All Questions
2,633 questions
2
votes
0
answers
211
views
Adjoint action of semi-direct product
Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions
$\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
0
votes
0
answers
129
views
A special Lie subalgebra
Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:
Let $L$ be a Lie algebra. We consider the Lie subalgebra $$...
- 366
2
votes
1
answer
406
views
Isomorphism between pull-backs of an F-crystal by different liftings of Frobenius
This might be a naive question. But since I haven't seen this in any reference, I'll try to ask it here. Let $T$ be a smooth scheme over the algebraically closed field $k$ of characteristic $p>0$ (...
- 637
1
vote
1
answer
178
views
Casimir of a three dimensional solvable lie algebra
Good morning everyone. I've encountered recently during my computations the following lie algebra
$$\mathfrak g=\text{span}(f_0,f_1,f_2),$$
with $$\begin{eqnarray}[f_2,f_1]&=&f_0+a f_2,\\ [...
user17697
1
vote
1
answer
323
views
translation functors in parabolic category $\mathcal{O}$
I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$.
I am mainly interested in the ...
- 8,597
4
votes
0
answers
144
views
When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?
Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...
- 373
2
votes
0
answers
219
views
A problem on 2 Lie (co)homology group and central extension
For a perfect Lie algebra $L$ over $C,$ the kernel of its universal central extension is isomorphic to $H_2(L,C),$ and its central extensions are in 1-1 correspondence to $H^2(L,C).$
Question (1): ...
- 73
10
votes
0
answers
269
views
differentiating positive energy LG reps
Background:Let $G$ be a cscsc¹ Lie group, and let $\widetilde{LG}$ be the universal central extension (center = $S^1$) of $LG:=Map_{C^\infty}(S^1,G)$, with the topology inherited from the $C^\infty$ ...
- 43.2k
1
vote
1
answer
552
views
How to draw a Littelmann path?
Littelmann path is a combinatorial tool to compute multiplicity. I have some questions about the definition of Littelmann path. It is said that a Littelmann path is a piecewise-linear mapping
$$\pi:...
- 6,211
0
votes
0
answers
150
views
Explicit calculation of module of derivations on noncommutative polynomial ring
Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$.
Explicitly how would one go about computing ...
- 5,405
5
votes
2
answers
315
views
Radical of projection equals projection of radical?
Given an Lie Algebra $L$ (of finite dimension and over an algebraically closed field with zero characteristic) and an ideal $I$, is it truth that
$rad\left(\dfrac{L}{I}\right)= \pi(rad(L))$,
where $...
- 349
1
vote
0
answers
166
views
Replacing the Lie commutator with something else [closed]
Take a vector space $V={A,B,C,...}$ (of matrices), and the commutator $[A,B]=AB-BA$, then a Lie algebra of $V$ is characterized by $[V,V]$ staying in $V$. (Loosely speaking.)
What happens ...
- 4,829
0
votes
1
answer
325
views
identify a curious subgroup in $U(n)$
Consider the following element $A$ in $U(n)$:
$$ \begin{pmatrix} 1/2(1+z) & 1/2(1-z) & \\\\
1/2(1-z) & 1/2(1+z) & \\\\
& &I_{n-2} \end{pmatrix},$$
where $|z| = 1$.
Now ...
2
votes
1
answer
371
views
Distribution algebras and loop algebras
The algebra of distribution and its relationship with the universal enveloping algebra is discussed in the Jantzen's book, as we can see a discussion in the question link (more specifically, the Jim ...
- 829
7
votes
0
answers
182
views
Deformation of Noether's first theorem
Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...
- 3,880
1
vote
1
answer
304
views
Periodic automorphism of nilpotent Lie algebra
Are there a non-abelian nilpotent Lie algebra $\mathfrak{n}$ over $\mathbb{R}$ and an automorphism $\alpha: \mathfrak{n} \to \mathfrak{n}$ such that:
$\alpha$ is periodic,
the fixed subspace of $\...
- 687
5
votes
0
answers
166
views
Adjoint orbit of two vectors
Let $G$ be a simple compact real Lie group and let $\mathfrak g$ be its Lie algebra. Let $u,v\in \mathfrak g$ be two distinct unit vectors and $H\subset \mathfrak g$ be a hyperplane with normal vector ...
