All Questions
2,633 questions
5
votes
1
answer
376
views
Translation of "le nilradicalisé de g"
I apologize for asking something that might well be found in a mathematical dictionary, but the similarity of the French word to an English one is frustrating my attempts to Google the answer (and the ...
- 25.8k
0
votes
1
answer
340
views
PBW-Theorem and multigraded Lie algebras
Fix a $\mathbb Z_+^n$-graded Lie algebra ${\frak a}=\oplus_{r \in\mathbb Z_+^n}^{} {\frak a}[r]$ such that ${\frak g}:={\frak a}[0]$ is a finite-dimensional semisimple Lie algebra over the complex ...
- 3
2
votes
1
answer
678
views
About localization theorem for affine Lie algebra?
Here is my question: how to define global section functor from D-module on affine flag variety to representation of affine Lie algebra?
Let's me explain the difficulty: it seems there doesn't exist ...
- 1,457
2
votes
0
answers
339
views
volume form in a symmetric space of real rank one
I want to compare the two canonical volume forms on a noncompact symmetric space fo real rank one.
The first one is the volume form induced by the Riemannian structure given by the Killing form ...
- 2,446
2
votes
0
answers
115
views
The condition of maximality in branching rules of $SO$ group representations
Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...
- 1,893
4
votes
1
answer
446
views
Dolbeault Operators for $CP^1$ as $\mathfrak{su}(2)$ Actions.
This question is related to a previous question of mine. More specifically, it results from my attempts to understand the simplest incarnation of a phenomenon mentioned therein.
Put a grading on the ...
- 3,409
1
vote
0
answers
378
views
Does Iwahori subalgebra correspond to any Cartan decomposition for affine Kac-Moody algebra?
Let $\hat{\mathfrak{g}}$ be an affine Kac-Moody algebra which is the central extension of $\mathfrak{g}[t,t^{-1}]$(polynomial version). Consider Iwahori subalgebra $I$. My question is whether $I$ ...
- 5,515
1
vote
0
answers
85
views
Analytic vectors of elliptic elements of the enveloping algebra in a strongly continuous representation of a Lie group
Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal ...
- 1,582
7
votes
0
answers
1k
views
Category O of Kac-Moody algebra
Category $\mathcal{O}$ of semisimple Lie algebra has been understood very well. One can decompose the category into different blocks by central characters, and evey block is Noetherian and Aritian, ...
- 1,457
0
votes
0
answers
349
views
cokernel for $L_\infty$-algebra morphisms
As I have asked a wrong question previously, I edited a bit.
It is obvious that the cokernel construction does not work well for the category of Lie algebras. The cokernel exists only for a normal ...
- 1,271
4
votes
0
answers
136
views
A subring of the Serre Swinnerton -Dyer ring of level N modular power series
Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that an element of (Z/...
- 5,422
2
votes
1
answer
524
views
Character formulas for non-integrable modules?
Let $\mathfrak{g}$ be a Kac-Moody Lie algebra (actually, I'd already be happy with an answer addressing the case where $\mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$).
1st ?: I'm wondering ...
- 1,335
6
votes
0
answers
344
views
Idempotent non-associative algebras
This is inspired by that question by Andreas Thom.
Let $L$ be a finitely generated Lie ring (or Lie algebra over a field) such that $L=[L,L]$, that is the Abelianization of $L$ is 0. Is it true ...
user6976
1
vote
0
answers
192
views
"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
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
3
votes
1
answer
288
views
Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?
I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...
- 16.9k
0
votes
1
answer
136
views
Reference request: Tensor products of modules for reductive Lie algebras
I am looking for a reference that describes how to decompose a tensor product of two finite dimensional simple modules for a reductive Lie algebra over $\mathbb{C}$.
In particular, I would like a ...
- 2,508
0
votes
1
answer
217
views
On Engel-anticommutative algebras
Let $\mu:\mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{R}^n$ be a alternating bilinear map, i.e. $\mu(X,Y)=-\mu(Y,X)$ (anticommutativity) and so, let $\mathfrak{a}=(\mathbb{R}^n,\mu)$ be a ...
7
votes
0
answers
509
views
Small sum of group elements acting as rank 1 matrix.
