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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Maximal tori in $SO(n,\mathbb{C})$

What are maximal tori in $SO(n,\mathbb{C})$? (not $SO(n,\mathbb{R})$) Can a maximal torus in $SO(n,\mathbb{C})$ be written as $T\cap SO(n,\mathbb{C})$ for some maximal torus $T$ in ...
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0answers
13 views

Is this a compact group?

Consider $$x(t)=e^{-iHt}x(0)$$ and define $$G=\{ e^{-iHt}\mid t\in \mathbb{R}_{\geq 0}\}$$ Also write $\bar{G}$ to the closure of $G$ wrt the Euclidean topology. Q: is $\bar{G}$ a compact group? ...
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1answer
57 views

Algebraic groups of multiplicative type in char 0

For a number field $k$ (so of char 0), are algebraic $k$-groups of multiplicative type always linear?
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1answer
32 views

Order of a group of matrices

I have to calculate the order of the group $G=\operatorname{GL}(n,\mathbb{Z}_m)$ with $m=p^k$ for $p$ prime and $k\in \mathbb{N}$. So I was thinking on the homomorphism $\det: G \rightarrow ...
2
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2answers
24 views

The set of one-parameter subgroup of the Multiplicative group $G_m$ is Z

Let $G_m= k^{*}=k-{0}$ be the multiplicative group. We know this is an Algebraic group also. How does one prove any algebraic group morphism $G_m \rightarrow G_m$ is of the form $t \mapsto t^{n}$ for ...
3
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1answer
39 views

Which p-adic groups are simply-connected?

Suppose that we are working over a nonarchimedean local field $F$, for instance $\mathbb{Q}_p$. Which semisimple algebraic groups (or Lie groups) over $F$ are simply-connected? In particular, I am ...
4
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1answer
47 views

Closed immersion factors through closed immersion

I'm currently working through the proof of Theorem 1, III.12 in Mumford's "Abelian Varieties". Let $G$ be a finite $k$-group scheme acting on an affine $k$-scheme $X:=\text{Spec}~A$ and let $\phi: ...
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1answer
37 views

Field of Definition of an Algebraic Group

Linear Algebraic Groups- James E. Humphreys Chapter-XII Let $K$ be an algebraically closed field and $k$ be a arbitrary sub-field of $K.$ A closed set X in $A^n=K\times ...$(n times)$\times K$ is ...
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36 views

Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
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38 views

$SU(n)$ as a variety

Consider the algebraic group $SU(n)$ as an algebraic group scheme over $\mathbb R$. Is it birational to an affine space over $\mathbb R$?
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18 views

Example of algebraic group of non simply connected nor adjoint type?

I expect this question has been asked but I can't find it. What are some examples of simple linear algebraic group that are not semisimple nor adjoint type? Are there "good" descriptions of them?
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1answer
38 views

How to characterize elements in the Bruhat open cell?

This might be an elementary question. For simplicity, let's assume $G=GL(n,F)$, where $F$ is a local field. Let $U$ be the subgroup of upper triangular unipotents, $A$ the subgroup of diagonal ...
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0answers
19 views

What are the elements in $U/U(w)$?

Let $U$ be the maximal upper unipotent subgroup of $GL_n$. Let $U(w) = U \cap wUw^{-1}$. Then $$ U(w) = \{(a_{ij}) \in U: a_{ij} = 0, \text{ if } i<j, w^{-1}(i) < w^{-1}(j) \}. $$ My question ...
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1answer
22 views

How to compute $U \cap wUw^{-1}$?

Let $U$ be the upper unipotent subgroup of of $GL_n$. It is said that $$ U \cap wUw^{-1} = \{ (a_{ij}) \in U \mid a_{ij}=0, i<j, w^{-1}(i) > w^{-1}(j) \}. $$ How to prove this? I try to compute ...
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0answers
22 views

Factorization of parabolic subgroups.

Let $P$ be a parabolic subgroup of an algebraic group $G$. How to prove that $P = L_P U_P$? Here $L_P$ is the Levi of $P$ and $U_P$ is the unipotent radical of $P$. Thank you very much. Edit: I think ...
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1answer
100 views

Find a closed subset of an algebraic group, closed under products, which does not contain $e$.

