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Questions tagged [algebraic-groups]

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

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How to obtain the coroots of $\operatorname{SL}_2$?

Let $G = \operatorname{SL}_2(\mathbb{C})$ and fix the maximal torus $T$ being the diagonal matrices in $G$. Let $\mathfrak{g} = \operatorname{Lie}(G)$, $\mathfrak{h} = \operatorname{Lie}(T)$ as usual, ...
Ray's user avatar
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1 answer
21 views

Is the product of reductive algebraic groups reductive?

I know some cases in which the product of reductive algebraic groups is reductive. For example, $\text{GL}_n\times\text{GL}_m$ is known to be reductive. A product of algebraic tori $T\times T'$ must ...
Siegmeyer of Catarina's user avatar
-1 votes
0 answers
31 views

Is the quotient of a linear complex group by a normal finite subgroup linear? [closed]

Lt $N$ be a finite normal subgroup of a complex linear group $G$. Is the quotient $G/H$ also a linear group?
Ronald's user avatar
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2 votes
0 answers
30 views

Cohomology class of automorphism group of Galois form

Let $\Gamma$ be the Galois group of a finite Galois extension $K/k$ of fields of characteristic zero. Let $G$ be an algebraic group defined over $k$. Let $G'$ be another algebraic group over $k$. We ...
gimothytowers's user avatar
0 votes
0 answers
90 views

A property of an irreducible root system

Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
jack's user avatar
  • 362
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0 answers
42 views

Local Action by Group Scheme (Milne's Algebraic Groups)

I have a question about following proof from Milne's book "Algebraic Groups: the theory of group schemes of finite type over a field", Chapter 8, proposition 8.9: PROPOSITION 8.9. Let $G \...
user267839's user avatar
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1 vote
1 answer
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étale fundamental group of a connected algebraic group over an algebraically closed field of characteristic p

I am looking for an explanation of the fact that the étale fundamental group of a connected unipotent algebraic group $G$ over an algebraically closed field $k$, where $char(k)=p$ has no non-trivial ...
Pambra iskra's user avatar
3 votes
0 answers
33 views

Left-invariant derivations of *continuous* functions on a Lie group [duplicate]

I am aware that the definition (or one of several equivalent definitions) of the Lie algebra $\mathfrak g$ of a Lie group $G$ is as follows: $\mathfrak g$ is the set of left $G$-invariant derivations $...
Andrea B.'s user avatar
  • 754
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Is there a scheme-theoretic definition for the center of an algebraic group (or group scheme)?

Since this is the case for other usual group-theoretic concepts like normal subgroups or derived subgroup (i.e. commutator), I was wondering if there exists a similar construction for an object ...
Siegmeyer of Catarina's user avatar
4 votes
0 answers
54 views

Products of algebraic groups and sums of Lie algebras

Let $G$ be a connected solvable algebraic group. Then we know $G = T \ltimes G_u$ is a semi-direct product, where $T$ is a maximal tori, $G_u$ is the unipotent part of $G$. Main Question. Why does ...
zh'nil's user avatar
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2 votes
1 answer
114 views

Theorem of Highest Weight for Reductive vs Semisimple complex algebraic group

Let $G$ be a connected, reductive, complex algebraic group. It is well known that if $G$ is moreover semisimple, i.e. its Lie algebra $\mathfrak{g}$ is semisimple (or equivalently, its center $Z(G)$ ...
user267839's user avatar
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1 vote
0 answers
58 views

Cohomology Theory for Flag Varieties

I'm reading these notes about central results of cohomology theory of flag varieties, ie those having the form $G/B$ for $G$ semisimple, simply-connected, complex algebraic group, and $B$ it's (up to ...
user267839's user avatar
  • 7,201
0 votes
0 answers
22 views

Hasse principle for quadratic forms vs algebraic groups

Can the Hasse-Minkowski theorem for quadratic forms over a number field be recovered from the Hasse principle for algebraic groups? More specifically, by the former I mean the theorem that two ...
Cyclicduck's user avatar
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0 answers
10 views

Decomposition of a general (not necessarily solvable) real algebraic group $G=(S\times T)\ltimes U$

