All Questions
283 questions
9
votes
1
answer
737
views
Del Pezzo surfaces and Picard-Lefschetz theory
Let $X$ be a smooth compact del Pezzo surface. For instance, one can consider the most classical case of a cubic surface. It is well known that the Picard lattice of $X$ is related to a root system (...
- 4,316
9
votes
1
answer
338
views
Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$?
Let $n\in\mathbb N$. Let $k$ be a commutative ring in which $1,2,3,\ldots$ are invertible. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., $...
- 33.8k
8
votes
1
answer
774
views
A variant on characteristic $p$ de Rham cohomology
I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction.
Let $k$ be a perfect field ...
- 156k
8
votes
2
answers
373
views
The minimum codimension of Lie subalgebra of $\chi^{\infty}(M)$
Assume that $M$ is an arbitrary manifold.
Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?
If not, what is a counter ...
- 366
8
votes
2
answers
2k
views
Lie algebras to classify Lie groups
What does the classification of Complex Semi-simple Lie algebras buy us in terms of classifying Lie groups? Certainly it classifies complex semi-simple lie groups but can we get any better? I know we ...
- 81
8
votes
0
answers
230
views
Integral Milnor-Moore theorem
Given a field K of char. zero the theorem of Milnor Moore
states that taking the enveloping hopf algebra defines an embedding
$\mathcal{U} $ from Lie algebras over K into hopf algebras over K.
Taking ...
- 1,361
8
votes
1
answer
452
views
What is the relation between Coxeter transformations of generalized Cartan matrices and Coxeter transformations of finite-dimensional algebras?
Note: This question now has a sister :-)
The Coxeter transformation of a generalized Cartan matrix:
In the paper
The spectral radius of the Coxeter transformations for a generalized Cartan matrix, ...
- 253
8
votes
1
answer
375
views
Motivation behind Panyushev's "constant-averages-along-orbits" conjecture
In his article "On orbits of antichains of positive roots" (European Journal of Combinatorics 30 (2009) 586–594, Dmitri Panyushev discusses an interesting self-map on the set of antichains of a finite ...
- 19.7k
8
votes
3
answers
1k
views
Simple Subalgebras of Simple Lie Algebras
Given a complex simple Lie algebra $\mathfrak{g}$ of rank $n\in\mathbb{N}$ with $n$ sufficiently large (say $n\ge10$), is there a way to determine whether $\mathfrak{g}$ contains a simple subalgebra ...
user117370
8
votes
2
answers
448
views
How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?
In Chern-Simons theory, one has modular fusion categories that are labelled by a Lie algebra and a "level", e.g. $SU(2)_2$ ("$SU(2)$ level $2$").
Physically this modular fusion category describes the ...
- 3,001
8
votes
1
answer
1k
views
Commutativity and Kostant sections
Let $(e, h, f)$ be an $\operatorname{SL}_2$-triple in $\mathfrak g$. My understanding is that $e + C_{\mathfrak g}(f)$ is called a Kostant section only in case $e$ is regular; but I don't impose this ...
- 12.9k
8
votes
3
answers
1k
views
Failure of Jacobson-Morozov in positive characteristics
The Jacobson-Morozov theorem that any nilpotent element $e$ in the Lie algebra of a simple algebraic group $G$ can be embedded in an $\mathfrak{sl}_2$-triple, has a restriction (in terms of the ...
- 115
8
votes
1
answer
406
views
Characterizations of Jacobson-Morozov parabolics associated to a nilpotent
Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a Jacobson-Morozov parabolic ...
- 1,423
8
votes
1
answer
650
views
Harish-Chandra isomorphism for compact symmetric spaces
I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
Below ...
- 21.8k
8
votes
1
answer
562
views
The parity of the full automorphism group order of finite non-abelian groups of prime exponent
Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?
- 3,738
8
votes
1
answer
1k
views
Maximal Submodule of a Verma Module
Let $\mathfrak{h}$ be a Cartan subalgebra of a $\mathbb{C}$-semi simple Lie algebra $\mathfrak{g}$. Given $\lambda \in \mathfrak{h}^*$, $M(\lambda)$ the Verma module of highest weight $\lambda$ and $N(...
- 83
8
votes
2
answers
1k
views
Lefschetz on étale fundamental group for quasi-projective varieties
If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, ...
- 483
8
votes
3
answers
721
views
References about Hasse diagrams of root systems
This is to ask about references of Hasse diagrams of irreducible root systems. I found here and there nice pictures of root systems of type $E$. I would like to ask for Hasse diagrams of classical ...
- 307
7
votes
0
answers
142
views
When is an algebra derived indecomposable?
Call a finite dimensional (acyclic) quiver $K$-algebra A derived indecomposable in case $A$ is not derived equivalent to an algebra of the form $B \otimes_K C$.
For example when the number of simples ...
- 26.5k
7
votes
0
answers
510
views
Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space
Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
- 1,942
7
votes
4
answers
1k
views
Lie algebra admitting some hyperbolic automorphism is nilpotent
Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $\mathbb{R}$ and $\phi:\mathfrak{g}\to\mathfrak{g}$ be a Lie algebra automorphism.
Viewing $\mathfrak{g}$ as a linear space and $\phi$ a ...
- 2,244
7
votes
2
answers
421
views
Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and $\phi_{2}:\mathfrak{sl}_2(\mathbb{C})\rightarrow\...
- 4,920
7
votes
1
answer
574
views
Does Aut(G) → Out(G) always split for a compact, connected Lie group G?
The outer automorphism group of a topological group $G$ is constructed by the short exact sequence
$$
1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \...