The accepted answer for this question proves the following statement: If $S$ is a closed subset of an algebraic group $G$ which contains $e$ and is closed under taking products in $G$, then $S$ is ...
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1answer
68 views

$SU(2)$ as an algebraic group

The $\mathbb R$-valued points of the algebraic group $SU(2)$ can be identified with the real 3-sphere. But how does one define $SU(2)$ over the base field $\mathbb R$ as an algebraic group? What are ...
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1answer
20 views

Hopefully easy Lang-Steinberg computation: for Weyl elements

How do you write an element of the Weyl group as $g^{-1} F(g)$? For instance, let $G = \langle x_1(t), x_2(t), x_{-1}(t), x_{-2}(t) : t \in K \rangle$ where $K$ is an algebraically closed field ...
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0answers
23 views

Some things about maximal tori

Let $G$ be a linear algebraic group over an algebraic closed field of characteristic $p\neq 2$. Suppose $\overline{G}_{\sigma}={G}_{\sigma}/Z({G}_{\sigma})$ where ${G}_{\sigma}$ is the set of fixed ...
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0answers
53 views

Regular elements of a module is open and dense

Let $\mathfrak{g}$ be an algebraic Lie algebra and $V$ be a $\mathfrak{g}$-module, then for each $v\in V$, define $\mathfrak{g}^v = \{x\in\mathfrak{g}:xv = 0\}$. Let $V_{reg}$ be the set of all $v$ ...
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0answers
17 views

Non-central proper normal subgroups of unitary groups over fields

Short version: Can someone give an example of an anisotropic Hermitian form over a field such that its corresponding projective unitary group is not simple? Let $F$ be a (commutative, associative, ...
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2answers
175 views

Lie Groups/Lie algebras to algebraic groups

I am reading some lie groups/lie algebras on my own.. I am using Brian Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction I was checking for some other references on ...
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1answer
30 views

Computing a differential on a derivation

Let $\varphi:G\to G'$ be a morphism of algebraic groups over an algebraically closed field $k$, so that $d\varphi:\mathscr{L}(G)\to\mathscr{L}(G')$ is a morphism of Lie algebras. Here I view ...
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0answers
16 views

Group structure at $k$-valued points induces a group scheme?

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $R$ be a finitely generated (possibly non-reduced) $k$-algebra. Let $\mathbf{S} = \mathrm{Spec} (R)$ be the affine scheme of ...
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0answers
22 views

Are there some examples of Kac-Moody groups which are not reductive?

Are there some examples of Kac-Moody groups which are not reductive? Thank you very much.
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1answer
32 views

Split groups and quasi-split groups.

By definition, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. A split group is a quasi-split group which has split torus ($T = \mathbb{G}_m^n$, ...
2
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1answer
32 views

Promise of the hidden subgroup problem for $\mathbb{Z} mod 2$

I am going through the talk, Graph isomorphism, the hidden subgroup problem and identifying quantum states, by Pranab Sen. On slide 3, the promise of the problem is defined as follows. Here is the ...
6
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1answer
95 views

Airy differential equation and Galois group

Consider the Airy equation $y^{(2)}=ry$ where $r \in \Bbb{C}(z)$ but not constant. How do you show that $G^0=G$, where $G$ is the galois group of the picard vessiot extension of solutions over ...
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1answer
45 views

proof of basic fact that torus actions are diagonalizable

Suppose a torus $T=(\mathbb{C}^\ast)^n$ acts on a finite dimensional vector space $W$, and define for $m \in M$ ($M$ is the character lattice of $T$) the eigenspace $W_m$ by $$W_m = \{w \in W \mid ...
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0answers
11 views

Group schemes decomposition

Given an abelian group scheme of finite type $(G,+)$ over $\mathbb{F}$ connected, and given two connected closed subgroup schemes of finite type $G$ over $\mathbb{F}$ connected $H$, $N$ of $G$. ...
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1answer
39 views

Is $\mathbb{C}[N]$ isomorphic to $U(\mathfrak{n})$?

Let $G$ be an algebraic group and $N$ its maximal unipotent subgroup consisting of all upper triangular unipotent matrices. Let $\mathfrak{n}$ be the Lie algebra of $N$. It is said that ...
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0answers
20 views

Is supercuspidal representation the same as cuspidal representation?

I found that both supercuspidal representation and cuspidal representation are defined as representations which are not subrepresentations of induced representations. Is supercuspidal representation ...
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1answer
45 views

How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$?

