All Questions
2,633 questions
3
votes
0
answers
64
views
p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,986
2
votes
1
answer
145
views
Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$
Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring.
There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
- 542
3
votes
1
answer
306
views
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$ as $\operatorname{GL}(n,\mathbb{C})$-modules
Consider the natural representations of $\operatorname{GL}(n,\mathbb{C})$ in the spaces
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$.
Is it known how to ...
- 21.8k
9
votes
1
answer
161
views
Eigenfunctions of the Laplace–Beltrami operator on the coadjoint orbit of $\mathfrak{su}(n)$
$\DeclareMathOperator\SU{SU}$For $\mathfrak{su}(2,\mathbb{C})$, the generic coadjoint orbit is $\mathbb{S}^2$, and the Laplace–Beltrami operator on it is given by
$$
\Delta \equiv \frac{1}{\sin\theta} ...
- 195
8
votes
0
answers
203
views
Logarithm of a $p$-group in $\mathrm{GL}_n(p)$
$\def\GL{\operatorname{GL}}\def\ZZ{\mathbb{Z}}\def\FF{\mathbb{F}}\def\Id{\mathrm{Id}}\def\fu{\mathfrak{u}}$Let $p$ be prime, let $n<p$, let $U_n(\FF_p)$ be the group of $n \times n$ upper ...
- 156k
3
votes
1
answer
111
views
Lie subalgebra annihilated by all derivations
Let $k$ be a field and $\mathfrak{g}$ a Lie algebra over $k$. Put $K(\mathfrak{g}) = \bigcap_{f\in\mathrm{Der}(\mathfrak{g})} \mathrm{Ker}(f)$, which is a Lie subalgebra of $\mathfrak{g}$.
Question. ...
- 985
2
votes
0
answers
128
views
A property of matrices formed by pairing of roots and coroots
Let $A$ be an $n\times n$ integral matrix, define its level $l(A)$ as
$$l(A) := \begin{cases}0 &\det A = 0 \\ \text{smallest integer } N \text{ such that } NA^{-1} \text{ is integral} &\det A \...
- 528
5
votes
2
answers
301
views
Non-semisimple Lie groups and Higgs bundles
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $X$ be a compact Riemann surface. Let $G$ be a real reductive Lie group, $H$ be a maximal compact subgroup of $G$ ...
- 151
6
votes
0
answers
103
views
Computer program for free restricted Lie polynomial
I am conducting numerical experiments involving the Gröbner–Shirshov Basis for restricted Lie algebras. At each step of the computation, I need to work with restricted Lie polynomials. Specifically, I ...
- 1,013
4
votes
0
answers
239
views
Jacobian of exponential map
I am playing around with the coarea formula and came across the problem of finding the Jacobian of the exponential map.
Let $G$ be a compact, semisimple Lie group with associated Lie algebra $\...
- 133
13
votes
0
answers
333
views
Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
- 357
7
votes
1
answer
132
views
Classification of modules all whose weight spaces are $1$-dimensional
In type $A$, the simple modules all of whose weight spaces are $1$-dimensional are the $L(n\varpi_1)$ and $L(\varpi_k)$. This can be seen from the fact that dimensions of weight spaces are given by ...
- 820
5
votes
0
answers
234
views
Avoiding Cartan subalgebra in a Lie algebra
Let $G$ be a simple complex algebraic group acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation.
What is the largest integer $d$ such that every subspace $U \subseteq \mathfrak{g}$ ...
- 269
3
votes
1
answer
113
views
Generalization of a result of Kostant related to Gauss decomposition and Toda lattices
I found myself needing a generalization of a result of Kostant in his famous paper
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math, Volume 34, 1979, ...
- 173
0
votes
0
answers
190
views
About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
15
votes
6
answers
671
views
Why, conceptually, does the torus normalizer in $G_2$ split?
Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension
$$ 1 \to T \to N \to W \to ...
- 835
4
votes
0
answers
106
views
Relationship between characteristic polynomials of a matrix and its adjoint representation
Let $A \in \mathrm{M}_n(F)$ be a matrix over a field $F$. Consider its adjoint representation $\mathrm{ad}_A \in \mathrm{End}(\mathrm{M}_n(F))$, defined by
$$
\mathrm{ad}_A(X) = [A, X] = AX - XA.
$$
I ...
- 269
1
vote
0
answers
75
views
Contragradient Module for the Kac-Moody Vertex Operator Algebra
For a vertex operator algebra $(V,Y,\left|0\right>)$ and a $\mathbb{Z}$-graded module $(M,Y_M)$, one can form the contragradient module $(M^{\lor},Y_{M^{\lor}})$ with underlying $\mathbb{Z}$-graded ...
