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.
1,517
questions
3
votes
0
answers
26
views
Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$
Let $G$ be a simply connected absolutely simple group
of one of the types $^1{\sf A}_{n-1}$ (inner) or $^2{\sf A}_{n-1}$
(outer) over a field $k$. All such groups are described on page 55
of Tits, ...
- 1,120
0
votes
1
answer
55
views
maximal subgroup of the general linear group
Maybe I'm being silly.
As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, is the conformal orthogonal group $CO_{2n}$ a maximal subgroup of $GL_{2n}$(or $CO_{2n+1}$ ...
- 1,443
2
votes
0
answers
78
views
+50
Significance of Chevalley bases, Chevalley groups, etc.
I am learning about semisimple algebraic groups, and came across the concept of Chevalley groups.
I think it is very exciting that you can define all semisimple complex linear algebraic groups over $\...
- 213
0
votes
0
answers
46
views
Product of unipotent subgroups is unipotent
Let $H,K$ be two closed normal unipotent subgroups of a linear algebraic group $G$, is their product $HK$ also a unipotent subgroup?
The unipotent radical of $G$ is defined to be the largest connected ...
- 477
4
votes
0
answers
63
views
Exponentiating a representation and Baker–Campbell–Hausdorff formula
Let $G = \{\rho \in \mathrm{Aut}(\mathbb{C}[[t]]) \,|\, \rho(t) \in t + t^2 \mathbb{C}[[t]] \}$ be a subgroup of continuos $\mathbb{C}-$automorphisms of $\mathbb{C}[[t]]$ and $\mathfrak{g} = t^2 \...
- 883
1
vote
0
answers
20
views
$\mathbb{G_a}$ is being linear algebraic group, but first diagram does not commute?
Question: Why they are all assume it trivial, but I cannot make it commute the diagram in the first axiom of being, linear algebraic group (for $A=k[s]$).
Definition 3.48.(ref.Mukai An Introduction to ...
- 2,295
3
votes
0
answers
35
views
Order of Chevalley group
Let $G$ be a semisimple algebraic group defined over $\mathbb{Q}$ (simply connected if it helps).
Then for almost all (maybe all) primes $p$, we can make sense of the group $G(p)$ (formed by reducing ...
- 213
0
votes
0
answers
75
views
Semisimple algebraic group defined over a number field
Below, a linear algebraic group stands for a Zariski-closed subgroup of $\operatorname{SL}_d(\mathbb{C})$ for $d\geq 1$.
I say that a linear algebraic group $G$ is realizable over a subfield $F$ of $\...
- 213
2
votes
0
answers
31
views
Surjective image of maximal torus is
I’m reading Humphrey’s Linear Algebraic Group. Here is a Corollary in 21.3:
Let $\phi:G \rightarrow H$ be a surjective morphism of linear algebraic groups. Let $T \subset G$ be a maximal torus: how to ...
- 477
4
votes
0
answers
59
views
When does an algebraic group have $PSL_2$ as a quotient?
The context for the question is due to a comment in Milne’s Introduction to Shimura Varieties. I have limited background in algebraic groups (i.e. just enough to know what it is when it is used).
...
2
votes
1
answer
78
views
General linear group inclusion
Do we have $\operatorname{GL}(n,F)\le \operatorname{O}(2n,F)$ where O means general orthogonal group and $F$ is an algebraically closed field?
I checked some finite group cases: $\operatorname{GL}(2,5)...
- 1,443
0
votes
0
answers
55
views
Is $\operatorname {GL}(n)\times \operatorname{Gr}(m,n)\to \operatorname{Gr}(m,n)$ closed?
Suppose $k$ is a field, and $m<n$ are nonnegative integers. Let $\operatorname{Gr}(m,n)$ be the Grassmannian (whose points are $m$-dimensional subspaces of a $n$-dimensional linear space). Then we ...
- 1,322
1
vote
0
answers
30
views
Group generated by matrices with algebraic entries is $ S $ arithmetic? [closed]
Let $ U_1, \dots, U_s \in SL(n,\mathbb{C}) $ be matrices with algebraic entries. Is the group
$$
\Gamma=\langle U_1,\dots, U_s\rangle
$$
always contained in some $ S $-arithmetic subgroup of $ SL(n,\...
