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. Consider using with the (group-theory) tag.

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0answers
42 views

Why is the ring of local-ring valued points of a ring scheme a local ring.

I'm confused on a supposedly easy claim: Let $T$ be a base scheme, and let $\mathbf{R}$ be a ring scheme over $T$, i.e. a scheme $\mathbf{R} \to T$ such that for all $E \in \operatorname{Sch}_{/T}$ $\...
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22 views

Galois Cohomology and Loop Groups

I am trying to understand problem 8.5 in Kac's Infinite dimensional Lie algebras. It goes as follows. Let $G$ be a semisimple algebraic group, let $\alpha$ be an automorphism of $G$ of order $m$, and ...
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1answer
70 views

Pushforward of equivariant sheaf

I work over an algebraically closed field of characteristic zero. Let $G$ be an algebraic group, $X,Y$ varieties with $G$-actions, and $\phi:X\to Y$ a $G$-equivariant morphism. Let $\mathcal{F}$ be ...
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1answer
81 views

Induced torus action on tangent and normal bundle

Consider a smooth, projective, irreducible variety $X$, defined over $\mathbb{C}$, and consider a $\mathbb{C}^*$-action on $X$, that is a map $$\mathbb{C}^*\times X\rightarrow X, \hspace{1cm} (t,x)\...
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1answer
24 views

Closedness of the product of two subgroups

Let $G:=\text{SL}(3,\mathbb R)$, equipped with the usual subspace topology, acting on $\mathbb R^3$ by the canonical action. Consider subgroups $\Gamma:=\text{SL}(3,\mathbb Z)$ and $Q_1:=\{g\in G:ge_1=...
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0answers
12 views

Weyl group of Parabolic subgroup

Let $G$ be an algebraic group over an algebraically closed field. Let $P$ be a standard parabolic subgroup of $G$ and $L$ be it's Levi subgroup. Do we have that weyl group of P equals weyl group of L? ...
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0answers
12 views

Is the unipotent part of the intersection of all Borel subgroups containing a torus connected?

I found this question in Humphreys algebraic groups exercise 26.14 Let $G$ be a connected algebraic group and $T$ a maximal torus. Is the unipotent part of the intersection of all Borel subgroups ...
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23 views

Action of algebraic group on quasi-projective variety

The action of an algebraic group $G$ on a quasi-projective variety $X$ is defined as a usual group action with the property that $$ G \times X \to X$$ is a morphism of quasi-projective varieties. But ...
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1answer
44 views

Is a K3 surface a spherical variety?

Do there exist any K3 surfaces that admit the structure of a spherical variety? That is, does there exists a K3 surface X, a reductive algebraic group G, and a Borel subgroup B of G, such that X ...
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1answer
30 views

Dimension of union of orbits

Assume that we are in characteristic $0$. Let $G$ be an algebraic group acting on a quasi-projective variety $Y$. Let $G_0$ be the connected component of $G$ that contains the identity element. Let $P ...
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0answers
31 views

Solve $\mathbb{Q}$[X] /$\mathbb{Q}$[X]($X^3$+$X$) $\cong$ $\mathbb{Q}$ x $\mathbb{Q}$[X]/($X^2$ $+$ $1$)

I need to solve the following 2 questions: 1) $\mathbb{Q}$$[X]$/$\mathbb{Q}$$[X]$($X^3$+$X$) $\cong$ $\mathbb{Q}$ x $\mathbb{Q}$$[X]$/($X^2$ $+$ $1$) 2) $\mathbb{R}$$[X]$/$\mathbb{R}$$[X]$($X^4$$-...
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1answer
41 views

Definition of Normalizers in Algebraic Groups

Let $G$ be an affine algebraic Group, $H \subseteq G$ a closed subgroup. The Normalizer of $H$ is usually defined as $\text{N}_G(H) := \{ g \in G | gHg^{-1} = H\}$. However, some authors define it ...
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15 views

How can the integral of the elliptic kernel $K_{\mathcal O}(x,x)$ over $Z(\mathbb A)G(F) \backslash G(\mathbb A)$ be written in the given form?

Let $G$ be a connected, reductive group over a number field $F$. Let $\gamma_1 \in G(F)$ be an elliptic element, i.e. one which is not a member of any proper parabolic subgroup of $G$. Let $\mathcal ...
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0answers
14 views

Finiteness of the volume $Z(\mathbb A) G(\mathbb Q) \backslash G(\mathbb A)$ for $G = \operatorname{GL}_2$

I am reading Gelbart's lectures on the trace formula and am confused on how the Siegel domain is used to prove the finiteness of the volume of $Z(\mathbb A) G(\mathbb Q) \backslash G(\mathbb A)$ for $...
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14 views

subgroups containing $T(3,K)$

Let $K$ be a field which is algebraically closed. $T(3,K)\subset GL(3,K)$ denotes the subgroup of upper-triangular matrix. Can we decide all subgroups of $GL(3,K)$ which contains $T(3,K)$?
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11 views

subgroups of a connected solvable group consisting of semi-simple elements

In Humphreys' linear algebraic groups, section 19.4, it says that: Let $G$ be a connected solvable algebraic group and $H$ an abstract commutative subgroup of $G$ consisting of semi-simple elements, ...
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0answers
16 views

exp and log bijection between nilpotents and unipotents. Also, log of elements of a unipotent group are in the Lie algebra.

