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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2
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159 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 ...
2
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1answer
51 views

Integer subgroup of indefinite orthogonal group

I'm interested in the subgroup of the indefinite orthogonal group consisting of integer matrices i.e. $$ZO(p,q):=\{M \in GL(p+q, \mathbb{Z})| M^TI_{p,q}M=I_{p,q}\},$$ where $I_{p,q}$ denotes the ...
1
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0answers
31 views

Exactness of Hom functor for torus representations?

Given a reductive algebraic group $G$ and a maximal torus $T$. Is it true that the functors $$ Hom_T(-,\lambda) $$ are exact, where $\lambda$ denotes one of the the simple one-dimensional ...
4
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1answer
91 views

Is a finite normal subgroup of a reductive algebraic group central?

In a proof I am reading, the author considers the situation where $G$ is a reductive algebraic group (variety) over the complex numbers $\mathbb C$ and $N\trianglelefteq G$ is a closed, normal ...
2
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1answer
53 views

Finding the Hopf Algebra Coproduct coming from an Affine Group Scheme

I was wondering if anyone could help with how to, strictly from Yoneda's Lemma, obtain the coproduct map on the Hopf Algebra for an Affine Group Scheme. Particularly for something like $\text{SL}_2$ ...
1
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1answer
60 views

Automorphisms of algebraic groups

I am working through Humphreys' Linear algebraic groups, and I am stuck on the following exercise ( ex 4 on pg 57) I need to show that the only automorphisms of $G_m$ (as an algebraic group) is $x ...
3
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1answer
50 views

Torsion Subgroups and Periodicity

I am trying to piece together elliptic curves in FLT and would greatly appreciate corrections to my summary (or attempts thereof). Mazur's paper "Number Theory as Gadfly" states, "there is a natural ...
4
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0answers
87 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 ...
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0answers
12 views

Questions about the differential of Ad.

In the book linear algebraic groups by Humphreys. Page 73, Lemma B says that for $x \in GL(n, K)$, $\mathbf{x} \in gl(n, K)$ we have $Ad x(\mathbf{x}) = x\mathbf{x}x^{-1}$. In the proof, the first ...
2
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1answer
122 views

Mal'cev completion of nilpotent groups

Is the $\mathbb{R}$-Mal'cev completion of a finitely generated torsion free nilpotent group connected and simply connected?. Thanks!
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0answers
39 views

Bounding the inner product in root systems.

Let $R$ be a root system (irreducible if that makes this easier) in the real vectorspace $E$. Let $\lambda$ and $\mu$ in $E$ with $w_0(\lambda)\leq \mu \leq \lambda$ where $w_0$ is the longest ...
1
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1answer
126 views

what does a “rational structure” mean in algebraic group theory

For an algebraic group $G$ and a finite field $\mathbb F_q$, what does an "$\mathbb F_q$-rational structure of $G$" mean? Is it always related to a Frobenius map? I encountered this while reading ...
3
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0answers
87 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 ...
4
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1answer
134 views

When does unitary group become compact?

This might be a silly question, but I got confused sometime. Let $F$ be a local field of characteristic zero, and let $E/F$ be a quadratic extension. Use $\sigma$ to denote the nontrivial element in ...
1
vote
1answer
61 views

Automorphisms of $\mathbb{G}_m$.

Let $\mathbb{G}_m$ be the multiplication group whose underlying set is $k^*$, where $k$ is a field. How to show that as an algebraic group there are only two automorphisms of $\mathbb{G}_m$? How many ...
1
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1answer
63 views

The Hopf algebra structure of $GL(n, K)$.

Let $K$ be an algebraic closed field and $G=GL(n, K)$. There is a Hopf algebra structure on $K[G]=K[T_{11}, T_{12}, \ldots, T_{nn}, d^{-1}]$, $d=\det (T_{ij})$ given by $e^*(T_{ij})=\delta_{ij}$, ...
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0answers
19 views

Questions about tangent maps of tangent spaces of algebraic groups.