- 7,032
4
votes
1
answer
124
views
Toric vector bundle - Klyachko's classification
I am trying to understand Klyachko's classification of toric vector bundles on a toric variety ( his article: Equivariant vector bundles on toric varieties and some problems of linear algebra). I am ...
- 163
3
votes
1
answer
51
views
Adjoint of the derived group
Firstly, let $G$ be a nice linear algebraic group (for instance connected reductive group) over $\mathbb{Q}$. I shall first define the two other groups I require
$\textbf{Definition:}$ If $Z(G)$ is ...
- 1,279
1
vote
0
answers
76
views
Defining a torus $T$ a certain way, prove that $T\cong (k^*)^n$.
The Question:
Suppose we are given the definition that a torus $T$ of a linear algebraic group over $k$ is a subgroup isomorphic to $\Bbb D_n$, show that $$T\cong(k^*)^m$$ for some $m\in\Bbb N$.
NB: ...
- 43k
2
votes
0
answers
51
views
More examples of non-split algebraic groups
I'm reading Reductive Groups over Local Fields by Tits (from the Corvallis proceedings), and I'm having trouble making sense of several of the definitions, especially when it comes to the local Dynkin ...
- 967
0
votes
0
answers
10
views
Explicit calculation of Bala-Carter correspondence
I am wondering where can I find examples of explicit calculation of Bala-Carter correspondence for algebraic groups of classical types of small ranks.
Would you please give me some clues?
Thank you ...
- 1,073
2
votes
0
answers
46
views
Is the category of finite flat commutative affine group scheme over a notherian ring Abelian?
We know that the category of finite flat commutative affine group scheme over field is Abelian. Through out the proof, I think it can be generated to the case over the noertherian ring.
As in the page ...
- 131
1
vote
1
answer
65
views
Geometry of homogeneous spaces $G/T$
Let us work over $\mathbb C$.
It is well known that if $G$ is a (connected) complex reductive group and $B$ a Borel subgroup, then the homogeneous space $G/B$ is a smooth projective variety. For ...
- 33
2
votes
1
answer
76
views
Free product of finite groups that is outside graph theory
The free product of finite groups $ A * B $ naturally acts on a biregular graph see Free Product of two finite groups. This seems like one of the only places that free products of finite groups appear ...
- 7,032
1
vote
0
answers
27
views
Are the integer points of a simple linear algebraic group 2-generated?
Set Up:
Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
- 7,032
1
vote
0
answers
34
views
Is $ G(\mathcal{O}_K[1/p]) $ dense in $ G(\mathbb{R}) $?
Set Up: Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ p $ be a prime in $ \mathcal{O}_K $. Consider the subring $ \mathcal{O}_K[1/p] $ of $ K ...
- 7,032
1
vote
1
answer
45
views
Properties of a specific subgroup of $GL_2(k)$
Question:
Let $k$ be a field. Consider the subgroup $B\subset GL_2(k)$ where
$$B=\left\{\begin{bmatrix}a&b\\0&d\end{bmatrix}: a,b,d\in k, ad\neq 0\right\}$$
a) Let $Z$ be the center of $GL_2(k)...
- 499
1
vote
0
answers
86
views
Intuition behind the spectrum of a ring or $k-$algebra.
There are two places where I have come across the notion of a spectrum.
The first is when $R$ is a ring, then $\text{Spec}(R)$ is defined to be the set of all prime ideals, and additionally one can ...
- 862
0
votes
1
answer
87
views
Question on the Borel subgroup of $SO(2n)(\mathbb{R})$
I am wondering what it looks like the maximal torus of the Borel subgroup of $SO(2n)(\mathbb{R})$.
I guess that $(2 \times 2)$ matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \...
- 357
4
votes
0
answers
69
views
Classical groups generated by tensor products of subgroups
Let $ G $ denote a classical group.
Question:
Is it the case that
$$
\langle G_n \otimes G_m,G_m \otimes G_n\rangle=G_{nm}
$$
as long as $ n \neq m $?
For example, if $ G $ is the classical group $ GL(...
