All Questions
2,633 questions
22
votes
1
answer
2k
views
Reference request: a tale of two mathematicians
I've heard tell the following anecdote involving Pierre Gabriel and Jacques Tit at least twice in a lapse of four years or so:
When P. Gabriel presented the theorem in a conference [sometime around ...
- 8,842
7
votes
0
answers
221
views
Strange formula for the dimension of a certain space of noncommutative polynomials
Consider a vector space $V_r(n)$ spanned by (noncommutative) monomials in variables $x_1,\ldots,x_r$
$$
x_{1}^{n_1}x_{2}^{n_2}\ldots x_{r}^{n_r}
$$
of total degree $n.$ Inside this space consider a ...
- 4,316
4
votes
1
answer
977
views
parabolic subalgebras and Cartan decomposition
Let $\mathfrak{g}$ be a complex simple Lie algebra and $\mathfrak{k}$ its complex subalgebra such that $(\mathfrak{g},\mathfrak{k})$ is a Hermitian symmetric pair; $\mathfrak{g}= \mathfrak{k}\oplus\...
- 11.2k
8
votes
2
answers
436
views
Can we count the number of simple modules for a reduced enveloping algebra?
Let $G$ be a reductive algebraic group over a field of positive characteristic $p$, which I'll assume to be very good for $G$. Then the Lie algebra $\mathfrak{g}$ is restricted and each simple $\...
- 13k
5
votes
1
answer
277
views
Infinite dimensional Lie algebras with trivial homology
The basic question is:
Does vanishing of homology with trivial coefficients imply triviality of an infinite-dimensional Lie algebra?
My question is motivated by acylic groups in group theory. In ...
- 63.9k
1
vote
1
answer
325
views
Nakajima quiver varieties for ADE quiver with one dimensional framing
Let $Q$ be a quiver of type $ADE$, $I$ is the set of vertices of $Q$. Let $\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$ be a Nakajima quiver variety for such quiver (here ${\mathbf{v}}=(v_i)_{i \in I}$ is ...
- 3,061
3
votes
0
answers
118
views
Twisting an L_{\infty} module quasi-isomorphism with a sufficiently small Maurer-Cartan element
I was wondering if someone could help me understand this result, or point me towards a reference. Suppose that $M$ and $N$ are $L_{\infty}$ modules over a dgla $L$. Suppose that $\phi: M \rightarrow N$...
- 295
2
votes
0
answers
63
views
What is the "correct" notion of a perfect graded commutative algebra?
My question is rather simplistic. While trying to dualize some statements about rational homotopy algebra of a space I got stuck with the following problem.
We have a notion of a perfect Lie algebra ...
- 1,374
2
votes
0
answers
287
views
Frobenius endomorphism is not flat
I am actually going through "Twenty four hours of local cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve:
Find a ...
- 63.9k
4
votes
0
answers
213
views
The Weyl group of Kac-Moody algebra and Coxeter group
Let $\mathfrak{g}$ be a Kac-Moody algebra, $W$ be its Weyl group, generate by fundmental reflections {$r_1,...,r_n$}. For $i,j$, we have relation $(r_ir_j)^{m_{ij}}=1$, $1\le i,j\le n$. Let $W^{'}$ be ...
- 63.9k
2
votes
0
answers
213
views
Positive roots and the longest element of the Weyl group
Take $\frak{g}$ a complex semisimple Lie algebra and its Weyl group $W$. Is it true that the number of positive roots of $\frak{g}$ is equal to the length of the longest element of $W$?
- 393
1
vote
1
answer
349
views
General strategy of error bound of matrix exponential
I want to ask General strategy of the error bound of the matrix exponential.
For example, suppose, $A, B$ are finite dimension $n \times n$ matrices with complex coefficients. Using Baker–Campbell–...
- 149k
5
votes
0
answers
244
views
Which symplectic manifolds are coadjoint orbits of finite-dimensional $G$ on $\mathfrak{g}^*$ with the Kostant–Kirillov–Souriau Poisson structure?
Given some finite-dimensional symplectic manifold $(M,\omega)$, can we answer whether or not it arises as the coadjoint orbit of some finite-dimensional $G$ acting on the dual $\mathfrak{g}^*$ to its ...
- 13k
4
votes
1
answer
150
views
A condition on minimal restricted subalgebras of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. Assume that the following condition holds:
For every restricted ideal $I$ of $L$, the minimal restricted subalgebras ...
- 5,878
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
2
answers
745
views
Is there any example of a Lie algebra which is not a derivation algebra?
I'm just studying Lie algebras. If $A$ is a $k$-algebra (not necessarily Lie or associative, just a bilinear law), it is straightforward to check that any derivation algebra of $A$ is a Lie algebra. I ...
- 5,878
7
votes
2
answers
777
views
Why is the generalized flag variety a “variety”?
In several places (for example, Chriss & Ginzburg’s book “Representation Theory and Complex Geometry”), the author says that the set $X$ of Borel subalgebras of a semi-simple Lie algebra $\...