Let $G, H$ be algebraic groups and $K$ an algebraic closed field. Suppose that we have a morhpism $f:G \to H$ and the corresponding morphism of functions $f^*:K[H] \to K[G]$ given by $\varphi \mapsto ...
2
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0answers
192 views

Construction of line bundles on the flag variety

Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags... I am a bit confused about the subject in the title. For ...
2
votes
1answer
50 views

Question about $S_\infty$ or $Aut(\mathbb{N})$

I have been reading a little Kechris and other random Polish group books, and have come across a question I just can't wrap my mind around. Show that $S_\infty$ or $Aut(\mathbb{N})$ the set of all ...
0
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1answer
47 views

Reference of the polynomials satisfying $f(t+u, v)+f(t,u)=f(u+v, t)+f(u,v)$.

Are there some reference of the polynomials satisfying $f(t+u, v)+f(t,u)=f(u+v, t)+f(u,v)$? I would like to know some of there properties. I am asking this question because these polynomials appear in ...
0
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1answer
30 views

Why the kernel of $\phi_0: Der_R(B, N) \to Der_R(A, N)$ is $Der_{A}(B, N)$?

I have some questions of the book Linear algebraic groups by T.A.Springer. On page 57, it is said that the kernel of $\phi_0: Der_R(B, N) \to Der_R(A, N)$ is $Der_{A}(B, N)$. Let $Der_R(A, M)$ be the ...
2
votes
1answer
50 views

How to show that $(\delta'f)(e)=\delta(\phi^* f)(e)$?

I am reading linear algebraic groups. I have a question in line 4 of the third paragraph of Page 66. How to show that $(\delta'f)(e)=\delta(\phi^* f)(e)$ for all $f\in K[G]$? Here $G$ is an algebraic ...
3
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1answer
130 views

How to show that $G_a$ and $G_m$ are connected?

Let $K$ be a field and $A^1$ the affine line. Let $G_a$ be the affine line $A^1$ with group law $\mu(x, y)=x+y$. Let $G_m$ be the affine open subset $K^* \subset A^1$ with group law $\mu(x,y)=xy$. How ...
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2answers
91 views

Linear algebraic group of $GL(V)$ is independent of choice of basis

This is an easy question, but I am confused now. Please see the definition in this book In the last line. If we have $f(g)=0$, where $f(y)=p(x_{11}(y),...,x_{nn}(y))$, then after choosing different ...
2
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0answers
34 views

Why the finite subgroups of $GL_n$ is closed with respect to Zariski topology?

Let $GL_n$ be the group of all $n$ by $n$ invertible matrices. Why the finite subgroups of $GL_n$ is closed with respect to Zariski topology? Are the zeros defined by some equations? Thank you very ...
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0answers
83 views

Questions about the book linear algebraic groups by Springer.

I am reading the book linear algebraic groups by Springer. I have a question on Page 53, on line 3, it is said that $d-h \geq p$ implies that ${d-h \choose p} \not\equiv 0 \pmod p$ by Lemma 3.4.2. But ...
3
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0answers
105 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
87 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 ...
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1answer
55 views

Questions about the proof of the isomorphism $k[X] \to \mathcal{O}(X)$ in the book Linear algebraic groups by Springer.

I am reading the book linear algebraic group by Springer. I have some questions on page 8. Theorem 1.4.5 is: $\phi: k[X] \to \mathcal{O}(X)$ is an isomorphism. (1) Line 8-9 of the proof of Theorem ...
3
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1answer
47 views

$GL(-)$ as a k-group functor

My question is essentially may lye simply in a notational obstruction. For a k-algebra M, Jantzen J. defines the k-group functor $GL(M)$ as: $GL(M)(A):=(End_A(M\otimes_{\mathbb{k}} A)^*$. My ...
5
votes
2answers
264 views

Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group?

By a complex reductive algebraic group I mean the group of complex points of a (possibly disconnected) affine algebraic group defined over $\mathbb{C}$ whose unipotent radical (maximal connected ...
2
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0answers
44 views

Relationship between invariants of a simple algebraic group

Let $G$ be a simple algebraic group over an algebraically closed field $k$. I believe all of the following invariants are well-defined. Besides the coxeter number, I haven't read about the others, ...
5
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0answers
102 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. ...
3
votes
1answer
67 views

The identity component of an algebraic group is always parabolic

Essentially I was wondering if the quotient of an algebraic group $G$ by its identity component $G^0$ is necessarily always parabolic. My argument: This seems right since $G^0$ is a closed ...
3
votes
1answer
81 views

Why is the map: $GL_n(K)\times GL_n(K) \to GL_n(K)$ regular?