Let $U$ be the positive unipotent radical of $SL_n$ and $\mathfrak{n}$ the Lie algebra of $U$. How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$? Here $\mathcal{O}_q[U]$ is the ...
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0answers
16 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 ...
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0answers
14 views

Local systems on algebraic groups

Two questions about local systems: Given an algebraic group G over a perfect field $k$ and given a colection of vectorial spaces $(F_y)_{y\in G}$ why there is an unique local system $F$ such that the ...
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0answers
18 views

Does taking $G$-invariant sections commute with infinite $\oplus$?

Let $X$ be a projective scheme over an algebraically closed field $k$, and let $G$ be a reductive algebraic group acting on $X$. If $R=\bigoplus_{n\geq 0}H^0(X,L^{\otimes n})$, then $G$ acts on $R$ ...
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0answers
26 views

The homogeneous coordinate ring attached to a $G$-linearized action

I am reading Huybrechts-Lehn's book on moduli of sheaves, and on page $85$, when they discuss GIT, the setup is as follows: $X$ is a projective scheme over an algebraically closed field $k$, there is ...
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0answers
28 views

Unipotent Group Scheme

Is this definition correct: An smooth unipotent group scheme $G$ over a perfect field $k$, with $\operatorname{char}(k) >0$ , is isomorpic to the affine scheme $k[x_1,...x_n]$.
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1answer
25 views

How to show that a representation is supercuspidal?

Let $G$ be a reductive group. If we know a representation $\pi$ of $G$ explicitly, how could we determine that $\pi$ is supercuspidal or not? Are there some references about this? Thank you very much. ...
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0answers
28 views

Intuition and examples of weil restriction

It is hard for me to catch the basic ideal of weil restriction, I need some simple examples to understand it, the following exercises are from the Springer's book: Linear Algebraic Groups 11.4.20(3). ...
1
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1answer
35 views

About the simply-connectedness of algebraic groups

Let $G$ be a simply-connected algebraic group. Is it necessarily true that its derived subgroup $G'$ is also simply-connected?
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2answers
124 views

Determining if these surjections have sections

Let $\pi: GL(2,k)\rightarrow PGL(2,k)$ be the canonical homomorphism, and pick some finite subgroup $G\subset PGL(2,k)$. Then we have an exact sequence $$1\rightarrow \{\alpha I\mid \alpha \in k\} ...
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0answers
10 views

Borel - Regular elements

In Borel's Linear Algebraic Groups (2ed) page 160 a regular element is defined in terms of its semisimple part, “thus $g$ is regular if and only if $g_s$ is regular.” A unipotent element $g$ has ...
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1answer
48 views

Are semisimplicity and regularity closed or open conditions in an algebraic group $G$?

Let $G$ be a connected algebraic group over an algebraically closed field. I'm trying to understand the phrase "the subvariety of semisimple elements in $G$ which are not regular." This tacitly ...
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1answer
51 views

Equivalent definitions of a regular element in an algebraic group

Let $G$ be a connected algebraic group over an algebraically closed field $k$. I'd like to see that the following two definitions are equivalent. An element $g\in G$ is regular if $\dim(C_G(g))$ is ...
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1answer
29 views

Conditions for a matrix in $Bw_0B$.

Let $B$ be the set of all upper triangular matrices in $GL_n$. What are the conditions for a matrix in $GL_n$ lies in $Bw_0B$ (What do the matrices in $Bw_0B$ look like)? Thank you very much.
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0answers
26 views

Question about a notation.

I have a question about the notation in the paper. On page 8, the 3-rd line from bottom, it is said that $h_J(t)$ is the corresponding diagonal matrix in $GL_4$. I think that $J$ is some set such that ...
0
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0answers
45 views

Questions about unipotent matrices.

I have a question about the notation in the paper. On page 8, the 5-th line from bottom, it is said that $u_J(t)$ denotes the upper triangular unipotent matrix with $t$ in position corresponding to ...
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1answer
36 views

What is the unipotent radical of a parabolic subgroup?

What is the unipotent radical of a parabolic subgroup? For example, let $$ P=\{ \left( \begin{matrix} a & b & k & c \\ d & e & f & g \\ 0 & 0 & h & i \\ 0 & 0 ...
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0answers
30 views

Coset variety of an algebraic group

Let $k$ be an algebraically closed field of characteristic $p\geq0$. An affine algebraic group $G$ is an affine algebraic variety (a Zariski closed subset of $k^m$ for some $m$) such that ...