- 15.5k
4
votes
1
answer
234
views
Reductive Levi decomposition of a parabolic subalgebra
Let $\mathbb{K}$ be a field (not assumed to be algebraically closed, but we can assume characteristic 0 if necessary), and let $\mathfrak{g}$ be a semisimple Lie algebra.
A Lie subalgebra $\mathfrak{p}...
- 954
6
votes
0
answers
105
views
Explicit formula for star product on the symmetric algebra of a Lie algebra via standard ordering
There is a well known vector space isomorphism $\phi:\mathcal{S}(\frak{g})\rightarrow U(\frak{g})$ given by the symmetrization (or Weyl ordering), i.e.
$$ \phi(t_{i_1}\dots t_{i_k})=\frac{1}{k!}\sum_{\...
- 63.9k
2
votes
0
answers
78
views
Examples of curvature-adapted subgroups of semi-Riemannian groups
Let $G$ be a semi-Riemannian group, i.e., a Lie group equipped with a bi-invariant semi-Riemannian metric. I am looking for examples of Lie subgroups that are curvature adapted to $G$.
First, allow me ...
- 985
16
votes
2
answers
1k
views
Is the tangent space functor from PD formal groups to Lie algebras an equivalence?
The previous version of this question was rather badly broken, and I hope this version makes some sense.
There have been a few questions on MathOverflow about how much representation-theoretic ...
- 35.5k
1
vote
3
answers
840
views
On the derivative of the exponential of adjoint action on a Lie algebra
I am studying the Drinfel'd-Sokolov hierarchies. And here is what confuses me. Let $\mathfrak{g}$ be a (simple) finite-dimensional Lie algebra, let $u(x), v(x) \in C^\infty(\mathbb{R},\mathfrak{g})$. ...
- 63.9k
2
votes
0
answers
115
views
Lie derivations of algebra of smooth functions in a symplectic manifold
Let $(M,\omega)$ be a finite-dimensional symplectic manifold. The algebra $C^\infty(M)$ of smooth functions is a Poisson algebra. Derivations $D : C^\infty(M) \to C^\infty(M)$ of the Poisson ...
- 33.3k
24
votes
4
answers
7k
views
Formal geometry
[Edit (June 20, 2010): I posted an answer to this which summarizes one that I received verbally a few weeks after posting this question. I hope it is useful to someone.]
I am presently seeking ...
- 63.9k
2
votes
0
answers
95
views
Local decomposition of semisimple Lie groups at the identity into the product of centralizer, unstable and stable horospherical subgroups
First let us look at an example. Let $G=\text{SL}(d,\mathbb R)$ and $A$ be its subgroup consisting of diagonal matrices with positive entries.
Let $a\in A$ be an element. We define the stable ...
- 63.9k
4
votes
0
answers
231
views
How big are small inverse powers of 2 mod powers of 3?
Let $a \bmod b$ take value as an integer in $[0, b)$. For any $T \ge 1$, for what $R \in [0, 3^n)$ is
$$\min \{2^{-t}\bmod 3^n: t =1, \dotsc, T\} := \min A_T > R?$$
When $T$ is fixed as $n$ ...
- 459
5
votes
0
answers
139
views
Liouville property in the Bost theorem on foliations
Let $X$ be a smooth algebraic variety over a number field $K$, and let $\mathcal{F}$ be an involutive coherent subsheaf of the tangent bundle. After a pullback to $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
- 485
0
votes
1
answer
187
views
Matrix representations of Lie groups of type $B_n$
For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices:
\begin{align}
\left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\...
- 13k
2
votes
0
answers
561
views
What is the precise definition of connect semisimple Lie groups "without compact factors" in the literature?
I frequently see in the literature (of homogeneous dynamics, in particular) of the notion "semisimple Lie groups without compact factors" without seeing its precise definition.
I have three (...
- 1,565
8
votes
2
answers
449
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 ...
- 2,821
3
votes
1
answer
485
views
Relationship between the representation theory of $\operatorname{Spin}(n)$ and $\operatorname{SO}(n)$
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$What is the exact relationship between the finite dimensional representations of the group $\SO(n)$ and its covering group $\Spin(n)$? More ...
- 156k
3
votes
1
answer
176
views
Example of a supersolvable Lie group/algebra whose nilradical does not have a complement
What is an example of a real solvable simply-connected Lie group $G$ whose nilradical does not have a complement (that is, $G$ is not a semidirect product of the nilradical and another subgroup)? Is ...
- 13k
2
votes
1
answer
1k
views
Derivative of adjoint action of exponential map
Let $X(t)$ be a $C^1$ (continuously differentiable) path in the Lie algebra (actually I just need finite-dimensional matrices). It is well-known (from Wikipedia page of Derivative of the exponential ...
