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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10
votes
0answers
147 views

a closed subset of an algebraic group with a constant tangent space is a coset

Let $G$ be an algebraic (not necessarily linear) group and let $Z \subset G$ be a Zariski closed irreducible subset. Since tangent bundle of $G$ is trivial, we may identify tangent spaces at all ...
1
vote
0answers
139 views

Questions about affine Weyl group and extended affine Weyl group for SL2.

Let $G=SL_2$. Then the Weyl group is generated by $s_1$. On page 3 of the lecture notes, it is said that the affine Weyl group is generated by $s_0, s_1$. (1) The element $s_0s_1$ can be identified ...
1
vote
0answers
76 views

P/B is isomorphic to the projective line $\mathbb{P}^1$

Suppose that $P \subset G$ is a parabolic subgroup containing a Borel subgroup $B$. Moreover, let $P$ be a minimal parabolic subgroup properly containing B, i.e., one corresponding to a single root ...
5
votes
2answers
227 views

How many elements of order $k$ are in $S_n$?

I need to find how many elements of order $k$ are in $S_n$ (where $k \leq n$). So if $k$ is prime, it's easy: $k$ can't be the $\mathrm{lcm}$ of any integers besides itself and one's (which we're ...
2
votes
1answer
69 views

Factor/Quotient Group $M/\mathbb{Z}$

I've posted another question here a few days ago asking what a factor/quotient group is because I couldn't wrap my mind around it. Although I have an idea what it means, I still don't fully understand ...
3
votes
0answers
35 views

Closed subgroups of algebraic group have DCC?

Are there any infinite descending chains of closed subgroups of the general linear group over a field? More specifically, is my argument ok? Can you fill in some of the details? Prop: No. Proof: If ...
3
votes
1answer
155 views

Minimal parabolic subgroups of a reductive group - Bruhat type decomposition

Let $G$ be a reductive group, $B$ a Borel subgroup, $P$ a minimal parabolic subgroup having a Levi decomposition $P = UL$, let $\alpha$ be one of the two roots of $L$ relative to $T$, and $U_\alpha, ...
3
votes
1answer
66 views

Non-uniqueness of group structure for affine algebraic groups

We know that every abelian variety has a unique group structure, but in the affine case, is that every affine algebraic group has more than one (up to isomorphism) group structure?
1
vote
0answers
31 views

Baily Borel Compactification: choice of boundary

In Borel/Ji " compactifications of symmetric and locally symmetric spaces " the Baily Borel compactification of a locally symmetric space is defined as $$\Pi\backslash(X\coprod_{\bf{P}}X_{P,h})$$ ...
1
vote
0answers
53 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
1
vote
0answers
48 views

Orbits and rational points in a $G$-variety

Let $K/k$ be a field extension, let $V_0$ be a variety over $k$, and let $V=V_0\times_k\mathrm{Spec}\;K$, so that we can speak of the $k$-rational points of $V$ as morphisms $\mathrm{Spec }\;k\to ...
0
votes
0answers
37 views

How does Weyl group acts on coroots?

We know that $W=N/T$ ($N = \{n \in G \mid nTn^{-1} = T \}$) acts on $T$ by $w(t)=wtw^{-1}$ (since $T$ is commutative, this action is well-defined). In the case of $GL_n$, by direct computation we know ...
4
votes
0answers
67 views

What is $\rho^{\vee}(-1)$?

What is $\rho^{\vee}(-1)$? By definition, $\rho^{\vee}$ is the sum of all positive coroots. I have some difficulty in computing $\rho^{\vee}(-1)$. For example, in the case of $SL_3$, all positive ...
1
vote
0answers
135 views

Possible small mistakes in Springer's *Linear Algebraic Groups*

In the 2nd edition of Springer's Linear Algebraic Groups, the proof of 16.2.2(i) (p. 271) begins with the assertion that the reductive group G is quasi-split over F iff the restriction map of ...
2
votes
0answers
122 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
7
votes
1answer
282 views

Geometric intuition for linearizing algebraic group action

I am reading an overview of geometric invariant theory and find myself stuck when we begin linearizing the action of an algebraic group on a variety. The definition given in my notes is that given an ...
0
votes
1answer
40 views

Rigidity of Diagonalizable Algebraic Groups

This question is about a result in the section 16.3 of the book Linear Algebraic Groups from Humphreys. The follow can be deduced from a proposition in the section 16.3 of the book: Corollary: Let ...
3
votes
0answers
57 views

Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
0
votes
1answer
49 views

Tate module of linear algebraic group

Let $G$ be a (smooth, connected, geometrically integral) commutative linear algebraic group over $\mathbf F_q$. Just as for abelian varieties, we can define the $\ell$-adic Tate module $$ T_\ell G ...
3
votes
1answer
285 views

How to compute radical and unipotent radical of an algebraic group?

By definition, an algebraic group $G$ is reductive if its unipotent radical is $\{e\}$. The radical of an algebraic group is the identity component of its maximal normal solvable subgroup. The ...
1
vote
3answers
249 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
2
votes
0answers
26 views

Example of algebraic group of type $G_2$

Can anyone point me to a concrete realization of a reductive algebraic group of type $G_2$ over a field of positive characteristic? I have some questions about how the adjoint action permutes certain ...
1
vote
0answers
66 views

why is Borel subgroup not nilpotent?

Let $G$ be a simple linear group group over an algebraically closed field $k$, and let $B$ be a maximal solvable subgroup. If things are happening over $\mathbb{C}$ then I know how to show that $B$ ...
4
votes
0answers
110 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
3
votes
2answers
101 views

“$L/K$ forms of each other”

In section $4$ of these notes, the author says two algebraic groups $G$ and $H$ defined over a field $K$ are "$L/K$ forms of each other" if they are "isomorphic over $L$", where $L$ is a finite field ...
2
votes
1answer
59 views

Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
3
votes
0answers
63 views

Computing the fundamental groups of simple algebraic groups of type $A$

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$. Let ...
1
vote
0answers
54 views

Isomorphism types of stabilizers of vectors in linear representations of the special linear group

Suppose we have a linear representation of the group $SL_d$ over $\mathbb{C}$. i.e. a finite dimensional vector space $V$ with a linear action of $SL_d$ on it. Let $v\in V$ be some vector and let ...
3
votes
0answers
179 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
votes
1answer
61 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
vote
0answers
34 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
votes
1answer
118 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
votes
1answer
58 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
vote
1answer
65 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
votes
1answer
54 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
votes
0answers
89 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 ...
1
vote
0answers
14 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
votes
0answers
158 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!
1
vote
0answers
41 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
vote
1answer
164 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
votes
0answers
88 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
votes
1answer
139 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
67 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
vote
1answer
70 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 $e^*(T_{ij})=\delta_{ij}$, ...
1
vote
0answers
22 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
votes
0answers
216 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
53 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
votes
1answer
50 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
votes
1answer
31 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
52 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 ...