- 473
7
votes
2
answers
670
views
Lie algebras with unique invariant scalar product
Every 1-dimensional or simple complex Lie algebra admits an invariant, symmetric and non-degenerate bilinear form. This form is unique up to multiplication by a nonzero constant (which in Yang-Mills ...
- 427
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
2
answers
1k
views
Dimension of the nilpotent centralizer of a nilpotent matrix
Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the $n\...
- 768
7
votes
1
answer
1k
views
When is the Ad (Adjoint Representation) Morphism a Closed Map
Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...
- 1,161
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
- 4,450
7
votes
1
answer
1k
views
Explicit convergence of Baker-Campbell-Hausdorff
Let g be a finite dimensional simple Lie algebra over C. The Baker-Campbell-Hausdorff series defines a (multivariable) analytic function from a neighborhood of 0 in g \times g \to g. What is the ...
- 71
7
votes
2
answers
418
views
About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$
Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) $...
- 9,019
7
votes
2
answers
499
views
Submanifolds of Lie groups with abelian normal bundle
Let $M$ be a submanifold of a symmetric space $Q$. The normal bundle $NM$ is called abelian if $\exp(N_{p}M)$ is contained in some totally geodesic and flat submanifold of $Q$ for all $p \in M$; see ...
- 985
6
votes
1
answer
675
views
Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups
I've recently read the following line in an interesting paper:
It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...
- 3,409
6
votes
0
answers
340
views
Asymptotically nilpotent Lie sets of matrices
A matrix $A\in\textbf{Mat}_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}_{k\to\infty}A^k(v) = 0$.
Question 1. Assume that $\mathcal{A}$ is the subset of $\textbf{...
- 291
6
votes
2
answers
514
views
Constructing real forms of the Tits-Freudenthal magic square for (Rosenfeld) projective planes
If $\mathbb{K},\mathbb{L} \in \{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$ then the Rosenfeld projective ("elliptic"?) plane $\mathbb{P}^2(\mathbb{K}\otimes\mathbb{L})$ is "the" compact Riemannian ...
- 32.5k
6
votes
1
answer
372
views
Classification of quasi-lisse vertex algebras
Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in Quasi-lisse vertex algebras and modular linear differential equations . They satisfy the property that the normalized character ...
user35360
6
votes
2
answers
1k
views
Non-faithful irreducible representations of simple Lie groups
For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group.
...
- 835
6
votes
3
answers
2k
views
Which linear combinations of simple roots are roots
Let $\Delta$ be the root system of a complex simple Lie algebra, $\Delta^+$ be positive roots and $\Pi$ be simple roots. We view $\Pi$ as nodes of the Dynkin diagram.
Then for any two simple roots $\...
- 1,804
6
votes
1
answer
572
views
A vector space associated with a vector field on a symplectic manifold
$\DeclareMathOperator\Div{Div}$Edit: The correct formulation of the vector space $S(X)$ which is defined in this question is the following:$$S(X)=\{Y\in \chi^{\infty}(M)\mid X.\omega(X,Y)=(1/...
- 366
6
votes
3
answers
1k
views
Simple question in the representation of SL(2,C)
Let $V$ the standard two dimensional representation of SL(2,C). The Fulton's book in representation theory say in pag 156 that $Sym^3(Sym^2V)=Sym^6(V) \oplus Sym^2(V)$.
In the excercises 11.23, the ...
- 943
6
votes
2
answers
376
views
$(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$?
Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental methods, that if $\nabla$ is an integrable connection on a vector bundle $L$ on $A$ ...
user61522
6
votes
1
answer
317
views
Which Lie groups are covers of matrix groups?
I would like to ask a variation on a question (not yet answered) I previously asked on math.SE, namely:
Which Lie groups are covers of matrix Lie groups? That is, which Lie groups $G$ admit discrete ...
- 3,115
6
votes
3
answers
773
views
Existence of a weight of a representation in the fundamental Weyl chamber
Let $\mathfrak g$ be a complex simple Lie algebra.
Fix a Cartan subalgebra $\mathfrak h$ of $\mathfrak g$, let $\Delta$ denote the corresponding root system.
Pick a partial order on $\mathfrak h$, ...
- 2,446
6
votes
1
answer
1k
views
Generic Smoothness Type of Results in Positive Characteristic
Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth.
We know that ...
- 313
6
votes
2
answers
891
views
Why can the Dolbeault Operators be Realised as Lie Algebra Actions
I've been looking at an example in the non-commutative geometry literature and I'm having trouble figuring out what the classical motivation is. I'll just describe the classical case here: Recall that ...
- 3,409
5
votes
2
answers
1k
views
Lie Groups and Lie Algebras
What is the exact relationship between Lie groups and Lie algebras? I know it's not bijective because all commutative Lie groups have isomorphic Lie algebras.
- 3,409
5
votes
2
answers
671
views
How to find faces of polytope defined by a Weyl orbit
A few days ago I asked the following question at MSE and received no answer. I thought I would try here.
Let $\xi$ be an integral dominant weight of an irreducible root system $\Delta$, and let $\...
- 474
5
votes
0
answers
460
views
Chern-Simons theory with non-compact gauge groups G
This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general ...
- 10.5k
5
votes
2
answers
1k
views
Heisenberg subalgebras of affine Lie algebras
It seems to be "well-known" that (infinite-dimensional) Heisenberg subalgebras of an affine Lie algebra $\hat{\mathfrak{g}}$ corresponding to a finite-dimensional simple Lie algebra $\mathfrak{g}$ of ...
- 618
5
votes
2
answers
1k
views
Lie's theorem in characteristic $p$
Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...
- 3,741
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53