I am interested in constructing small (as possible) group $G$ with large dimensional irreducible representation $\rho,V$ such that exist three elements of $g_1,g_2,g_3\in G$ such that for some $c_1,...
- 2,179
7
votes
0
answers
404
views
Reference for the Thick Affine Grassmanian
Let $G$ be a reductive group and $LG$ be the algebraic loop group of $G$; i.e. $LG(k) = G( k((t)) )$. There is a fair amount of literature on the affine Grassmanian $LG(k)/G(k[[t]])$ and its Picard ...
- 3,968
0
votes
0
answers
153
views
Nontrivial copies of SO(r) in SO(n)
If $G=SO(n)=SO(\mathbb R^n)$ and $r\leq n$, it is easy to find a closed subgroup $H\leq G$ that is isomorphic to $SO(r)$, just let $S\subseteq\mathbb R^n$ be an $r$-dimensional subspace and let $H=\{g\...
- 1,987
1
vote
0
answers
253
views
Generalizing groups via the Hall-Witt identity
In studying the integrability problem for Lie algebra representations, I have been led to wonder whether generalizing the notion of group by dropping associativity, while keeping the Hall-Witt ...
- 393
6
votes
0
answers
207
views
The meaning of a "subcomplex" of the Cartan-Eilenberg of a Lie algebra
Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^* $ be the dual vector space. We have the standard Cartan-Eilenberg complex
$(\wedge^{\cdot} \mathfrak{g}^* ,\text{d}_{...
- 9,019
3
votes
1
answer
723
views
A strange logical implication in algebraic geometry
So there's an old theorem of Lang and Weil showing that the Riemann hypothesis for curves over finite fields implies a kind of quasi-riemann hypothesis for surfaces over finite fields.
I am wondering:...
- 13.1k
3
votes
0
answers
308
views
Invertible Hasse-Witt for non-ordinary curves
Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...
- 395
4
votes
0
answers
203
views
The Killing form on quantized enveloping algebras and reduction to the classical case
Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
- 3,637
3
votes
1
answer
361
views
A functor that comes from a morphism in a bigger category
My loose question is like this: what would you say about an equivalence of categories where both are concrete categories, and the equivalence functor is induced from a set-theoretic bijection at the ...
- 7,320
4
votes
0
answers
174
views
Number of submodules in $\wedge^2 V$ and $S^2V$ isomorphic to $\mathfrak{g}$
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let
$\mathfrak{g}\subset\mathfrak{so}(V)$ be an orthogonal
irreducible representation. It can be shown that the number of
$\mathfrak{g}$-...
- 59
6
votes
0
answers
418
views
The Hochschild-Serre spectral sequence relative to an ideal containing the derived subalgebra
Is the Hochschild-Serre spectral sequence $$H_\bullet(\mathfrak g/\mathfrak h,H_\bullet(\mathfrak h,k))\Rightarrow H_\bullet(\mathfrak g,k)$$ for an extension of Lie algebras $$0\to\mathfrak h\to\...
- 47.3k
1
vote
0
answers
871
views
Centre of a Lie algebra
Let $f\in \mathbb{C}[x_1,\dots, x_n]$ be a reduced homogeneous polynomial of degree n.
Let $\mathfrak{g}=\{ \delta \in Der_{\mathbb{C}^n}|\delta(f)\in (f)\mathcal{O}_{\mathbb{C}^n} \text{ and weight}(...
- 671
7
votes
0
answers
207
views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
- 1,199
1
vote
1
answer
415
views
Is this an identity in Lie bialgebras?
Perhaps this will be a trivial question. For this post, everything is over your favorite field of characteristic $0$.
Definitions and notation
Recall that a Lie algebra is a vector space $\mathfrak ...
- 54.6k
1
vote
0
answers
218
views
Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.
I have a few questions on an application of the Weyl character formula.
To start with we work with the $\mathbb{Q}$ version of Hamilton's quaternions and consider the maximal order $\mathfrak{O} = \...
- 562
2
votes
1
answer
195
views
Fixed points of quantised enveloping algebra for affine $\mathfrak{sl}_n$
Consider the automorphism of the algebra $U_q(\widehat{\mathfrak{sl}}_n)$ induced by the obvious diagram automorphism of the extended type A Dynkin diagram. More precisely, if the vertices of the ...