Let $G$ be a connected real algebraic group. I have seen the following two facts: Based on this notes 1.If $G$ is nilpotent, then $G_s$ is a closed, connected subgroup (hence a torus). $G_u$ is the ...
taylor's user avatar
  • 579
2 votes
1 answer
52 views

Decomposing central simple algebras

Assume $A$ is a central simple algebra of degree $2^nl$ for some odd number $l$ and of index $2^m$ with $m$ dividing $n$. Question: Can we write $A$ as $B \otimes C$, with $C$ being a central simple ...
nxir's user avatar
  • 313
2 votes
1 answer
87 views

Homogeneous spaces for unipotent groups are isomorphic to $\mathbb{A}^n$

I am looking for a proof of the following fact: Let $U$ be a unipotent algebraic group over $\mathbb{C}$. Then any $U$-homogeneous space is isomorphic to $\mathbb{A}^n$. Context: Let $G$ be a ...
zygomatic's user avatar
  • 103
0 votes
1 answer
44 views

Question about a specific argument in the proof of Geometric Satake

Suppose $\tilde{G}_{\mathbb{Z}_p}$ is a flat affine group scheme over $\text{Spec}(\mathbb{Z}_p)$ (the p-adic integers) such that the fiber over the generic point $\tilde{G}_{\mathbb{Q}_p}$ is known ...
I'm Representable's user avatar
2 votes
0 answers
87 views

(Algebraic) monodromy group of $V \otimes V^*$

The algebraic monodromy group of a continuous semisimple (or irreducible further) representation $$\rho: G \to GL(V)=GL_n(\mathbb C)$$ is, by definition, the Zariski closure of the image of $\rho$ in ...
LWW's user avatar
  • 766
0 votes
1 answer
38 views

Lie-Kolchin theorem, unipotent matrices and Ad-unipotent elements

Let $G\le \text{SL}(n,\mathbb R)$ be a connected real matrix Lie group, not necessarily solvable The two basic questions I have, and I have been told by people that they "follow from" Lie-...
taylor's user avatar
  • 579
0 votes
0 answers
31 views

Is the Frobenius twist of a representation a fully faithful functor?

Let $k$ be a perfect field of characteristic $p$. Let also $G$ be a (affine) group scheme over $k$, and $V$ be a representation of $G$. Given a scheme $X$ over $k$, one can define its Frobenius twist $...
JeCl's user avatar
  • 511
1 vote
0 answers
69 views

Using étale fundamental group to show unramifiedness of Tate module

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with good reduction at $p$, i.e., there exists an elliptic curve $\mathcal{E}$ over $\mathbb{Z}_p$ whose generic fiber is isomorphic to $E$. I have ...
WLOG's user avatar
  • 1,306
0 votes
0 answers
19 views

Precise statement for "orbit of algebraic group is open in its Zariski closure" and source of proof

My question is about the precise clarification (and source of proof) of the following statement I have heard of: Let $H$ be an algebraic group action on a variety $X$. Let $x\in X$. Then $H.x$ is open ...
taylor's user avatar
  • 579
0 votes
0 answers
29 views

Examples of reductive group $G$ and primes $p$ where $G$ ramifies at $p$

I'm looking for an example (as simple as possible) of a reductive group $G$ (over a number field) for which we can compute each prime number at which $G$ ramifies.
Marsault Chabat's user avatar
0 votes
0 answers
24 views

On the computation of local adjoint $L$-function of unramified representation

Let $F$ be a p-adic local field and $W$ be a 2n-dimensional symplectic space over $F$. Let $G_n$ be the isometry group of $W$ and $B_n$ be the Borel subgroup of $G_n$. Then the maximal torus $T_n$ of $...
Kery's user avatar
  • 357
2 votes
1 answer
49 views

Is an extension of an algebraic group by a multiplicative group a semidirect product?

