All Questions
2,633 questions
11
votes
0
answers
1k
views
Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?
It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
- 1,764
2
votes
0
answers
130
views
(Graded) Lie algebras with "nice" irreps
"(The only Lie algebras for which) all finite-dimensional representations are completely reducible (are the semisimple Lie algebras)" (Chapman, here on MO). Evidently this can't no longer hold when ...
- 4,829
2
votes
0
answers
128
views
On Eigenvalues of the symmetric linear transformation related to a lie algebra's representation?
Let $\mathfrak{g}$ be a quadratic (finite dimensional) lie algebra and $\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(W)$ be an anti-symmetric representation of $\mathfrak{g}$ on a finite dimensional ...
- 327
0
votes
0
answers
53
views
Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action
I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
- 2,099
1
vote
0
answers
63
views
Reference for using an algebra of meromorphic functions to extend a Lie algebra
For example, let $\mathfrak{g}=\mathfrak{sl}_{2}\left(\mathbb{C}\right)$, let $s_{0}=1$, $s_{1}=-1$, $s_{2}$=0, $s_{3}=\infty$ in $\mathbb{P}_{1}\left(\mathbb{C}\right)$ and $\mathcal{R}$ is the ...
- 357
1
vote
0
answers
216
views
polynomial representation of $sl_{2}(k)$
Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
\...
1
vote
1
answer
230
views
Generic Stabilizers in a Nilpotent Lie Algebra
Let $\frak{g}$ be a finite-dimensional nilpotent complex Lie algebra, and consider the adjoint Lie algebra representation of $\frak{g}$. What is known about the Lie-algebraic structure of the ...
- 4,920
4
votes
0
answers
189
views
Does this space of mod 2 modular forms admit a (Z/8)* degree decomposition?
Fix an odd N>0. Let M consist of all odd elements of Z/2[[x]] that are the mod 2 reductions of elements of Z[[x]] arising as the Fourier expansions of modular forms for (Gamma_0)(N); it's easy to see ...
- 5,422
4
votes
1
answer
627
views
Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...
- 33.8k
2
votes
1
answer
939
views
Weyl group Invariants
What are the generators of $\mathbb C[V^m]^W$, where $W$ is the Weyl group
of type $E_6, E_7, E_8$, V^m denote 'm' (m > 1) copies of the Cartan subalgebra
and the action is the diagonal action?
Is ...
- 21
4
votes
1
answer
228
views
Equivariant Levi subalgebras.
Suppose $\mathfrak g$ is a finite dimensional Lie algebra over a field on characteristic zero and $G$ is a finite group of automorphisms of $\mathfrak g$.
Does there necessarily exist a Levi ...
- 47.3k
2
votes
0
answers
243
views
simple roots of a reflection subgroup
Consider a Hermitian symmetric pair of complex Lie algebras $(\mathfrak{g},\mathfrak{k})$ and split the set of roots into compact roots (i.e. roots of $\mathfrak{k}$) and noncomapt roots $\Delta = \...
- 8,597
4
votes
1
answer
412
views
F_q-structures on schemes
Let $k|\mathbb{F}_q$ be a field extension. An $\mathbb{F}_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}_q$-subalgebra $A _0$ of $A$ such that $A _0 \otimes _{\mathbb{F}_q} k \cong A$ via the ...
- 5,243
15
votes
0
answers
779
views
Lifting varieties from char. $p$ to char. 0 after alterations
The question is related to this MO question:
Lifting varieties to characteristic zero.
Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an ...
- 4,265
4
votes
0
answers
164
views
Is there an analogue of distributions in characteristic p?
Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is ...
- 1,488
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 ${\...
- 16.9k
0
votes
1
answer
103
views
A property of compact involutions of semi-simple Lie algebras?
I need to prove the statement below. Since my background on Lie theory is rather weak, I post it here.
Let $\frak{g}$ be a complex semi-simple Lie algebra. Fix a Cartan subalgebra $\frak{h}$ with ...
