All Questions
2,543 questions
3
votes
2
answers
489
views
Hopf algebra of Chevalley group from the root system
Has anyone worked out a uniform way of constructing the Hopf algebra of a Chevalley group out of the root system (or, more precisely out of the root datum for reductive groups).
By "uniform", I mean ...
- 13.3k
3
votes
2
answers
1k
views
Connectedness of Centralizers in $GL_n$
I was wondering if there is any obvious reason or quick proof that for every $g\in GL_n$ the centralizer $Z_{GL_n}(g)$ is connected. Also I wanted to see why for any semisimple $s\in Sp_{2n}$ the ...
- 52.9k
2
votes
5
answers
1k
views
Product of two algebraic subgroups of a (solvable) group = another algebraic subgroup?
Let $G$ be a linear algebraic group over a field $K$. (Say $K=\mathbb{F}_q$ or
$K=\mathbb{C}$; do not assume $K$ is algebraically closed or of characteristic $0$.) Let $H_1$, $H_2$ be algebraic ...
- 324
4
votes
2
answers
604
views
Adem-Wu relations from Bullett-Macdonald identities
Question. Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; ...
- 20.7k
3
votes
0
answers
379
views
Maximal compact subgroup of a real semisimple Lie group of "quasi-adjoint" type.
Let $\mathfrak{g}$ be a semisimple complex Lie algebra, and $G$ the adjoint group of $\mathfrak{g}$.
Let $\sigma:\mathfrak{g}\rightarrow \mathfrak{g}$ be a complex conjugation and $\mathfrak{g}^{\...
- 59
9
votes
2
answers
1k
views
Action on the highest weight vector of a representation of a semisimple linear algebraic group
Let $G$ be a semisimple linear algebraic group, $V$ a $G$-representation and $v \in V$ a vector of highest weight $\lambda$. Is it true, that for any positive root $\alpha \in R^+$ the one dimensional ...
- 52.9k
8
votes
2
answers
3k
views
Lie algebras of algebraic groups
Where can i find material about the definition of the exponential morphism from the Lie algebra of an algebraic affine group to the group?
- 1,602
2
votes
2
answers
329
views
Existence of proper invariant subset in an irreducible action
Let $G<\rm{GL}_n(\mathbb{k})$ be a linear group, where $\mathbb{k}$ is an algebraically closed field. Assume that the linear action of $G$ on $\mathbb{k}^n$ is strongly-irreducible (i.e. there are ...
- 3,246
2
votes
1
answer
249
views
unipotent group and translation invariant metric
Let $U$ be a unipotent upper triangluar group over a local field $K$ of characteristic
zero. Can we guarantee that there is a right translation invariant metric on $U$ such
that any ball of finite ...
- 11.1k
4
votes
2
answers
694
views
Ample line bundle and Frobenius morphism on smooth toric variety
Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\...
- 16.2k
9
votes
1
answer
756
views
Is every reductive group scheme etale locally trivial?
Let $S$ be a scheme over a field $k$, and let $G$ be a reductive group scheme over $S$. Let us call it trivial, if it is a pull-back of a group scheme over $k$ via the structure morphism $S\to k$. Is ...
- 1,602
3
votes
2
answers
708
views
$k$ structures on $K$ vector spaces
The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.
Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is ...
- 52.9k
2
votes
1
answer
332
views
Ample bundle under Frobenius morphism
Let $k$ be an algebraically closed field of char($k$)=p>0, $X$ a smooth projective variety over $k$, $F:X\rightarrow X^{(1)}$ the relative Frobenius morphism. Let $E$ be an ample vector bundle on $X$. ...
- 35.2k
6
votes
1
answer
2k
views
Margulis normal subgroup theorem
Margulis' normal subgroup theorem states that any normal subgroup of a higher rank lattice is either finite or of finite index. The obvious question is: can one classify finite normal subgroups of ...
CommunityBot
- 1
7
votes
1
answer
2k
views
Parametrization of 2-dimensional torus
The units with norm $+1$ in a pure cubic number field $K$ generated
by a cube root of $m = ab^2$, where $a$ and $b$ are coprime and
squarefree integers, correspond to integral points on the torus
$$ ...
