All Questions
2,633 questions
2
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224
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$(\mathfrak{g},K)$-modules and parabolic category $\mathcal{O}$
I am trying to get acquainted with various infinite dimensional representations of Lie groups. So a general reference would be appreciated. Right now I am trying to figure out the following question.
...
- 8,597
12
votes
0
answers
716
views
Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
3
votes
0
answers
234
views
Generators and relations for the enveloping algebra of a unipotent radical
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0 and let $B \subseteq G$ be a Borel subgroup with unipotent radical $U$. Let $P \supseteq B$ be a ...
- 3,637
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
3
votes
0
answers
359
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Does Branching in the Weight Diagram affect an embedding?
All groups here are compact semisimple Lie groups. Out of laziness I will use $B_7$ to mean $Spin(15)$.
Suppose that one has a group $H$ and a subgroup $G$. The embedding determines the decomposition ...
- 5,191
0
votes
1
answer
175
views
An inseparable lift of a regular variety.
Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure ...
- 16.9k
4
votes
1
answer
270
views
Invariant forms
Is there a classification of pairs $(\mathfrak g,V)$, with $\mathfrak g$ a Lie algebra and $V$ a $\mathfrak g$-module, such that $(\Lambda^3V)^{\mathfrak g}\neq0$?
- 47.3k
6
votes
0
answers
524
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Approximating the radical of a Lie algebra by Killing radicals
The following question might be filed under "idle curiosity", but I'm hoping some expert on Lie algebras can answer it.
Background
One of the fundamental structural results about Lie algebras is the ...
- 33.3k
3
votes
0
answers
281
views
What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?
I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the ...
- 4,265
1
vote
0
answers
189
views
Exotic Chains for Group Homology of a Complex Lie Group
Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group
Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
- 18.7k
4
votes
0
answers
184
views
Exotic Chains for Group Cohomology of a Complex Lie Group
Related Question: Exotic Chains for Group Homology of a Complex Lie Group
Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
- 18.7k
4
votes
0
answers
537
views
Working with quadratic Lie algebras
A quadratic Lie algebra is a Lie algebra with an invariant inner product and the main examples are semisimple Lie algebras. This definition then makes sense in any linear symmetric monoidal category. ...
- 9,132
4
votes
0
answers
534
views
Why are Lie Algebras/ Lie Groups so much like crossed modules, and not?
A crossed module consists of a pair of groups $G$ and $H$ with a group homomorphism, $t:H \rightarrow G$, and $\alpha: G \times H \rightarrow H$ that defines an action of $G$ on $H$, $\tilde{\alpha}$: ...
- 5,264
4
votes
1
answer
282
views
Adjoint orbits of small subspaces in Lie algebras
I've been studying isometric quotients (by compact Lie groups) of compact simply connected homogeneous spaces $G/H$ and their inherited curvature. One of the issues that continually arises is the ...
- 6,556
1
vote
0
answers
75
views
Is it possible to compute the Iwahori Decomposition using the Chavalley Commutator Formulas?
Ideally, I would like a constructive, algorithmic proof of this fact. I have convinced myself that it is true, but my "proof" is not pretty. I would like to know if a more attractive or intuitive ...
- 11
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
3
votes
1
answer
121
views
Lower bound on the degree of a product of elements in a hyperalgebra/enveloping algebra
Background:
Fix a linear algebraic group $G$ over an algebraically closed field $k$ of arbitrary characteristic and let $B \subseteq G$ be a Borel subgroup with unipotent radical $N$. Let $\Delta^+$ ...
- 3,637
1
vote
0
answers
269
views
exactness of n homology functor
Let $G$ be a real reductive group, $P=MN$ a parabolic subgroup with its Levi decomposition, let $\mathfrak{n}$ be the nilpotent Lie algebra of $N$. Now given a smooth representation $(\pi,V)$ of $G$, ...
- 2,709
5
votes
0
answers
234
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Modular reduction of exceptional complex reflection groups
I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $...
- 10.7k
3
votes
1
answer
139
views
What's the name of an algebra? is it isomorphic to $w_\infty \times w_\infty$?
Just as we know,
$w_\infty$:=span {${z^\alpha }\partial _z^\beta|\alpha,\beta\in\mathbb{Z}, \beta\geq0$ }.