- 1,804
0
votes
1
answer
212
views
L a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L).
Let L be a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L). (Der is short for derivation). In order to show that ad(L)=Der(L), the proof I followed proves that that the ...
2
votes
1
answer
278
views
What is the family derived from the absolute Frobenius on the Hilbert scheme?
Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...
- 27.9k
1
vote
1
answer
189
views
Dominant weights appear in Discrete Series
If $\lambda$ is a Harish-Chandra paramater. Let $\pi_\lambda$ it's associated discrete series, it's known by the minimal K-type thm that every K-type of $\pi_\lambda\mid_K$ has highest weight of the ...
- 11
2
votes
1
answer
256
views
The real group orbits on the flag variety always contains the holomorphic directions?
Let $G$ be a real semisimple Lie group and $\mathfrak{g}$ be its complexified Lie algebra. We have the flag variety $\mathcal{B}$ of $\mathfrak{g}$ which is the set of all Borel subalgebras of $\...
- 9,019
4
votes
1
answer
222
views
Why is there a discrepancy between the normalizations of the central terms for the commutation relations of the Virasoro versus Neveu-Schwarz Lie algebras?
Following the standard conventions in the literature, the commutation relations of the Virasoro Lie algebra are given by
$$[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}\frac1{12}(m^3-m)c,$$
$$[c,L_n]=0.$$
...
- 43.2k
5
votes
0
answers
381
views
Lie-infinity structure in Lagrangian Floer theory ?
Is there (besides the A-infinity structure) also a L-infinity structure in Lagrangian Floer theory (forming together a G-infinity structure) - like in Hochschild cohomology ?
9
votes
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
3
votes
2
answers
1k
views
Representations of reductive Lie group
Let $G$ be a reductive algebraic group and $\varrho$ a representation of $G$ in $GL(n)$. Is it true that $\varrho$ is completely reducible? Moreover, how are related the representations of the Lie ...
- 671
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 ...
- 4,265
4
votes
1
answer
534
views
Can a simple lie algebra be determined by weights of its representation?
Suppose you are given a linear combination of points of $\mathbb{Z}^n$ which corresponds to the weight datum of a nontrivial representation $V$ of some simple Lie algebra $g$. Is it possible to ...
- 7,972
0
votes
1
answer
184
views
why no Lie algebra degenerate to a rigid algebra? Why the closure of a rigid algebra forms the irreducible component of variety of Lie algebras?
Hi
I just started working on degeneration and contractions, I would like to know:
why no Lie algebra degenerate to a rigid algebra?(rigid algebra:an algebra whose orbit is zariski open)
Why the ...
2
votes
0
answers
93
views
Locus maximizing the holomorphic sectional curvature in a non-compact Hermitian symmetric space
Is there a quick way to prove the following statement, if possible without resorting to the classification of simple Lie groups?
Let $G$ be a simple Lie group of non-compact Hermitian type of rank $r$...
- 301
5
votes
0
answers
887
views
Classification of generalized Cartan matrices (GCMs)
A GCM is square matrix $A = (a_{ij})$ satisfying: (1) $a_{ij} \in \mathbf{Z}$ (2) $a_{ii} = 2$ for all $i$. (3) $a_{ij} \leq 0$ for $i \neq j$. (4) $a_{ij} = 0$ iff $a_{ji} = 0$. There is a standard ...
- 751
1
vote
1
answer
390
views
nullity of the second fundamental group of a Lie group
Can anyone tell me why it is that Lie groups seem to have their second fundamental group $\pi_2(G)$ equal to $0$, or provide me with a link to an article or a book reference?
I came across this fact ...
- 2,751
1
vote
0
answers
120
views
Vanishing theorems that work in positive characteristic
Let $X$ be a smooth projective variety over a field of characteristic $p>0$ of dimension at least $2$. I am looking for some examples when $H^2(\mathcal{O}_X)$ vanishes. Is there any standard way ...