- 834
15
votes
1
answer
518
views
Pairs of matrices for which traces of powers are independent of the order
Let $A,B$ be $n\times n$ matrices over ${\mathbb C}$ such that, for all $m,k$ and all partitions $(i_1,\ldots ,i_r)$ of $m$ and $(j_1,\ldots ,j_r)$ of $k$ (perhaps with some zero parts),
$${\rm tr}\, (...
- 1,336
4
votes
1
answer
232
views
What is known about string functions for $\widehat{\mathfrak{sl}_2}$?
Let $\Lambda$ be a dominant integral weight for $\widehat{\mathfrak{sl}_2}$. The string function associated to a maximal weight $\lambda$ of $L(\Lambda)$ is the series $$ a^{\Lambda}_\lambda = \sum_{k ...
- 820
12
votes
3
answers
802
views
The orders of the exceptional Weyl groups
Who first calculated the orders of the Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$? How were these orders calculated?
- 467
5
votes
0
answers
50
views
Surjectivity of irreducible representation of Chevalley algebras
Let $A$ be an associative algebra over an algebraically closed field $k$. The following theorem holds (see, for instance, Theorem $2.5$ on page $24$ in Introduction to Representation Theory by Pavel ...
6
votes
2
answers
506
views
Group of diffeomorphisms and its tangent space i.e. its Lie algebra
So I feel like there are many questions and also many sources on what I am asking, but I still don't understand what I think is a very basic thing in my head:
It is known, that for a Lie group $G$ (...
- 63
4
votes
0
answers
105
views
A question about decomposing root system $A_{n}$
Denote $\Phi(n)$ as the root system of Lie algebra $\mathfrak{g}$ of type $A_{n}$. Call a disjoint union $\Phi(n) = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if each $\Phi_{k}$ is ...
- 93
0
votes
0
answers
60
views
Bounding number of commutators of a certain type
Say I have a vector space $V$ over $\mathbb{C}$, generated by $2n$-letters, $(x_1,...,x_n,y_1,...,y_n)$. Let $C_d$ denote the commutators on $V$ of length $d$, and let $(B_1,...,B_r)$ denote a basis ...
- 2,907
11
votes
0
answers
183
views
Are algebras with rational structures dense in varieties of real Lie nilpotent algebras?
One says that a real nilpotent Lie algebra has $\mathbb Q$-structure if it has a basis with rational structure constants. It is well known that there are nilpotent Lie algebras without $\mathbb Q$-...
- 723
3
votes
1
answer
110
views
Roots of polynomial $\sum_{\sigma \in W} x^{l(\sigma)}$
Let $W$ be Weyl group of a root system $\Phi$ (of finite dimensional simple Lie algebra). For $\sigma\in W$, $l(\sigma)$ be the its length. Consider the following polynomial
$$P_\Phi(x) = \sum_{\sigma ...
- 528
6
votes
2
answers
794
views
Tensor algebra and universal enveloping algebra
Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
- 673
1
vote
1
answer
64
views
What is $\left[ \begin{array}{c} K_i;0\\ \ell\\ \end{array} \right] _{\varepsilon _i}$ in the restricted specialization in QUE algebras?
I have a question about the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. It comes from section $9.3$ on page $300$ of this book
In Section 9.1, the authors define ...
- 137
8
votes
0
answers
145
views
Asymptotics of generalized exponents of highest weight modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $H^k$ be the space of homogeneous degree $k$ harmonic polynomials in $\mathrm{Sym}(\mathfrak{g}^*)$ and $H\subset\mathrm{Sym}(\mathfrak{g}^*)...
- 413
3
votes
1
answer
140
views
Is it true that all abelian p-nilpotent restricted Lie algebras are mirrors of finite abelian p-groups?
Let $k$ be a perfect field of characteristic $p$. A $p$-nilpotent restricted Lie algebra $\mathfrak{g}$ is a Lie algebra with $[p]$-restriction mapping such that $\mathfrak{g}^{[p]^n} = 0$ for some $n ...
- 542
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
3
votes
1
answer
160
views
Embedding flag manifolds of real semisimple lie group
I want to know given a connected (maybe we can assume it to be simply connected or linear) real semisimple lie group $G$ and one of its maximal parabolic group $P$, how can we embed the flag variety $...