- 487
2
votes
1
answer
206
views
Extending representations of Lie subalgebras to the whole Lie algebra
Let $\frak{g}$ be a complex simple Lie algebra and let $\frak{k}$ be a non-zero semisimple Lie subalgebra of $\frak{g}$. Is it possible to realize every simple $\frak{k}$-module $W$ as a $\frak{k}$-...
- 435
2
votes
1
answer
191
views
Normalizer of SU$(2)$ in SU$(6)$
Consider the $\mathfrak{su}(2)$ subalgebra of $\mathfrak{su}(6)$ embedded as
$$\mathfrak{su(2)}=\text{Span}\{\mathbb{1}_3 \times \sigma^i\}, \quad i=1,2,3$$
with $\sigma^i$ the Pauli matrices and $\...
- 11.2k
6
votes
0
answers
173
views
Orlik-Solomon algebra and hyperplane complements in positive characteristic
Let $k$ be an algebraically closed field of characteristic $p\geq 0$, $\underline H:=\{H_1,\dots, H_m\}$ a set of hyperplanes in $\mathbb A_k^n$ and $X:=\mathbb A^n-(\bigcup H_i)$.
Given a ring $R$ ...
- 728
6
votes
1
answer
243
views
Do weight vectors live between the highest and lowest weights?
For a simple complex Lie algebra $\frak{g}$, let $V$ be an irreducible $\frak{g}$-module. Is it true that the weights of the non-zero weight vectors in $V$ are less than the highest weight vector and ...
- 13k
11
votes
4
answers
3k
views
What does ramification have to do with separability?
Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
- 411
3
votes
1
answer
69
views
Action of Coxeter element on mod $2$ root lattice is semisimple
Let $\Lambda$ be a simply laced root lattice and $w$ a Coxeter element of the Weyl group of $\Lambda$.
Question: Is it true that the action of $w$ on the $\mathbb{F}_2$-vector space $\Lambda/2\Lambda$ ...
- 3,186
3
votes
0
answers
250
views
For which representations of a Lie algebra is the induced trace form basic?
Let $\mathfrak{g}$ be a simple Lie algebra. Let $\rho$ be a representation of $\mathfrak{g}$ on a finite-dimensional vector space $E$.
Consider now the bilinear form on $\mathfrak{g}$:
\begin{equation}...
- 556
1
vote
1
answer
316
views
A nice/simple relationship between the Chevalley generators of $\mathfrak{sp}_n$ and the Chevally generators of $\mathfrak{sl}_n$?
The Lie algebra $\mathfrak{sl}_n$ is defined to be the trace free matrices in $M_n(\mathbb{C})$. The Lie algebra $\mathfrak{so}_n$ is defined to be the matrices $A$ in $M_n(\mathbb{R})$ satisfying $A +...
- 63.9k
0
votes
0
answers
229
views
Coordinate ring of a flag variety
Edited:
[If G here is a simply connected semismple complex algebraic group.
A partial flag variety $G/P$ can be naturally embedded as a closed subset of $\prod_j \mathbb{P} (L(\omega_j)^*)$.
The ...
- 201
2
votes
1
answer
182
views
Gelfand-Naimark and Peter-Weyl for the unitary group
Consider the compact Lie groups $U(l)$ (the unitary group) and $U(1) \times SU(l)$ for some natural number $l$. Both the groups have the same Lie algebra $\frak{gl}_l$. Which means that they both have ...
- 2,385
6
votes
0
answers
113
views
$S_n$-invariant polynomials on the dual of reflection representation
Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (...
- 1,336
21
votes
5
answers
5k
views
Mirror symmetry mod p?! ... Physics mod p?!
In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that?
(Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than $\mathbb{C}$...
- 35.5k
9
votes
1
answer
373
views
Is the Magnus Lie algebra of a finitely presented group finitely presented
Let $G$ be a finitely presented group and let $L(G)$ be the Magnus Lie algebra associated to the lower central series of $G$. This $L(G)$ is a graded Lie ring generated by its degree 1 piece $L_1(G) =...
- 63.9k
1
vote
1
answer
96
views
Indecomposable modules for Virasoro algebra whose weights are bounded
It is clear that the Verma modules are indecomposable modules for the Virasoro algebra with the $L_0$-weights bounded. I am wondering if the Verma modules exhaust all such indecomposable modules. Does ...
- 378
0
votes
5
answers
3k
views
Examples of finite dimensional non simple non abelian Lie algebras
Hello, I have recently started reading about Lie algebras. However all the examples I have encountered so far are simple and semisimple Lie algebras. Thus I would love to see ...
- 19.6k
7
votes
0
answers
219
views
On the center of Koszul Lie algebras
The short question is the following: If a positively graded Lie algebra $\mathfrak g$ over a field $F$ is Koszul, is the center of $\mathfrak g$ concentrated in degree $1$?
Let us be more precise. A ...
2
votes
0
answers
165
views
Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not?
Is it true or false that the Inner (Inn), Outer (Out) and Total (Aut) Automorphism of a Lie group $G$ is the same as the covering group of the Lie group, say $\tilde G$ (regardless of how many types ...
- 10.5k