- 7,032
3
votes
1
answer
205
views
Classifiying algebraic group extensions of $\mathbb{Z}/p\mathbb{Z}$ by $\mu_p$
I am trying to solve exercise 2.6 in Milne's book on Algebraic Groups, which states the following:
Let $k$ be a a field of characteristic $p$, show that the isomorphism class of extensions
$0 \...
- 33
0
votes
1
answer
51
views
Show that the identity component $G^\circ\sqsubseteq G$ (i.e., is characteristic) in a linear algebraic group $G$.
This is Exercise 7.6.6 of Humphreys', "Linear Algebraic Groups".
The Question:
Show that the identity component $G^\circ$ of a linear algebraic group $G$ is a characteristic subgroup.
The ...
- 43k
2
votes
0
answers
53
views
Is a closed, nonempty subset of a (linear) algebraic group $G$ that is closed under taking products a subgroup of $G$? [duplicate]
This is a question based on Exercise 7.6.5 of Humphreys', "Linear Algebraic Groups". For a solution to that particular problem, see:
A closed subset of an algebraic group which contains $e$ ...
- 43k
-1
votes
1
answer
75
views
Is the quotient group of $2$-th root of unity compact? [closed]
Let $F$ be a number field and $\mathbb{A}$ its adele ring.
Let $\mu_2$ be the group scheme of 2-th root of unity defined over $F$.
I guess that $\mu_2(F) \backslash \mu_2(\mathbb{A})$ is compact ...
- 357
3
votes
1
answer
64
views
Let $\mathfrak{A}$ be a finite dimensional $k$-algebra for alg. closed $k$. Prove ${\rm Aut}(\mathfrak{A})$ is a closed subgroup of $GL(\mathfrak{A})$
This is Exercise 7.6.3 of Humphreys', "Linear Algebraic Groups".
The Question:
Let $\mathfrak{A}$ be a finite dimensional $k$-algebra for algebraically closed field $k$. Prove that ${\rm ...
- 43k
2
votes
1
answer
133
views
Integrating Homomorphisms of Borel Subalgebras
Let $G$ be a connected simple complex Lie group and $\mathfrak{g}$ be its Lie algebra. Let us fix a root decomposition, let $\mathfrak{b}_\pm$, $\mathfrak{n}_+$ and $\mathfrak{h}$ be the corresponding ...
- 99
0
votes
0
answers
30
views
Quotient of algebraic group by finite subgroup
Let $G$ be an affine algebraic group defined over a field of charactersitic $0$. Let $H$ be a finite subgroup of $G$. Left translations by geometric points of $H$ yield a finite group of automorphisms ...
- 31
0
votes
0
answers
26
views
When is unitary group over ring of integers dense?
Let $ SU_n(O_d) $ denote an integral unitary group of $ n \times n $ matrices over a totally real number field $ K_d:=\mathbb{Q}(cos(\frac{2\pi }{d})) $ where $ O_d $ is the ring of integers of $ K_d $...
- 7,032
1
vote
0
answers
56
views
Special orthogonal group and general spin group for *non-regular* (degenerate) quadratic spaces
For a quadratic space $(V, q)$ over a field $F$ (of characteristic $\neq 2$), we can define is associated special orthogonal group $\mathrm{SO}(V, q)$ and the general spin group $\mathrm{GSpin}(V, q)$....
- 14.7k
1
vote
1
answer
114
views
The torsion subgroup of a diagonalisable linear algebraic group $G$ with ${\rm char}(k)=p$ (alg. closed $k$) is dense in $G$
This is Exercise 3.2.10(5b) of Springer's, "Linear Algebraic Groups (Second Edition)".
The Question:
Let $p$ be the characteristic exponent of an algebraically closed field $k$. Let $G$ be ...
- 43k
1
vote
0
answers
98
views
Complete resource of Ngô's course notes on Algebraic Groups and Group Schemes
I'm looking for Ngô's M2 course notes on "Groupes algébriques et schémas en groupes".
The Wayback Machine has an incomplete capture here. However, it apparently lacks chapter 1, 3, and 5.
...