This is probably a very simple question with a negative answer, but I somehow cannot find a counterexample. Let $X$ be a smooth algebraic variety over an algebraically closed field $k$. Assume that $X$...
L_b's user avatar
  • 684
1 vote
0 answers
41 views

Affine group schemes

I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups. My question Let $k$ be an ...
Tommk's user avatar
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1 vote
1 answer
57 views

How is the centralizer in $\mathrm{GL}_n(k)$ a torus

I have the following definition: Suppose that $g \in \mathrm{GL}_n(k)$ is regular and semisimple. Define $T_g := \mathrm{Cent}_{\mathrm{GL}_n(k)}(g)$ to be the centralizer of $g$ in $\mathrm{GL}_n(k)$....
Gargantuar's user avatar
-1 votes
1 answer
70 views

Zariski closure of an algebraic subgroup of finite index [closed]

Let $G$ be an algebraic group and $H \subset G$ a subgroup of $G$. Let $H_0$ be a subgroup of $H$ of finite index. Then I guess the Zariski closure of $H_0$ is exactly the Zariski closure of $H$ in $G$...
finiteness's user avatar
0 votes
1 answer
33 views

Fibre bundle over Borel subgroup with fibre the complete flag

Let $G = \mathrm{GL}_n$, let $B$ be a Borel subgroup of $G$. The set $\mathcal{B}$ of $G$-conjugates of $B$ can be given an algebraic variety structure and is known as the variety of Borel subgroups ...
Gutiérrez's user avatar
0 votes
0 answers
55 views

Connected components of subgroup of torus

Consider a finite field $\mathbb{F}_q$ of characteristic $p>0$. Let $A=(a_{ij})$ be an integer matrix with $k$ columns and a finite number of rows. Consider the algebraic subgroup $\pmb{H}_A$ of ...
user avatar
2 votes
1 answer
81 views

Classifying representations of GL(n): what's a good reference?

What's a good reference for the classification of finite-dimensional algebraic representations of $\mathrm{GL}(n,k)$ when $k$ is an arbitrary field of characteristic zero? For $k = \mathbb{C}$, ...
John Baez's user avatar
  • 1,667
0 votes
0 answers
41 views

Center of Levi subgroups in SLn

Fix a partition $\lambda$ of $n$ of length $l$. Consider a Levi subgroup $L_{\lambda}$ in $GL_{n}(\mathbb{C})$. There is the decomposition $L_{\lambda} = \prod_{i} GL_{\lambda_{i}}$. Now the ...
arczn's user avatar
  • 51
1 vote
0 answers
51 views

Surjective maps between algebraic groups induce surjective maps between connected components

I am reading the paper of F. Beukers, A refined version of the Siegel-Shidlovskii theorem (here is the link). The author mentions the following result in Algebraic group theory without proof. Lemma 2....
Khainq's user avatar
  • 384
2 votes
0 answers
141 views

Involutions in PCO

In the algebraic group $G=\operatorname {PCO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there in $G \setminus \...
scsnm's user avatar
  • 1,293
4 votes
1 answer
83 views

Is a subgroup of $\operatorname{GL}(n,\mathbb{R})$ semialgebraic if and only if all its orbits are?

A subset $X \subseteq \mathbb{R}^n$ is called semialgebraic if it is of the form $$ X = \bigcup_{finite} \bigcap_{finite} \{ x \in \mathbb{R}^n \colon f_{i,j}(x) \star 0 \} $$ where $\star$ represents ...
Strichcoder's user avatar
  • 1,962
3 votes
0 answers
92 views

Is galois cohomology invariant under inner forms and not just pure inner forms?

Let $G, G'$ be smooth algebraic groups over $k$ (absolute Galois group $\Gamma$) which are etale inner forms of each other, that is, there exists an isomorphism $G_{k_s} \cong G'_{k_s}$ and the ...
C.D.'s user avatar
  • 1,591
2 votes
1 answer
44 views

Character group and tate module of algebraic group

Let $k$ be a field and $\bar k$ a separable closure. For an algebraic $k$-torus, denote by $G_* = \mathrm{Hom}(\mathbb{G}_{m, \bar k}, G_{\bar k})$ its group of cocharacters. This is a finitely ...
Erich's user avatar
  • 235
3 votes
0 answers
97 views

Non abelian algebraic subgroup of $GL_n(\mathbb{C})$ consisting of diagonalizable matrices

Can someone give me an example of an affine algebraic group over characterstic $0$, which consists of semisimple elements but is not commutative. Preferably given as a closed connected algebraic ...
Fabio Neugebauer's user avatar
2 votes
0 answers
99 views

To what extent does learning scheme theory help with more classical algebraic geometry?

