear algebraic group scheme $G$ always contained in a Borel subgroup of $G$? I have an argument for why the answer is yes, and I ...
3
votes
0answers
157 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 ...
3
votes
0answers
37 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
58 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
70 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
189 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 ...
3
votes
0answers
91 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
162 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 ...
3
votes
0answers
95 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 ...
3
votes
0answers
42 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
38 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
163 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
155 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
123 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
169 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
257 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
26 views

Abelian hypersurfaces

Is there any way to tell from its defining equation when an affine hypersurface has a group law? I am aware that all such groups are matrix groups; in that case I might want to ask when a variety over ...
2
votes
0answers
28 views

How to find the affine coordinate algebra of $n$-torsion points of elliptic curve?

In particular, I'm wondering about the affine coordinate algebra of $E[3]$ where $E$ is the elliptic curve $y^{2}=x^{3}-D$ over $\mathbb{Q}$ with $D=2^{8}3^{5}5^{2}$. I think we can view $E[3]$ as ...
2
votes
0answers
37 views

Different definitions of projective matrix groups, with one giving an algebraic group but not the other

Recently my professor told me that the usual definition of PSL(2,$\mathbb{R}$) = SL(2,$\mathbb{R}$)/{$\pm$I} does not give an algebraic group, but the following definition does: PSL(2,$\mathbb{R}$) ...
2
votes
0answers
15 views

Understanding $SL_3(D)$ where D is a central division algebra

Suppose that $K$ is a non-archimedean local field of positive characteristic and $D$ is a four-dimensional central division algebra over $K$. The group $SL_{3}(D)$ can be embedded as a $K$-form of ...
2
votes
0answers
31 views

Inclusion of Tori induces surjection of character groups?

Let $k$ be an algebraic closed field. Let $T, T'$ be algebraic Tori in the classical sense, meaning $T \cong \mathbb{A}_k^n \setminus V(X_1 \cdots X_n)$, $T' \cong \mathbb{A}_k^{n'} \setminus V(X_1 ...
2
votes
0answers
26 views

Order of elements in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
2
votes
0answers
21 views

Are linear reductive algebraic groups closed under extensions?

Say we have a ses of algebraic groups $1 \to A \to B \to C \to 1$ where $A,C$ are linear reductive algebraic groups. Does it follow that $B$ is also a linear reductive algebraic group? In other ...
2
votes
0answers
29 views

PGL_1(A) as a rational group.

Let $A$ be a central simple algebra over a field $F$. How one can see that the group scheme ${\bf\rm{PGL}}_1(A)$ is embedded as an open subgroup of $\mathbb{P}(A)$, the projective space over $A$. ...