But, what's the name of the following algebra,
span {$\{{z^{\alpha_1}}{y^{\alpha_2}}\partial ...
- 67
3
votes
0
answers
487
views
Is the Cartan matrix a complete invariant of a Kac-Moody algebra?
In chapter 1 of Kac's book "Infinite dimensional Lie algebras" it is mentioned that two Kac-Moody algebras are isomorphic if and only if their Cartan matrices are isomorphic (i.e. they are the same up ...
- 5,243
6
votes
0
answers
456
views
On periods of algebraic integers modulo rational primes
I run, somewhat indirectly, into the following problem and I have no hints where to look in the literature in search for answers or clues.
Let $K$ be a number field, which we may assume Galois if it ...
- 797
2
votes
0
answers
321
views
Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
- 3,545
0
votes
0
answers
289
views
Good and/or standard notation for the abelianization of a Lie algebra
I'd like to solicit good notations for the abelianization of a Lie algebra $\mathfrak g$. One could write $\mathfrak g/[\mathfrak g,\mathfrak g]$, or even $H_1(\mathfrak g)$ but I'd like something ...
- 4,898
0
votes
0
answers
105
views
approximation in Lie algebras
Let $x_{1}$, $x_{2}$, $x_{3}$ three disctinct closed points of a curve $X$ over an algebraically closed field k.
Let G a connected reductive group and $\mathfrak{g}$ his Lie algebra.
I fix a Borel $...
- 3,472
3
votes
0
answers
264
views
Is it possible to construct category $\mathcal{O}^{\mathfrak{p}}$ with non-standard parabolic subalgebra
The usual definition of the parabolic category $\mathcal{O}^{\mathfrak{p}}$ is the following. We consider Lie algebra $\mathfrak{g}$ of the rank $r$ with the root system $\Delta$ and the set of ...
- 214
4
votes
0
answers
287
views
Characters of Kac-Moody algebra from orbifold
In a Wess–Zumino–Witten model on some Lie group G, the character of a particular integrable representation is the same as the specialized character from the corresponding Kac-Moody algebra. Suppose ...
- 248
3
votes
1
answer
298
views
In what way do exact sequences of Lie ideals integrate to the category of groups?
Please excuse, very naive question:
Suppose $g$ is a topological Lie algebra over Q and $G$ = $exp(g)$ the associated group
(take free group on formal symbols $exp(X)$, X $\in$ $G$, and impose all ...
- 934
2
votes
1
answer
197
views
Polytopes related to the conjugation action of a Lie group on multiple copies of itself?
Let G be a finite dimensional real Lie group. As I understand it, the quotient space of G acting on itself by conjugation is a well studied polytope which can be identified with the fundamental alcove ...
- 27.5k
1
vote
0
answers
124
views
Approximated characters
Is it possible to construct series of groups $G_i, |G_i|\mapsto \infty$ and functions $f_i:G_i\mapsto$ {$ 1,0,-1$} such that $f_i(1)=0$, $f_i(g) \in ${$-1,1$} for $g\neq 1$ such that dimension of $C[...
- 2,179
0
votes
0
answers
23
views
Existence of a subregular element with abelian centralizer in a quadratic Lie algebra
All Lie algebras here will be finite dimensionnal complex Lie algebra.
We say that such an Lie algebra $\mathfrak{g}$ is quadratic if there exist a skew-symetric, non-degenerate bilinear form or ...
- 188
0
votes
0
answers
61
views
Representation and Laplacian on the Heisenberg group
Let $\pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H=\mathbb C\times \mathbb R$. For $\varphi\in L^2(\mathbb R)$, we have
$$\pi_{\lambda} (z,t)\varphi(\xi)=e^{i\...
- 650
0
votes
0
answers
90
views
Some details about Kirillov-Kostant Poisson bracket
Let $G$ be a finite dimensional Lie group with Lie algebra $\mathfrak{g}$. The Kirillov-Kostant Poisson bracket on $\mathfrak{g}^*$ is defined as $$\{\cdot ,\cdot \} :C^{\infty}(\mathfrak{g}^*)\times ...
- 287