This is a question that has come to mind as I have been studying Vakil’s notes on algebraic geometry. I am studying this in my own time out of interest and afterwards wish to look at other books (...
Nethesis's user avatar
  • 3,996
2 votes
1 answer
76 views

Proof that principal congruence subgroups $\Gamma(N)$ are torsion free.

I have read the fact that the principal congruence subgroups $\Gamma(N)$ of $\mathrm{GL}_n({\mathbb{Z}})$ are torsion free for $N \geq 3$ several times, but only saw proofs for very specific ...
Staub und Dreck's user avatar
1 vote
0 answers
30 views

Every ring $R$ with $K ⊆ R ⊆ L$ (subrings) is a field $\Leftrightarrow$ $L/K$ is algebraic

I have this exercise where I am having some problems solving it: Show that for a field extension $L/K$ the following are equivalent: (a) $L/K$ is algebraic. (b) Every ring $R$ with $K ⊆ R ⊆ L$ (...
Marco Di Giacomo's user avatar
2 votes
0 answers
39 views

Can Borel subgroups be partitioned into sets of roots?

I'm doing a project in algebraic geometry where Borel subgroups play a very important role, but my supervisor made a comment that confused me. Let $G$ be a semisimple algebraic group and let $T\subset ...
nspace's user avatar
  • 75
3 votes
0 answers
117 views

Connection between maximal tori and roots

I've been studying algebraic groups and there is a confusion that I have been unable to resolve. Let $G$ be a semisimple algebraic group and $T\subset G$ a maximal torus. If we let $T$ act on ${\rm ...
nspace's user avatar
  • 75
2 votes
0 answers
55 views

Centralizer generators

This is a question posted on overflow but no reply has been received. In the algebraic group $G=\operatorname {PSO}(4,K)<\operatorname {PCGO}(4,K)$ where $K$ is an algebraically closed field of an ...
scsnm's user avatar
  • 1,293
0 votes
0 answers
85 views

Does $Spec(k[G/[P,P]])$ have worse than quotient singularities?

Let $k$ be an algebraically closed field of characteristic zero, let $G$ be a connected reductive linear algebraic group over $k$, and let $P$ be a parabolic subgroup of $G$. So we have the flag ...
Dave's user avatar
  • 13.6k
0 votes
0 answers
67 views

Why is Serre duality compatible with $G$-actions of linear algebraic groups?

Im currently reading Jantzens "Representations of Algebraic Groups" 2nd Edition. On page 203 he explains how a $G$-linearized sheaf $\mathcal L$ induces a $G$-module structure on the ...
Capotasto's user avatar
0 votes
1 answer
91 views

GL$_n$ schemes representation

Let $n \geq 0$ be a natural number. Consider the schemes $$ \mathrm{GL}_n:=\operatorname{Spec}\left(\mathbb{Z}\left[X_{i, j} \mid 1 \leq i, j \leq n\right]_f\right),\quad \mu_m :=\operatorname{Spec}\...
Mario's user avatar
  • 727
1 vote
1 answer
254 views

Tensor product of irreducible representation of $\mathfrak{sl_3}$

Let $\omega_1$ and $\omega_1$ be fundatmentl weights and $V(\lambda)$ be the unique irreducible representations of highest weight $\lambda$ of $\mathfrak{sl_3}$. I want to decompose, for $m>n$, $V(...
Rick's user avatar
  • 391
0 votes
1 answer
78 views

Algebraic Torus is a group scheme

I am taking a course on toric varieties this semester, and I am a little confused by how the algebraic torus is a group scheme, as we didn't really define what a group scheme is. I was given the ...
Chris's user avatar
  • 3,411
0 votes
0 answers
74 views

On the exactness of diagonalisable linear algebraic groups

in Herzig's lecture notes on linear algebraic groups (page 24/25, see https://www.math.toronto.edu/~herzig/lin-alg-groups17-seaton-notes.pdf), there is a quite interesting fact on the exactness of ...
max_121's user avatar
  • 759

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