- 1,016
1
vote
1
answer
669
views
orbit of a Dynkin diagram automorphism
Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$.
What is a set of representatives of the orbits of $\Delta$ under $f$ ?
Thanks!
- 829
2
votes
0
answers
866
views
dual Coxeter number, affine algebra, invariants under twisting
Sometime ago we came across invariant quantities under twisting of all affine algebra. (See the appendix E of http://arxiv.org/abs/hep-th/0403076 .) Choose the convention so that the longest root has ...
- 191
5
votes
1
answer
446
views
More questions involving characteristic 2 theta series identities
In my answer to my earlier question, "Existence of certain identities involving characteristic 2 thetas", I established some curious identities when the thetas have prime "level" congruent to 1 mod 4 ...
- 5,422
2
votes
0
answers
284
views
universal enveloping algebras and commutator subalgebras
Let $A$ and $B$ are Lie subalgebras of a Lie algebra $L$. $U(A)$,
$U(B)$ and $U(L)$ are the universal enveloping algebras of $A$, $B$
and $L$, respectively. Let $[A, B]$ be the Lie subalgebras ...
- 491
5
votes
2
answers
389
views
Building Lie-like algebras from modules over semisimple Lie algebras
Here is a construction of a very broad class of "Lie-like" algebras, and I want to know more about them.
Here is the main definition: Suppose $\mathfrak{g}$ is a complex semsimple Lie algebra and $\...
- 1,277
3
votes
1
answer
256
views
Topologic or geometric mean of the structure constants of a semi simple lie algebra
Let $G$ be a semi simple Lie group (or real reductive), $\mathfrak{g}$ its lie algebra and $B$ its killing form. We can defined the 3-form $k$ by
$$k(X,Y,Z)=B([X,Y],Z).$$
with $X,Y,Z\in \mathfrak{g}$.
...
- 1,111
1
vote
0
answers
51
views
Iwasawa decomposition and Non-Abelian Centraliser of A
I'm studying Knapp's book "Representation Theory of Semisimple Groups" and am trying to understand the structure theory of non-compact groups. Namely, let $G=KAN$ be the Iwasawa decomposition, $\...
- 51
2
votes
1
answer
1k
views
on a characterization of parabolic subgroups
Over a base field $k$, linear $k$-groups stand for affine algebraic $k$-groups. For simplicity take $k$ to be a field of characteristic zero, as in this case one has the correspondence between ...
- 1,305
1
vote
1
answer
99
views
Hermitian symmetric structure on a homogeneous subspace
Let $G$ be a semisimple group over $Q$ and $K$ a maximal compact subgroup of $G(R)^+$.
I am assuming that $G(R)^+/K$ has a structure of a non-compact Hermitian symmetric domain.
Let $g= p + k$ be ...
- 547
1
vote
2
answers
756
views
reductive Lie subalgebra
Suppose to have a Lie algebra L with a reductive lie subalgebra G. Let l an element of L such that [l,g] is in G for every g in G, is it true that l is an element of G?if not, there are some ...
- 671
11
votes
1
answer
483
views
3/4-Lie superalgebras: how much of a theory can one develop?
Let $\mathfrak{s} = \mathfrak{s}_0 \oplus \mathfrak{s}_1$ be a real Lie superalgebra. (The ground field does not matter much, but at least one formula will not work as written if the characteristic ...
- 33.3k
10
votes
1
answer
411
views
Is there much theory of superalgebras acting on manifolds by alternating polyvector fields?
Usual story: vector fields on $M$, with their Lie bracket, form a Lie algebra. We can consider "actions" of some other Lie algebra ${\mathfrak g}$ on $M$ by looking at Lie homomorphisms ${\mathfrak g}\...
- 27.9k
8
votes
0
answers
873
views
Resolution of singularities in positive characteristic
I am currently trying to make some small parts of the minimal model program work for some very explicit varities in positive characteristic. I have such a variety $X_1$ and I know that there is a ...
- 783
2
votes
0
answers
189
views
About the Lie algebra of polyvector fields
I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...
- 1,609
7
votes
2
answers
536
views
What are the polynomial relations between these characteristic 2 "thetas" ?