Let $K$ be an algebraically closed field of characteristic zero. I am struggling to solve two exercises in Humphreys Linear algebraic groups: 15.8 and 15.10. The first asks us to show that the maps $...
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0answers
6 views

Semisimple and unipotent parts of an automorphism of a finite $k$-algebra are also automorphisms

Let $A$ be a finite (as a $k$-module) $k$-algebra for $k$ an algebraically closed field, $x$ an automorphism of $A$. Humphreys Linear algebraic groups exercise 15.2 asks us to show that the semisimple ...
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1answer
39 views

The Lie algebra of the normalizer of a closed subgroup of a linear algebraic group

Let $G$ be a connected algebraic group over an algebraically closed field of characteristic zero, and $H \subseteq G$ a closed connected subgroup. In Humphreys Linear algebraic groups Exercise 13.1 we ...
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0answers
50 views

Explicit description of the representation of $A/\mathfrak m^n$ given by self conjugation of the algebraic group $G=\operatorname{Spm}A$

Let $G=\operatorname{Spm} A$ be the algebraic group $\mathrm{GL}_2(k)$, where $k$ is some field. In particular, $A=k[x_1,x_2,x_3,x_4,(x_1x_4-x_2x_3)^{-1}]$. Then $G$ acts on itself by conjugation, and ...
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3answers
70 views

Abstract Algebra ( tutoring suggestions )

I'm looking for some good Abstract Algebra courses on youtube but i couldn't find any , so if someone has any suggestions please help me ( or a book with some nice problems would be good also ).
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1answer
47 views

Connected components of an algebraic group

Let $V$ be a subvariety in the complex projective space $\mathbb{CP}^n$ (with finite number of connected components). Is it true that the group of projective automorphisms preserving $V$ is a linear ...
1
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1answer
58 views

Isomorphic elliptic curves (abelian varieties) over $\mathbb{C}$ and $\overline{\mathbb{Q}}$

For two elliptic curves $E_1, E_2$ defined over $\mathbb{Q}$, we assume $E_1$ and $E_2$ are isomorphic over $\mathbb{C}$, then how to prove they are isomorphic over $\overline{\mathbb{Q}}$? Also, can ...
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0answers
32 views

Group extension of $G$ by $\mathbb{G}_m$ and the Picard group.

I was reading about algebraic groups and I ran into a proof that I don't understand. The setup is as follows: $k$ is a separably closed field, and $G$ is a simply connected semi-simple group over $k$. ...
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0answers
27 views

A remark about $\rho$ and $G$ being simply connected

Let $G$ be a reductive complex algebraic group, $H \subset G$ a Cartan subgroup and $R^+$ a set of positive roots, and $X_+(H)$ the set of dominant weights. Let us also assume that $G$ is simply ...
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0answers
12 views

Harish Chandra induction of regular semisimple character

Let $q=p^n$ be a power of a prime number. I'm interested in the following situation:$G=Gl(n,\bar{\mathbb{F}_q})$ and $L \subseteq G$ is a Levi subgroup stable for the action of the Frobenius (which I'...
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0answers
10 views

relation between weyl group of reductive group $W(G,T)$ and weyl group of the dual group $W(\hat{G},\hat{T})$

what is the relation between Weyl group of reductive group $W(G,T)$ and Weyl group of the dual group $W(\hat{G},\hat{T})$? I think they must be isomorphic but I'm not sure.
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1answer
37 views

Descent of commutative algebraic group

Let $G$ be a smooth affine algebraic group over a field $k$. If $G_{\bar{k}}$ is commutative over $\bar{k}$, is it necessary that $G$ is a commutative algebraic group over $k$? I think the answer is ...
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0answers
26 views

Invariant Differential Operators on Homogeneous Spaces

Let $G$ be a reductive complex algebraic group and $H$ a reductive algebraic subgroup. Then there is a well-known isomorphism $$\text{Diff}^G(G/H)\cong \text{Dist}(G/H,eH)^H$$ where $\text{Diff}^G(G/...
4
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1answer
74 views

Dimension problems regarding action of reductive group on variety

Let $k$ a field and $X$ a $k$ algebraic variety. Let $G$ a $k$ reductive group acting on $X$. I denote with $X_d$ the set of points in $X$ which have stabilizer of dimension $d$. It is a fact that $...
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1answer
54 views

Are rational representations of an (affine) algebraic group the same as modules over some ring?

Given an algebraic group $G$, are the rational representations of $G$ in natural correspondence with modules (of some type?) of a ring associated to the group? Certainly rational representations give ...
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0answers
36 views

Is the union of orbits of a closed set open in its closure?