- 1,981
6
votes
0
answers
436
views
Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?
As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on '...
- 16.9k
1
vote
0
answers
220
views
Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q
Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...
- 407
7
votes
1
answer
498
views
Does some version of U_q(gl(1|1)) have a basis like Lusztig's basis for \dot{U(sl_2)}?
There's a non-unital algebra $\dot{U}$ formed from $U_q (sl_2)$ by including a system of mutually orthogonal idempotents $1_n$, indexed by the weight lattice. You can think of this as a category with ...
- 1,756
2
votes
0
answers
177
views
Explicit Lie May structure on cosimplicial DG Lie algebras
In the paper "Homotopy Lie algebras", Schechtman and Hinich proved that any cosimplicial
differential graded Lie algebra has the structure of a 'Lie May algebra'.
If my understanding is right here, ...
- 624
2
votes
1
answer
646
views
Quotient by p-th roots of unity in characteristic p
Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu_p = {\rm Spec}\ k[\...
- 16.2k
2
votes
0
answers
464
views
understanding Milne's article "Duality in the flat cohomology of a surface"
I have trouble understanding a point of Milne's article "Duality in the flat cohomology of a surface" http://jmilne.org/math/articles/1976a.pdf
see the "Alternatively" on p. 177, paragraph before ...
user19475
6
votes
1
answer
494
views
Non-free projective modules for a Universal Enveloping Algebra?
Let g be a finite dimensional Lie algebra over k, and let U be its universal enveloping Lie algebra. Is there a left module M of U which is projective but not free? That is, is the Quillen-Suslin ...
- 13k
-9
votes
1
answer
504
views
Lia algebra strings [closed]
Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra $g$ of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the length of an $a$-string through $b$ is at most ...
- 829
5
votes
0
answers
454
views
If $p=0$ and $df=0$, is $f$ a $p$th power?
This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...
- 4,487
4
votes
0
answers
214
views
Harmonic analysis and non-symmetric Macdonald polynomials?
I have recently been reading a lot about Macdonald polynomials, the symmetric and the non-symmetric ones. One thing that strikes me is that the symmetric Macdonald polynomials admit a positive theory, ...
- 41
3
votes
1
answer
336
views
Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$.
The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$
has the bracket
$$[xt^r, yt^...
- 138
5
votes
0
answers
281
views
Is the "Toeplitz algebra" the representation ring of a Hopf algebra related to SU(2)?
More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...
- 118k
1
vote
0
answers
154
views
A Isomorphism between the extension group and cohomology group of Lie algebras [closed]
Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove it....
- 19
1
vote
2
answers
563
views
Reps of groups and reps of algebras
I've got what might be a couple of very basic questions on the fundamental representations of locally isomorphic semi-simple Lie groups and their relationship to representations of the corresponding ...
3
votes
1
answer
409
views
A question on the construction of finite W-algebras
In a well known construction of finite W-algebras, one first constructs a certain
nilpotent subalgebra $\mathfrak{m}$ along with a character $\chi:\mathfrak{m}\rightarrow \mathbb{C}$.
Then one defines
...
- 13.2k
10
votes
0
answers
547
views
Explicit change of basis for the Schur-Weyl basis
The Schur-Weyl duality states that $\bigotimes_{m=1}^n \mathbb{C}^k$ can be decomposed as a direct sum over the tensor product of irreductible representations of $SU(k)$ and of the symmetric group $\...
- 254
2
votes
0
answers
279
views
deRham cohomoloy of CM liftings of Jacobians
Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
- 637
5
votes
0
answers
762
views
Applications (and source) of Bourbaki exercise on root systems with two root lengths?
In Chapters 4-6 of Bourbaki's Groupes et algebres de Lie, Exercise 20 for Section
VI.1 concerns irreducible (reduced) root systems with roots of two lengths: in other
words, systems of types $B_\ell, ...
- 52.9k