- 175
3
votes
1
answer
231
views
Does a representation of the universal cover of a Lie group induce a projective representation of the group itself?
Suppose that $G$ is a connected Lie group, $\tilde{G}$ its universal cover, $p:\tilde{G}\to G$ the covering map. Does a representation $\rho$ of $\tilde{G}$ on a finite-dimensional vector space $V$ ...
- 3,115
2
votes
0
answers
48
views
Homomorphism to a finite p-group/Lie ring Q: estimate on |Q|
Let $L$ be a finitely generated, torsion-free nilpotent group. Furthermore, assume it is also a Lie ring, i.e. a Lie algebra but over $\mathbb{Z}$. The correspondance between $L$ as a group and $L$ as ...
- 61
1
vote
0
answers
60
views
Reference for Gröbner-Shirshov algorithm in free restricted Lie algebras
I am searching for a reference on the Gröbner-Shirshov algorithm specifically for free restricted Lie algebras. I have already consulted the textbook by Bokut et al (Gröbner–Shirshov Bases Normal ...
- 1,013
4
votes
0
answers
184
views
Does a dual basis for $U_h(\mathfrak{sl}_2(\mathbb{C}))$ exist?
Let $\mathcal{F}_h(\operatorname{SL}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $a, b, c, d$ subject to the following relations:
\begin{align*}
& ac = e^{-h}ca, \quad bd = e^{-h}...
- 255
4
votes
1
answer
101
views
K-types of a representation of the minimal Gelfand-Kirillov dimension
Let $G$ be a noncompact real simple Lie group not of Hermitian type, and $\mathfrak{g}_0$ its Lie algebra. Fix a maximal compact subgroup $K$ in $G$ with its Lie algebra $\mathfrak{k}_0$. Write $\...
- 951
3
votes
1
answer
129
views
Questions about the quotient of extended Weyl group and the isomorphism of extended Weyl group
When I am reading a paper An Algebraic Characterization of the Affine Canonical Basis by Beck, Chari, and Pressley, and I have some questions about some notations.
In the paper, we assume that $\...
- 137
1
vote
0
answers
281
views
A question about decomposition of irreducible root system
Fix an irreducible root system $\Phi$ with rank $r$ and a root base $\Delta$ (we only care type ADE). Call a disjoint union $\Phi = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if ...
- 93
3
votes
1
answer
162
views
Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
1
vote
0
answers
46
views
The difference between two description of affine Weyl groups
I have a question about the difference between two description of affine Weyl groups.
Let me write two descriptions of affine Weyl groups:
Let $\mathfrak{g}=\mathfrak{g}(A)$ be affine Lie algebras ...
- 137
0
votes
0
answers
111
views
Irreducible representations of $\mathfrak{sl}(m|n)$
It is well known that the irreducible representations of the Lie algebra $\mathfrak{sl}(n)$ are symmetric powers of a vector space of dimension $n$. This can be viewed for instance using Young ...
- 558
7
votes
1
answer
262
views
A name for the Weyl group of $\frak{so_{2n}}$
For the $A$-series the Weyl group is the symmetric group $S_n$. For the $B$ and $C$ series the Weyl group is the hyperoctahedral group $\mathbb Z_2 \wr S_n$.
A) Does the $D$-series Weyl group $S_n \...
- 467
1
vote
0
answers
52
views
How large can the normalizer of $\mathrm{Ad}(G)$ in $\mathrm{GL}(\mathfrak{g})$ be?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ad{Ad}$Let $G$ be a real Lie group with Lie algebra $\mathfrak g$ (say reductive/semisimple if it makes the question easier).
I am interested in ...
- 1,942
2
votes
0
answers
82
views
Explicit $K$-basis of a Lie subalgebra
$\newcommand{\Kbar}{{\overline K}}
\newcommand{\Q}{{\mathbb Q}}
$I consider Example 5.7 of Sansuc's paper Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres of ...
- 14.2k
5
votes
0
answers
123
views
Algebraic groups and formal group laws in characteristic p
In characteristic zero, there is a well-known equivalence between Lie groups, formal group laws and Lie algebras.
Let $p$ be a prime. The equivalence between Lie groups and Lie algebras has an ...
- 413
6
votes
4
answers
415
views
Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group).
By the Tannakian formalism, $G(k)$ can be ...
- 3,446
20
votes
3
answers
2k
views
Where do root systems arise in mathematics?
One often hears that root systems are ubiquitous in mathematics and physics. The most obvious occurrence of root systems is in the classification of complex simple Lie algebras. Where else do they ...