Let $K$ be a field and $GL_n(K)$ the set of all invertible $n$ by $n$ matrices over $K$. Let $m: GL_n(K)\times GL_n(K) \to GL_n(K)$ be the usual multiplication of matrices. Why the map $m$ is regular? ...
0
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1answer
46 views

How to show that the restriction of $\pi$ to the subrepresentation $W$ factor through $G/N$?

I am reading the lecture notes. On Page 16, Line 1, it is said that the restriction of $\pi$ to the subrepresentation $W$ factor through $G/N$. What does "factor through" mean? How to show that the ...
4
votes
1answer
70 views

Why $G/N$ is discrete?

I am reading the lecture notes. On page 15, Line -5, why $G/N$ is discrete? Thank you very much.
1
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1answer
62 views

How to show that two representations are equivalent?

I am reading the lecture notes. On page 14, example of $C_{c}^{\infty}(G)$. I am trying to show that the map $A$ takes $f$ to $g\mapsto f(g^{-1})$ is an invertible element of ...
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1answer
58 views

How to show that $\pi^*(g)=\chi(\det g)^{-1}$?

I am reading the lecture notes. On page 14, how could we show that $\pi^*(g)=\chi(\det g)^{-1}$? I think that $\langle \pi^*(g)\lambda, v \rangle = \langle \lambda, \chi(\det g)^{-1} v \rangle = ...
0
votes
1answer
52 views

Questions about reductive groups.

I am reading the lecture notes. Let $G$ be a reductive group and $(\pi, V)$ a representation of $G$. For $v \in V$, define $\operatorname{Stab}(v)=\{g\in G \mid \pi(g)v=v\}$ and $V^{\infty}=\{v\in V ...
4
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2answers
197 views

Automorphism Groups of Projective Algebraic Surfaces

When does a projective algebraic surface have an infinite automorphism group? Is there a simple criterion, or at least a sufficient condition?
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0answers
33 views

Is there any good books or papers about Coxeter groups with negative weight function?

As to Coxeter group with weight function, most people are concerned with positive weight functions, including Lusztig, writing Hecke algebras with unequal parameters. This book don't deny the negative ...
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0answers
34 views

Calculating this quotient?

What is the quotient of the algebraic Groups $GL_n$ by $Sp_n$ equal to? I was conisering a different example and would use the universal property to establish it, but I wasn't certain what it should ...
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0answers
36 views

Complexification of the inclusion $\text{U}_n\subset \text{GL}_n(\mathbb{C})$

What is the map $\text{GL}_n(\mathbb{C}) \to \text{GL}_n(\mathbb{C}) \times \text{GL}_n(\mathbb{C})$ named in the title? I guess it has something to do with the polar decomposition, but I can't manage ...
4
votes
1answer
275 views

What are the Borels/parabolics of the orthogonal or symplectic groups?

Does anyone know where I can find info about the Borel subgroups and parabolic subgroups of algebraic groups? I know what they are for the general linear group and for the special linear group you ...
3
votes
0answers
119 views

Are simple algebraic groups absolutely simple?

Let $k$ be a field. By a simple algebraic group over $k$ I mean an affine group scheme $G$ of finite type over $k$ such that $G$ is connected, non-commutative and every normal closed subgroup of $G$ ...
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0answers
51 views

The symplectic transvection group is isomorphic to what $A^1$, $A^2$?

The transvections generate the symplectic group, but to demonstrate its connectedness I need to establish their connectedness. I was thinking of doing this through an isomorphism of alg. groups with ...
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1answer
70 views

Representation Theory - Example of a not-G-stable V

I'm studying linear representations for algebraic groups for the moment. And I kind of got stuck on some theorem. The existence of a finite linear representation makes use of the fact that $V$ is ...
7
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1answer
137 views

Is taking inverse automatically well-defined?

By the usual definition, Lie group is a manifold $G$ with a group structure on it such that the multiplication $m\colon G\times G\to G$ and taking inverse $i\colon G\to G$ are both smooth maps. But it ...
4
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1answer
78 views

Is Bruhat cell dense in p-adic topology?

I've seen in literature a statement like 'there exists an open and dense Bruhat cell'. In $GL(2,F)$ for example, where $F$ is a p-adic field, let $\omega=\begin{pmatrix} & 1 \\ 1 & ...