Let $G$ be a linear algebraic group that acts morphically on an affine (irreducible) variety $X$. It is a standard result (Prop. 1.11) that for any $x \in X$, the orbit $G_x=\{gx : g \in G\}$ is ...
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0answers
64 views

Conjugacy of maximal algebraic tori

Suppose $G$ is a connected, reductive algebraic group over a nonarchimedian local field $F$, which splits over a finite extension $E/F$. I frequently see a result stating that "all maximal $F$-tori ...
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0answers
16 views

Dimension of the unipotent radical

Why is $\dim \mathfrak{p}- \dim \mathfrak{l}=\dim G/P$ for $P$ is a parabolic subgroup of $G$ and $L$ is a Levi factor.
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0answers
50 views

Tangent space of a product of algebraic group.

I need help with this problem from Shafarevich Basic Algebraic Geometry. Let $G$ be an algebraic group and $\Psi:G\times G\rightarrow G$ the regular map defined by the group law. Let $T_{e}G$ ...
4
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1answer
49 views

Lie algebra of a linear algebraic group acts on functions by derivations: what does this mean?

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic zero. Let $X$ be an affine variety on which $G$ acts. Then $G$ naturally acts on the coordinate ring $\...
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2answers
98 views

Do “enveloping algebras” for algebraic groups exist?

Let $K$ be a field, for simplicity algebraically closed of characteristic $0$. Let $G$ be a reductive group over $K$. Def. A finite-dimensional $K$-algebra $R$ is an enveloping algebra for $G$, if $R^...
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0answers
37 views

Prove $A^\times$ is an inner form of $GL_n$

Question Let $A$ be a central simple algebra over $F$ where $F$ is a field and $dim_FA=n^2$. $A^\times$ can be seen as an algebraic group over $F$ and let $G=A^\times$. Prove that $G$ is an ...
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0answers
31 views

Prerequisites for Serre's Algebraic Groups and Class Fields

What are the prerequisites for reading and understanding the book Algebraic Groups and Class Fields by Serre. Could you suggest some books to learn the prerequisites? Thanks
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41 views

Stabiliser in general position

Consider the linear representation $V=$ Sym$^3(\mathbb{C}^3)$ of SL$_3(\mathbb{C})$. Now Theorem 7.2 of Popov-Vinberg asserts the following (Richardson [1972a], Luna [1973]): For any action of a ...
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1answer
34 views

An ideal that contains the commutator of a solvable Lie algebra

Let $\mathfrak g$ be a real solvable Lie algebra and $\mathfrak n$ be an ideal of $\mathfrak g$ such that the commutator algebra $\mathfrak g'$ is contained in $\mathfrak n$. Now, let $\mathfrak m$...
3
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2answers
310 views

Subgroups of $GL_n$ containing upper triangular matrices

EDIT: I rephrased the claim for clarity. Let $k$ be a field (that we may assume to be algebraically closed, but I don't think it is necessary). Let $n\geq 1$ and $T$ denote the subgroup of $GL_n$ ...
2
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1answer
68 views

Show that $\mathfrak{sl}_n$ is the Lie algebra of the algebraic group $SL_n$

I am currently reading the book "Linear Algebraic Group" by Springer, more precisely in chapter 4 where Lie algebras of linear algebraic groups are introduced. I would like to prove that the Lie ...
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1answer
17 views

Expressing a cycle/set as odd or even.

I have a cycle here that I have broken down into 2 disjoint cycles, them being (1,5,6) and (2,8). I'm wondering what is the process of telling whether the set is even or odd. Is it the number of ...
1
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1answer
38 views

Semisimple algebras of simple Lie algebras and their quotients

Let $\mathfrak s$ be a complex semisimple Lie algebra, then $\dim _\mathbb C\mathfrak s\geq 3$. But, however, is it possible for $\mathfrak s$ to have a semisimple Lie subalgebra $\mathfrak h$ such ...
2
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1answer
110 views

Kleiman's theorem on intersection theory

I study the book 3264 & All That Intersection Theory in Algebraic Geometry by Eisenbud & Harris and I'm a little bit confused on on the proof of Kleiman’s theorem on pages 21: Theorem 1.7 (...
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1answer
31 views

Level set of characteristic polynomials closed in Zariski topology. Humphreys Proposition 18.2.

I've been reading through Humphreys Linear Algebraic Groups, and this question concerns the proof of Proposition 18.2 (page 117). I believe the essence of my confusion is the following claim: $\...
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0answers
42 views

Birationality and group actions

Given two complex algebraic varieties $X,Y$ which are birational. Moreover, let $G$ be an algebraic group acting on $X$ and $Y$ and assume that the geometric quotients $X/G$ and $Y/G$ exist. What are ...
2
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2answers
50 views

Reference Request; Existence of Limit

Background/thoughts Since $GL_d(\mathbb{R})$ is a topological group then the maps $A\mapsto A^{-1}$ and $(B,A)\mapsto A+B$ are continuous. Therefore, in the case where $X \in GL_d(\mathbb{R})$ then $...
1
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1answer
33 views

Conjugacy classes of unipotent algebraic groups

I want to prove the following statement: Let $G$ be a unipotent linear algebraic group acting on an affine variety $X$. Then all $G$−orbits are closed in $X$. The following is what I have done : Let ...

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