- 4,195
8
votes
1
answer
1k
views
Picard group of Schubert varieties
Let $G$ be a semisimple linear algebraic group, $P$ be a parabolic subgroup and $w$ be an element of the Weyl group of $G$. I want to calculate the Picard group of the Schubert variety $X_P(w):=\...
- 3,103
9
votes
2
answers
1k
views
Relative Lie Algebra cohomology and sheaf cohomology
(I apologize in advance for the vagueness of my question). Let $G$ be a reductive algebraic group over $\mathbb C$ with Lie algebra $\frak g$ and Borel $B$. I have seen casual references to the fact ...
- 52.9k
2
votes
1
answer
797
views
Rationality of flag varieties
Let $X$ be a (generalised) flag variety over an algebraically closed field $k$ of characteristic zero, that is to say, $X$ is a projective variety which is a homogeneous space for some algebraic group ...
- 52.9k
0
votes
0
answers
352
views
Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
23
votes
2
answers
3k
views
Why are Tamagawa numbers equal to Pic/Sha?
For a connected algebraic group $G$ over a global field $K$ with adeles $A$, the Tamagawa number of $G$ is the volume of $G(A)/G(K)$. It is conjectured (and often known) to be rational, namely the ...
CommunityBot
- 1
5
votes
2
answers
561
views
Diagonalizable subgroups in a simply connected group
This is a continuation of my previous question.
Let $G$ be a connected reductive group over an algebraically closed field $k$ of characteristic 0.
We assume that $\mathrm{Pic}\ G=0$.
This is the same ...
CommunityBot
- 1
7
votes
2
answers
513
views
Tameness for the Galois closure of a map of curves
Say we are working over some $K=\overline{K}$, of characteristic $p>0$. Let $\phi: Y\rightarrow X$ be a nonconstant map of smooth projective curves. To this map we can associate a map $\psi: Z\...
- 839
0
votes
0
answers
524
views
DeRham cohomology
The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...
- 1
6
votes
1
answer
545
views
Is it true that no one-dimensional group variety acts transitively on $\mathbb{P}^1$?
This question may be trivial to people with the right background, but I do not see the answer.
Let $\Bbbk$ be an algebraically closed field. Can any one-dimensional group variety (over $\Bbbk$) ...
- 52.9k
9
votes
0
answers
389
views
Twisted Springer fibers
In the study of certain moduli spaces of $p$-divisible groups I came across the following twisted version of a Springer fiber, and I was wondering whether some expert on algebraic groups/algebraic ...
- 1,645
4
votes
2
answers
339
views
Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?
Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
- 20.5k
6
votes
1
answer
825
views
More on universal homeomorphisms
I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).
Is a universal ...
- 27k
6
votes
1
answer
393
views
finite quotients of fundamental groups in positive characteristic
For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
CommunityBot
- 1
-1
votes
1
answer
2k
views
Semisimple elements of a lie algebra
Let $G\subset GL_n(\mathbb{C})$ be an algebraic group of dimension n, and let $\mathfrak{g}$ its Lie algebra.Is there a relations between the maximal number of independent semisimple elements of $G$ ...
- 52.9k
2
votes
2
answers
548
views
The product of non-commuting semisimple matrices need not be semisimple
In general when one proves that the product of semisimple (i.e. diagonalizable) matrices is semisimple one assumes they commute and are thus simultaneously diagonalizable, and then the result follows. ...
- 5,654
3
votes
2
answers
452
views
Is there an invariant theory explanation of the orbit structure of GL₂ acting on second-diagonal symmetric matrices by g∙X = gXJg^tJ ?
Statement of the Specific Result
Let $J$ denote the matrices with ones on the "second diagonal", meaning the diagonal between the (1,n) and (n,1) entry, and zeros elsewhere. So in the case $n=2$, ...
- 2,018
2
votes
1
answer
408
views
Is restriction of scalars of simply connected algebraic groups still SC?
Let $G$ be a simply connected semisimple algebraic $K$-group and $K$ be a finite extension of $k$.
Is $R_{K/k}G$ still a simply connected algebraic group?
We say $G$ is simply connected if for any ...
- 583
7
votes
2
answers
736
views
What does a homogeneous space of a linear algebraic group know about the group?