Suppose $\ell=2m+1$, $m>0$. Define $[i]$ in $\mathbb{Z}/2\mathbb{Z}[[x]]$ to be $$\sum_{n\equiv i\mod l} x^{n^2}.$$ Note that $[0]=1$, and that $[i]=[j]$ whenever $\ell$ divides $i+j$ or $i-j$.
...
- 5,422
1
vote
1
answer
470
views
Module given by generators and relations
Let $\frak G$ be a Lie algebra and let $M$ be a $\frak G$-module generated by a vector $v$ satisfying some set of defining relations denoted by $R$. I mean, $M = U(\frak G)/\langle R \rangle$, where $\...
- 13
5
votes
0
answers
530
views
Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?
Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$.
Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...
- 3,897
3
votes
0
answers
150
views
Physical relevance of either fundamental identity generalizing Jacobi [closed]
There are two fundamental identities for n-ary generalizations of the Jacobi identity.
One fundamental identity is right for Nambu mechanics and such, the other for L_\infty algebras as in CSFT.
Which ...
- 3,880
2
votes
1
answer
510
views
hyperalgebras (positive characteristic)
The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$.
How to ...
- 829
4
votes
1
answer
742
views
Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A_n$, that is, $\mathfrak{g} = \...
- 2,160
4
votes
2
answers
1k
views
Under what conditions will a unitary matrix fix a subspace which does not diagonalize the generating Hamiltonian?
Hello, this is my first post here. I hope that it is not too vague; I will be as precise as I can, but I have more of a meta-problem so please forgive me if this is inappropriate. My question is ...
7
votes
0
answers
286
views
Level p characteristic 2 modular forms and thetas
BACKGROUND
Let p be an odd prime. An element of Z/2[[x]] is "modular of level p" if it is the mod 2 reduction of a g in Z[[x]] with g the Fourier expansion of a modular form for gamma_0(p). In ...
- 5,422
0
votes
0
answers
167
views
Are Generalized Verma modules natural w.r.t isometries?
Let $H$ be a subgroup of $G$ with Lie algebras $\mathfrak{h}$ and $\mathfrak{g}$ respectively. If I have 2 representations $V, W$ of $\mathfrak{h}$ equipped with a $\mathfrak{h}$ invariant inner ...
- 257
4
votes
0
answers
185
views
Are these subspaces of $\mathbb{Z}/3[[x]]$ stable under the shallow Hecke algebra?
This is a characteristic $3$ analog of part of my earlier question, "Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?"
Notation
Fix a prime $N$ other than $3$. Let $F,G \in \mathbb{Z}/3[[x]]$...
- 5,422
4
votes
2
answers
402
views
lower bound for torsion of abelian varieties
Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ ...
9
votes
1
answer
566
views
algorithm for calculating the Chow groups of a variety over a finite field
Is there an algorithm for calculating the Chow groups of a variety over a finite field?
It is know that $H^{2i,i}_\mathrm{mot}(X,\mathbf{Z}) = CH^i(X)$. In how many cases does this help us?
user19475
5
votes
1
answer
461
views
Given a branched cover with branch cycle description $(g_1,...,g_r)$, does $g_i$ generate some decomposition group?
Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so $\alpha_1...\...
- 5,929
3
votes
0
answers
141
views
Examples of divisible Lie algebras
We say that a nonzero Lie algebra $L$ is divisible, if for all elements $a$ and $b$ with $a\neq 0$, there exists $x\in L$ such that $[a, x]=b$. What are examples of divisible Lie algebras?
- 2,233
2
votes
0
answers
255
views
Lang isogeny for group stacks
Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...
- 3,605
2
votes
1
answer
359
views
Characterization of the weight orbit in the projective space via second order Casimir.
This is the spin-off of the question I previously asked.
First, let me remind you some notation from that question:
$G_0$ - compact, simply connected Lie group giving rise (by complexification) ...
- 965
0
votes
1
answer
981
views
Name of upper triangular/lower triangular Lie Algebra decomposition
What is the name of the Lie algebra decomposition where the positive root vectors are upper triangular and the negative root vectors are lower triangular?
- 2,123
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
0
votes
0
answers
143
views
A question on lie groups( Lie algebras)
What is an example of a compact Lie group $G$ which is not isomorphic to a product of two nontrivial lie groups and satisfies the following property:
There are two non zero vector fields $X, Y \in ...
- 366