- 159
1
vote
0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
2
votes
0
answers
167
views
Borel (parabolic) subalgebras of twisted affine Lie algebras.
The notion of Verma-type modules for affine Lie algebras is related to the concept of Borel subalgebras. The literature is extensive when the affine algebra is untwisted and all constructions come ...
- 21
13
votes
0
answers
943
views
Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
- 3,637
4
votes
1
answer
221
views
Do permutation modules of solvable groups have self-dual socle in characteristic 2?
I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that ...
- 143
1
vote
0
answers
135
views
multiplicity of a weight in the basic representation of $\hat{sl_2}$
it is known by Weyl character formula that multiplicity of the weight $\wedge_{0}- n.\delta$
in the basic representation $L(\wedge_{0})$ of $\hat{sl_2} $is equal to the number of partitions of $n$.It ...
- 287
6
votes
0
answers
181
views
On an interesting subalgebra of the functions on the cotangent bundle of the flag variety
Setup
Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the ...
- 3,637
4
votes
0
answers
454
views
Norm in the fundamental representations of Lie algebras
Let $\mathfrak{g}$ be a complex simple Lie algebra, $\omega_0$ a dominant weight, $\rho$ a fundamental representation with highest weight $\Lambda$.
Fix some weight $w$ in this representation. Let $\...
- 41
5
votes
1
answer
263
views
Modeling free Lie algebras with matrix algebras
I am approximating some algebraic expressions of operators from a free Lie algebra. It is possible but messy to collect all independent operator objects of a given degree (same as grading?) that ...
- 731
6
votes
0
answers
304
views
How to decide if two surfaces occurring in Springer theory are isomorphic?
In the study of a simple algebraic group (say over $\mathbb{C}$) and related geometry of its flag variety associated with the Springer correspondence, one encounters pairs of surfaces which have some ...
- 52.9k
0
votes
0
answers
155
views
complex reductive Lie groups which are not defined over the real numbers
Hello
Someone knows something about the complex reductive Lie groups (the complexification of a compact Lie group) which are not defined over the real numbers. I would like to know, if is it possible,...
- 1
1
vote
0
answers
33
views
artinian quotients of U(g)
Suppose that $G$ is a connected Lie Group with Lie algebra $\mathfrak{g}$ and $\Gamma$ is a cocompact lattice, and you choose a left-invariant metric on $G/\Gamma$ and let $\Delta$ be the Laplacian. ...
- 2,125
1
vote
0
answers
346
views
Do all finite $W$-superalgebras have 1-dimensional representations?
Premet proved the famous KW-conjecture in modular Lie algebra.
After, Premet introduced the finite $W$-algebra $U(g, e)$.
Also, Premet proposed the conjecture every algebra $U(g, e)$ admits a $1$-...
- 93
1
vote
0
answers
308
views
Why Weyl group associated to Cartan matrix which defines positive definite bilinear form is finite?
In the book by V.Kac "Infinite dimensional Lie algebras" in proposition 4.9 it is argued that Weyl group constructed by Cartan matrix which defines positive-definite metric on Cartan subalgebra is ...
- 11
1
vote
0
answers
66
views
A canonical map Aut$_{\mathsf{Lie}_R}(\mathfrak{n} \rtimes_\pi \mathfrak{g}) \to$ Aut$_{\mathsf{Lie}_R}(\mathfrak{n})$
Let $\mathfrak{n}$, $\mathfrak{g} \in \mathsf{Lie}_R$ be two Lie algebras over a commutative ring $R$, s.t. $\mathfrak{g}$ acts on $\mathfrak{n}$ as a derivation: $\pi:\mathfrak{g} \to \text{Der}(\...
- 19
3
votes
0
answers
423
views
Cohomologies associated to residually torsion-free nilpotent groups
This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.
A group $G$ is ${\it residually \ torsion \ free \ ...
- 373
2
votes
1
answer
341
views
Restriction from $\mathfrak{gl}_{2n}$ to $\mathfrak{sp}_{2n}$
Hi,
I am faced with a finite-dimensional representation $V$ of $\mathfrak{gl}_{2n}$, whose character I know. I know how to use this character to determine the irreducibles for $\mathfrak{gl}_{2n}$ ...
- 6,131