Let $X=G/H$, where $G$ is a connected linear algebraic group over the field $\mathbf{C}$ of complex numbers
and $H\subset G$ is an algebraic subgroup.
In general, we can write the algebraic variety $X$...
- 52.9k
5
votes
1
answer
366
views
Differential of a nilpotent or semisimple element
Let $G$ be an algebraic connected subgroup of $GL_n(\mathbb{C})$ and let $\chi : G \to \mathbb{C}^*$ a character. Consider $d_e\chi : \mathfrak{g} \to \mathbb{C}$ the differential of $\chi$ at the ...
- 671
13
votes
3
answers
2k
views
Density question in algebraic group
Suppose G is an algebraic group defined over F, the algebraic closure of F is K. Consider the Zariski topology on G(K), is G(F) Zariski dense in G(K)?
- 3,246
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 423
10
votes
2
answers
1k
views
For what reductive groups $G$ over $K$ are the inner forms classified by $H^1(K, G^{ad})$?
Suppose $G$ is a connected reductive algebraic group over an arbitrary field $K$; let $Z$ be the center of $G$. The inner automorphisms of $G$ are given by $\operatorname{Inn}(G) = G / Z = G^{\...
- 2,178
12
votes
2
answers
929
views
Are representations of a linearly reductive group discretely parameterized?
Suppose $G$ is a linearly reductive group over a field (say $\mathbb C$). Does somebody know of a proof that any flat family of finite-dimensional representations of $G$ must be locally constant?
In ...
CommunityBot
- 1
3
votes
2
answers
2k
views
all parabolic subgroup of GL(3,K) and their Levi decomposition?
Who can tell me all parabolic subgroup of GL(3,K) and their Levi decomposition?
- 52.9k
6
votes
3
answers
590
views
Zariski-closed subsemigroups of SL_n(C) are groups
I would like to show that any Zariski-closed subsemigroup of $SL_n(\mathbb{C})$ is a group. If I understand correctly, this is consequence 1.2.A of http://www.heldermann-verlag.de/jlt/jlt03/BOSLAT.PDF ...
- 33.8k
7
votes
5
answers
869
views
Principal bundles over groups
If we have an extension of groups (say algebraic groups or group schemes) $1\to F\to P\to G\to 1$, then $P$ is a principal $F$-bundle over $G$ (is it locally trivial?). How about going in the opposite ...
- 35.5k
3
votes
2
answers
1k
views
Connectedness of centralizers and regular elements in unipotent groups
Let $G$ be a connected linear algebraic group over an algebraically
closed field $k$ of characteristic $p$. An element $x\in G$ is
called regular if its centralizer has minimal dimension among
all the ...
- 3,246
7
votes
2
answers
429
views
Multiplication tables for H*(G/P)?
Hi everyone. My recent work has me developing software to compute in $H^\ast(G/P)$, where $G$ is a complex connected semisimple algebraic group and $P$ is a standard parabolic subgroup (usually, $B$ ...
- 1,602
2
votes
1
answer
528
views
Is there an easy proof of the fact that the intermediate image functor respects weights?
It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially ...
- 1,576
4
votes
2
answers
402
views
lower bound for torsion of abelian varieties
Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ ...
- 5,050
7
votes
2
answers
1k
views
Reference request: representations of unipotent groups have a fixed point.
I'm looking for a reference for the following standard result:
Let $U$ be a unipotent algebraic group over an algebraically closed field $k$ (of any characteristic); then any algebraic ...
- 20.7k
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 91
9
votes
1
answer
447
views
Are affine groups over rings of integers finitely generated?
I'll begin by saying that I'm not sure what I want to ask specifically, but pretty sure what in general, so please don't hold my misunderstandings against me, but do comment on them.
I know that the ...
- 601
1
vote
0
answers
409
views
Pushforward of equivariant bundles via the Frobenius morphism
Let $G$ be a semisimple algebraic group over an algebraically closed field of positive characteristic $p$ and let $B \subseteq G$ be a Borel subgroup. Set $X := G/B$, the flag variety of $G$. Also let ...
- 3,637
1
vote
2
answers
393
views
Could the Kunneth decomposition of a motif depend on the choice of $l$?
Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...
- 1,576