- 349
12
votes
1
answer
829
views
+500
What is the connection between algebraic groups and topoi?
I have a longstanding interest in topos theory. (See this MSE search of my questions about topos theory.) I am studying for a postgraduate research degree in linear algebraic group theory. Naturally, ...
- 43k
2
votes
1
answer
73
views
$PGL_n(F)$ - Where to read about its (structure and) representation theory?
Ultimately, I'm interested in non-Archimedean local fields $F$, and am happy to take $\text{char}(F)=0$, and just focus on split $PGL_n(F)$. I suspect that the theory "more less" reduces ...
- 1,429
1
vote
1
answer
30
views
Maximal torus in the Borel subgroup is maximal in $G$.
Let $G$ be a linear algebraic group and $T\subset B\subset G$ where $T$ is torus and $B$ is a Borel subgroup of $G$.
Suppose that $T$ is maximal in $B$. Then, how can I show that $T$ is also maximal ...
- 81
1
vote
0
answers
64
views
The associated $X$-bundle to a principal $G$-bundle
We can assume that everything is over the complex numbers.
Let $ U $ be a scheme and $ G $ an (affine) algebraic group. Let $ \pi: P \rightarrow U $ be a principal $ G $-bundle. Suppose $ X $ is ...
- 2,394
3
votes
0
answers
41
views
Reference request, relation algebraic groups, real groups and their Lie algebras
Let's work in characteristic $0$. Let $G$ be a semi-simple algebraic group defined over a subfield of $\mathbb{R}$ (invariant under transposition). Let $\mathfrak{g}$ be its Lie algebra (Zariski-...
- 1,594
3
votes
1
answer
198
views
Salvaging Exercise 3.2.10(2) of Springer's, "Linear Algebraic Groups (Second Edition)".
This uses the soft-question tag because there might be more than one valid answer, and it's a matter of guesswork to some extent; but there is a right answer (in theory).
Thoughts and Motivation:
As a ...
- 43k
0
votes
0
answers
43
views
Why is $(Ad\, x)^*(u)(X) = u(Ad(x^{-1})X)$ for $u\in (T_eG)^*$ and $X\in T_eG$?
This question concerns Springer's "Linear Algebraic Groups" 2nd Edition page 69.
Let $G$ be a linear algebraic group. For $x\in G$ we have an automorphism $Int(x):G \to G$, $y \mapsto xyx^{-...
- 11
1
vote
0
answers
41
views
A Question about Matrix Computation in GL4
Let us consider $G=GL_4(k)$, where $k=\overline{\mathbb{F}_p}$. Consider the set $S$ of the following kind of elements in $G$:
$$
\left(
\begin{matrix}
1 & 0 & \ast &\ast \\
0 & 1 &...
- 161
1
vote
0
answers
52
views
Does an arbitrary algebraic group G possess a unique largest normal solvable subgroup?
On page 125 of Humphreys' "Linear Algebraic Groups," it states the following: "An arbitrary algebraic group G possesses a unique largest normal solvable subgroup." However, I am ...
- 81
3
votes
1
answer
191
views
If a hom. $\phi:G\to H$ of diagonalisable linear algebraic groups is injective, then the induced hom. $\phi^*:X^*(H)\to X^*(G)$ is surjective
This is Exercise 3.2.10(2) of Springer's book, "Linear Algebraic Groups (Second Edition)". According to Approach0, it is new to MSE.
The Question:
Let $\phi:G\to H$ be a homomorphism of ...
- 43k
1
vote
0
answers
68
views
Variety structure of the Prüfer group
The group $\mathbb{R}/\mathbb{Z}$ rather famously is isomorphic to the algebraic group given by the variety $X^2 + Y^2 =1$ over $\mathbb{R}^2$ with group operation given by $(X_1,Y_1)\cdot (X_2,Y_2) = ...
- 85
3
votes
1
answer
68
views
Weil restriction of a base change
$\DeclareMathOperator{\Hom}{Hom}$
$\DeclareMathOperator{\Res}{Res}$
Let $G$ be an algebraic group over $K$ and let $L/K$ be a finite separable extension. Consider the Weil restriction $G' := \Res^L_K(...
- 6,100